├── APEX_format.sty ├── Fundamentals_of_Matrix_Algebra_3rd_Ed.answers ├── Fundamentals_of_Matrix_Algebra_3rd_Ed.aux ├── Fundamentals_of_Matrix_Algebra_3rd_Ed.idx ├── Fundamentals_of_Matrix_Algebra_3rd_Ed.log ├── Fundamentals_of_Matrix_Algebra_3rd_Ed.out ├── Fundamentals_of_Matrix_Algebra_3rd_Ed.pdf ├── Fundamentals_of_Matrix_Algebra_3rd_Ed.synctex ├── Fundamentals_of_Matrix_Algebra_3rd_Ed.tex ├── Fundamentals_of_Matrix_Algebra_3rd_Ed.toc ├── README.txt ├── cover ├── amazon_createspace_cover.aux ├── amazon_createspace_cover.log ├── amazon_createspace_cover.pdf ├── amazon_createspace_cover.tex ├── back_cover.aux ├── back_cover.jpg ├── back_cover.log ├── back_cover.pdf ├── back_cover.tex ├── back_cover1.jpg ├── front_cover.aux ├── front_cover.bbl ├── front_cover.blg ├── front_cover.jpg ├── front_cover.log ├── front_cover.pdf ├── front_cover.tex ├── front_cover1.jpg ├── front_cover_in_text.log └── front_cover_in_text.tex ├── exercises ├── 01_01_ex_01.tex ├── 01_01_ex_02.tex ├── 01_01_ex_03.tex ├── 01_01_ex_04.tex ├── 01_01_ex_05.tex ├── 01_01_ex_06.tex ├── 01_01_ex_07.tex ├── 01_01_ex_08.tex ├── 01_01_ex_09.tex ├── 01_01_ex_10.tex ├── 01_01_ex_11.tex ├── 01_01_ex_12.tex ├── 01_01_ex_13.tex ├── 01_01_ex_14.tex ├── 01_01_ex_15.tex ├── 01_01_ex_16.tex ├── 01_01_ex_17.tex ├── 01_01_ex_18.log ├── 01_01_ex_18.tex ├── 01_01_ex_19.tex ├── 01_01_ex_20.tex ├── 01_01_ex_21.tex ├── 01_01_exercises.tex ├── 01_01_exercisestest.tex ├── 01_01_exset_01.tex ├── 01_01_exset_02.tex ├── 01_02_ex_01.tex ├── 01_02_ex_02.tex ├── 01_02_ex_03.tex ├── 01_02_ex_04.tex ├── 01_02_ex_05.tex ├── 01_02_ex_06.tex ├── 01_02_ex_07.tex ├── 01_02_ex_08.tex ├── 01_02_ex_09.tex ├── 01_02_ex_10.tex ├── 01_02_ex_11.tex ├── 01_02_ex_12.tex ├── 01_02_ex_13.tex ├── 01_02_ex_14.tex ├── 01_02_ex_15.tex ├── 01_02_ex_16.tex ├── 01_02_ex_17.tex ├── 01_02_ex_18.tex ├── 01_02_ex_19.tex ├── 01_02_ex_20.tex ├── 01_02_ex_21.tex ├── 01_02_ex_22.tex ├── 01_02_ex_23.tex ├── 01_02_ex_24.tex ├── 01_02_ex_25.tex ├── 01_02_ex_26.tex ├── 01_02_exercises.tex ├── 01_02_exset_01.tex ├── 01_02_exset_02.tex ├── 01_02_exset_03.tex ├── 01_02_exset_04.tex ├── 01_02_exset_05.tex ├── 01_03_ex_01.tex ├── 01_03_ex_02.tex ├── 01_03_ex_03.tex ├── 01_03_ex_04.tex ├── 01_03_ex_05.tex ├── 01_03_ex_06.tex ├── 01_03_ex_07.tex ├── 01_03_ex_08.tex ├── 01_03_ex_09.tex ├── 01_03_ex_10.tex ├── 01_03_ex_11.tex ├── 01_03_ex_12.tex ├── 01_03_ex_13.tex ├── 01_03_ex_14.tex ├── 01_03_ex_15.tex ├── 01_03_ex_16.tex ├── 01_03_ex_17.tex ├── 01_03_ex_18.tex ├── 01_03_ex_19.tex ├── 01_03_ex_20.tex ├── 01_03_ex_21.tex ├── 01_03_ex_22.tex ├── 01_03_exercises.tex ├── 01_03_exset_01.tex ├── 01_03_exset_02.tex ├── 01_03_exset_03.tex ├── 01_04_ex_01.tex ├── 01_04_ex_02.tex ├── 01_04_ex_03.tex ├── 01_04_ex_04.tex ├── 01_04_ex_05.tex ├── 01_04_ex_06.tex ├── 01_04_ex_07.tex ├── 01_04_ex_08.tex ├── 01_04_ex_09.tex ├── 01_04_ex_10.tex ├── 01_04_ex_11.tex ├── 01_04_ex_12.tex ├── 01_04_ex_13.tex ├── 01_04_ex_14.tex ├── 01_04_ex_15.tex ├── 01_04_ex_16.tex ├── 01_04_ex_17.tex ├── 01_04_ex_18.tex ├── 01_04_exercises.tex ├── 01_04_exset_01.tex ├── 01_04_exset_02.tex ├── 01_05_ex_01.tex ├── 01_05_ex_02.tex ├── 01_05_ex_03.tex ├── 01_05_ex_04.tex ├── 01_05_ex_05.tex ├── 01_05_ex_06.tex ├── 01_05_ex_07.tex ├── 01_05_ex_08.tex ├── 01_05_ex_09.tex ├── 01_05_ex_10.tex ├── 01_05_ex_11.tex ├── 01_05_ex_12.tex ├── 01_05_ex_13.tex ├── 01_05_ex_14.tex ├── 01_05_ex_15.tex ├── 01_05_ex_16.tex ├── 01_05_ex_17.tex ├── 01_05_ex_18.tex ├── 01_05_ex_19.tex ├── 01_05_exercises.tex ├── 01_05_exset_01.tex ├── 02_01_ex_01.tex ├── 02_01_ex_02.tex ├── 02_01_ex_03.tex ├── 02_01_ex_04.tex ├── 02_01_ex_05.tex ├── 02_01_ex_06.tex ├── 02_01_ex_07.tex ├── 02_01_ex_08.tex ├── 02_01_ex_09.tex ├── 02_01_ex_10.tex ├── 02_01_ex_11.tex ├── 02_01_ex_12.tex ├── 02_01_ex_13.tex ├── 02_01_ex_14.tex ├── 02_01_ex_15.tex ├── 02_01_ex_16.tex ├── 02_01_ex_17.tex ├── 02_01_ex_18.tex ├── 02_01_ex_19.tex ├── 02_01_ex_20.tex ├── 02_01_ex_21.tex ├── 02_01_exercises.tex ├── 02_01_exset_01.tex ├── 02_01_exset_02.tex ├── 02_01_exset_03.tex ├── 02_01_exset_04.tex ├── 02_02_ex_01.tex ├── 02_02_ex_02.tex ├── 02_02_ex_03.tex ├── 02_02_ex_04.tex ├── 02_02_ex_05.tex ├── 02_02_ex_06.tex ├── 02_02_ex_07.tex ├── 02_02_ex_08.tex ├── 02_02_ex_09.tex ├── 02_02_ex_10.tex ├── 02_02_ex_11.tex ├── 02_02_ex_12.tex ├── 02_02_ex_13.tex ├── 02_02_ex_14.tex ├── 02_02_ex_15.tex ├── 02_02_ex_16.tex ├── 02_02_ex_17.tex ├── 02_02_ex_18.tex ├── 02_02_ex_19.tex ├── 02_02_ex_20.tex ├── 02_02_ex_21.tex ├── 02_02_ex_22.tex ├── 02_02_ex_23.tex ├── 02_02_ex_24.tex ├── 02_02_ex_25.tex ├── 02_02_ex_26.tex ├── 02_02_ex_27.tex ├── 02_02_ex_28.tex ├── 02_02_ex_29.tex ├── 02_02_ex_30.tex ├── 02_02_ex_31.tex ├── 02_02_ex_32.tex ├── 02_02_ex_33.tex ├── 02_02_ex_34.tex ├── 02_02_ex_35.tex ├── 02_02_ex_36.tex ├── 02_02_ex_37.tex ├── 02_02_ex_38.tex ├── 02_02_ex_39.tex ├── 02_02_ex_40.tex ├── 02_02_ex_41.tex ├── 02_02_ex_42.tex ├── 02_02_ex_43.tex ├── 02_02_ex_44.tex ├── 02_02_ex_45.tex ├── 02_02_exercises.tex ├── 02_02_exset_01.tex ├── 02_02_exset_02.tex ├── 02_02_exset_03.tex ├── 02_02_exset_04.tex ├── 02_02_exset_05.tex ├── 02_03_ex_01.tex ├── 02_03_ex_02.tex ├── 02_03_ex_03.tex ├── 02_03_ex_04.tex ├── 02_03_ex_05.tex ├── 02_03_ex_06.tex ├── 02_03_ex_07.tex ├── 02_03_ex_08.tex ├── 02_03_ex_09.tex ├── 02_03_ex_10.tex ├── 02_03_ex_11.tex ├── 02_03_ex_12.tex ├── 02_03_ex_13.tex ├── 02_03_ex_14.tex ├── 02_03_ex_15.tex ├── 02_03_ex_16.tex ├── 02_03_ex_17.tex ├── 02_03_ex_18.tex ├── 02_03_ex_19.tex ├── 02_03_ex_20.tex ├── 02_03_ex_21.tex ├── 02_03_ex_22.tex ├── 02_03_ex_23.tex ├── 02_03_ex_24.tex ├── 02_03_ex_25.tex ├── 02_03_ex_26.tex ├── 02_03_ex_27.tex ├── 02_03_ex_28.tex ├── 02_03_exercises.tex ├── 02_03_exset_01.tex ├── 02_03_exset_02.tex ├── 02_03_exset_03.tex ├── 02_03_exset_04.tex ├── 02_04_ex_01.tex ├── 02_04_ex_02.tex ├── 02_04_ex_03.tex ├── 02_04_ex_04.tex ├── 02_04_ex_05.tex ├── 02_04_ex_06.tex ├── 02_04_ex_07.tex ├── 02_04_ex_08.tex ├── 02_04_ex_09.tex ├── 02_04_ex_10.tex ├── 02_04_ex_11.tex ├── 02_04_ex_12.tex ├── 02_04_exercises.tex ├── 02_04_exset_01.tex ├── 02_05_ex_01.tex ├── 02_05_ex_02.tex ├── 02_05_ex_03.tex ├── 02_05_ex_04.tex ├── 02_05_ex_05.tex ├── 02_05_ex_06.tex ├── 02_05_ex_07.tex ├── 02_05_ex_08.tex ├── 02_05_ex_09.tex ├── 02_05_ex_10.tex ├── 02_05_ex_11.tex ├── 02_05_ex_12.tex ├── 02_05_ex_13.tex ├── 02_05_ex_14.tex ├── 02_05_ex_15.tex ├── 02_05_ex_16.tex ├── 02_05_ex_17.tex ├── 02_05_ex_18.tex ├── 02_05_ex_19.tex ├── 02_05_ex_20.tex ├── 02_05_ex_21.tex ├── 02_05_ex_22.tex ├── 02_05_ex_23.tex ├── 02_05_ex_24.tex ├── 02_05_ex_25.tex ├── 02_05_ex_26.tex ├── 02_05_ex_27.tex ├── 02_05_ex_28.tex ├── 02_05_ex_29.tex ├── 02_05_ex_30.tex ├── 02_05_ex_31.tex ├── 02_05_ex_32.tex ├── 02_05_ex_33.tex ├── 02_05_ex_34.tex ├── 02_05_ex_35.tex ├── 02_05_ex_36.tex ├── 02_05_exercises.tex ├── 02_05_exset_01.tex ├── 02_05_exset_02.tex ├── 02_05_exset_03.tex ├── 02_06_ex_01.tex ├── 02_06_ex_02.tex ├── 02_06_ex_03.tex ├── 02_06_ex_04.tex ├── 02_06_ex_05.tex ├── 02_06_ex_06.tex ├── 02_06_ex_07.tex ├── 02_06_ex_08.tex ├── 02_06_ex_09.tex ├── 02_06_ex_10.tex ├── 02_06_ex_11.tex ├── 02_06_exercises.tex ├── 02_06_exset_01.tex ├── 02_06_exset_02.tex ├── 03_01_ex_01.tex ├── 03_01_ex_02.tex ├── 03_01_ex_03.tex ├── 03_01_ex_04.tex ├── 03_01_ex_05.tex ├── 03_01_ex_06.tex ├── 03_01_ex_07.tex ├── 03_01_ex_08.tex ├── 03_01_ex_09.tex ├── 03_01_ex_10.tex ├── 03_01_ex_11.tex ├── 03_01_ex_12.tex ├── 03_01_ex_13.tex ├── 03_01_ex_14.tex ├── 03_01_ex_15.tex ├── 03_01_ex_16.tex ├── 03_01_ex_17.tex ├── 03_01_ex_18.tex ├── 03_01_ex_19.tex ├── 03_01_ex_20.tex ├── 03_01_ex_21.tex ├── 03_01_ex_22.tex ├── 03_01_ex_23.tex ├── 03_01_ex_24.tex ├── 03_01_exercises.tex ├── 03_01_exset_01.log ├── 03_01_exset_01.tex ├── 03_02_ex_01.tex ├── 03_02_ex_02.tex ├── 03_02_ex_03.tex ├── 03_02_ex_04.tex ├── 03_02_ex_05.tex ├── 03_02_ex_06.tex ├── 03_02_ex_07.tex ├── 03_02_ex_08.tex ├── 03_02_ex_09.tex ├── 03_02_ex_10.tex ├── 03_02_ex_11.tex ├── 03_02_ex_12.tex ├── 03_02_ex_13.tex ├── 03_02_ex_14.tex ├── 03_02_ex_15.tex ├── 03_02_ex_16.tex ├── 03_02_ex_17.tex ├── 03_02_ex_18.tex ├── 03_02_ex_19.tex ├── 03_02_exercises.tex ├── 03_02_exset_01.tex ├── 03_02_exset_02.tex ├── 03_03_ex_01.tex ├── 03_03_ex_02.tex ├── 03_03_ex_03.tex ├── 03_03_ex_04.tex ├── 03_03_ex_05.tex ├── 03_03_ex_06.tex ├── 03_03_ex_07.tex ├── 03_03_ex_08.tex ├── 03_03_ex_09.tex ├── 03_03_ex_10.tex ├── 03_03_ex_11.tex ├── 03_03_ex_12.tex ├── 03_03_ex_13.tex ├── 03_03_ex_14.tex ├── 03_03_ex_15.tex ├── 03_03_ex_16.tex ├── 03_03_ex_17.tex ├── 03_03_ex_18.tex ├── 03_03_ex_19.tex ├── 03_03_ex_20.tex ├── 03_03_ex_21.tex ├── 03_03_ex_22.tex ├── 03_03_ex_23.tex ├── 03_03_ex_24.tex ├── 03_03_ex_25.tex ├── 03_03_exercises.tex ├── 03_03_exset_01.tex ├── 03_03_exset_02.tex ├── 03_03_exset_03.tex ├── 03_04_ex_01.tex ├── 03_04_ex_02.tex ├── 03_04_ex_03.tex ├── 03_04_ex_04.tex ├── 03_04_ex_05.tex ├── 03_04_ex_06.tex ├── 03_04_ex_07.tex ├── 03_04_ex_08.tex ├── 03_04_ex_09.tex ├── 03_04_ex_10.tex ├── 03_04_ex_11.tex ├── 03_04_ex_12.tex ├── 03_04_ex_13.tex ├── 03_04_ex_14.tex ├── 03_04_ex_15.tex ├── 03_04_ex_16.tex ├── 03_04_ex_17.tex ├── 03_04_ex_18.tex ├── 03_04_ex_19.tex ├── 03_04_ex_20.tex ├── 03_04_ex_21.tex ├── 03_04_ex_22.tex ├── 03_04_exercises.tex ├── 03_04_exset_01.tex ├── 03_04_exset_02.log ├── 03_04_exset_02.tex ├── 03_04_exset_03.tex ├── 03_04_exset_04.tex ├── 03_05_ex_01.tex ├── 03_05_ex_02.tex ├── 03_05_ex_03.tex ├── 03_05_ex_04.tex ├── 03_05_ex_05.tex ├── 03_05_ex_06.tex ├── 03_05_ex_07.tex ├── 03_05_ex_08.tex ├── 03_05_ex_09.tex ├── 03_05_ex_10.tex ├── 03_05_ex_11.tex ├── 03_05_ex_12.tex ├── 03_05_exercises.tex ├── 03_05_exset_01.log ├── 03_05_exset_01.tex ├── 04_01_ex_01.tex ├── 04_01_ex_02.tex ├── 04_01_ex_03.tex ├── 04_01_ex_04.tex ├── 04_01_ex_05.tex ├── 04_01_ex_06.tex ├── 04_01_ex_07.tex ├── 04_01_ex_08.tex ├── 04_01_ex_09.tex ├── 04_01_ex_10.tex ├── 04_01_ex_11.tex ├── 04_01_ex_12.tex ├── 04_01_ex_13.tex ├── 04_01_ex_14.tex ├── 04_01_ex_15.tex ├── 04_01_ex_16.tex ├── 04_01_ex_17.tex ├── 04_01_ex_18.tex ├── 04_01_ex_19.tex ├── 04_01_ex_20.tex ├── 04_01_ex_21.tex ├── 04_01_ex_22.tex ├── 04_01_ex_23.tex ├── 04_01_ex_24.tex ├── 04_01_ex_25.tex ├── 04_01_ex_26.tex ├── 04_01_ex_27.tex ├── 04_01_ex_28.tex ├── 04_01_exercises.tex ├── 04_01_exset_01.tex ├── 04_01_exset_02.tex ├── 04_01_exset_03.tex ├── 04_02_ex_01.tex ├── 04_02_ex_02.tex ├── 04_02_ex_03.tex ├── 04_02_ex_04.tex ├── 04_02_ex_05.tex ├── 04_02_ex_06.tex ├── 04_02_exercises.tex ├── 04_02_exset_01.tex ├── 05_01_ex_01.tex ├── 05_01_ex_02.tex ├── 05_01_ex_03.tex ├── 05_01_ex_04.tex ├── 05_01_ex_05.tex ├── 05_01_ex_06.tex ├── 05_01_ex_07.tex ├── 05_01_ex_08.tex ├── 05_01_ex_09.tex ├── 05_01_ex_10.tex ├── 05_01_ex_11.tex ├── 05_01_ex_12.tex ├── 05_01_ex_13.tex ├── 05_01_ex_14.tex ├── 05_01_exercises.tex ├── 05_01_exset_01.tex ├── 05_01_exset_02.tex ├── 05_01_exset_03.tex ├── 05_01_exset_04.tex ├── 05_02_ex_01.tex ├── 05_02_ex_02.tex ├── 05_02_ex_03.tex ├── 05_02_ex_04.tex ├── 05_02_ex_05.tex ├── 05_02_ex_06.tex ├── 05_02_ex_07.tex ├── 05_02_ex_08.tex ├── 05_02_ex_09.tex ├── 05_02_ex_10.tex ├── 05_02_ex_11.tex ├── 05_02_ex_12.tex ├── 05_02_exercises.tex ├── 05_02_exset_01.tex ├── 05_02_exset_02.tex ├── 05_02_exset_03.tex ├── 05_03_ex_01.tex ├── 05_03_ex_02.tex ├── 05_03_ex_03.tex ├── 05_03_ex_04.tex ├── 05_03_ex_05.tex ├── 05_03_ex_06.tex ├── 05_03_ex_07.tex ├── 05_03_ex_08.tex ├── 05_03_ex_09.tex ├── 05_03_ex_10.tex ├── 05_03_ex_11.tex ├── 05_03_ex_12.tex ├── 05_03_ex_13.tex ├── 05_03_ex_14.tex ├── 05_03_ex_15.tex ├── 05_03_ex_16.tex ├── 05_03_ex_17.tex ├── 05_03_ex_18.tex ├── 05_03_exercises.tex ├── 05_03_exset_01.tex ├── 05_03_exset_02.tex ├── 05_03_exset_03.tex ├── 05_03_exset_04.tex ├── 05_04_ex_01.tex ├── 05_04_ex_02.tex ├── 05_04_ex_03.tex ├── 05_04_ex_04.tex ├── 05_04_ex_05.tex ├── 05_04_ex_06.tex ├── 05_04_ex_07.tex ├── 05_04_ex_08.tex ├── 05_04_ex_09.tex ├── 05_04_ex_10.tex ├── 05_04_ex_11.tex ├── 05_04_exercises.tex ├── 05_04_exset_01.tex ├── 05_04_exset_02.tex ├── titles.txt ├── troy.tex └── troy1.tex ├── fundamentals_format.tex ├── headers ├── APEX_Format_Examples.answers ├── APEX_Format_Examples.aux ├── APEX_Format_Examples.idx ├── APEX_Format_Examples.log ├── APEX_Format_Examples.out ├── APEX_Format_Examples.pdf ├── APEX_Format_Examples.tex ├── APEX_Format_Manual.answers ├── APEX_Format_Manual.aux ├── APEX_Format_Manual.dvi ├── APEX_Format_Manual.idx ├── APEX_Format_Manual.log ├── APEX_Format_Manual.out ├── APEX_Format_Manual.pdf ├── APEX_Format_Manual.tex ├── APEX_Format_Manual.toc ├── APEX_Header.log ├── APEX_Header.tex ├── Header_APEX.idx ├── Header_APEX.log ├── Header_APEX.tex ├── Header_APEX_x.tex ├── Header_Environments.tex ├── Header_Environments_test.log ├── Header_Environments_test.tex ├── Header_Exercises.log ├── Header_Exercises.tex ├── Header_Fundamentals_Of_Matrix_Algebra.tex ├── Header_Header.tex ├── Header_Simple_Definitions.tex ├── Header_TikZ.tex ├── Matrix_Algebra_Text_Header.tex ├── Test_Header_All.tex ├── chapter_graphics.tex ├── copy_Header_Exercises.tex └── xAPEX_Header.tex └── text ├── 01_Existence_Uniqueness_Solutions.tex ├── 01_Gauss_Jordan_Elimination.tex ├── 01_Introduction_Linear_Equations.tex ├── 01_Solving_Systems.tex ├── 01_Using_Matrices.tex ├── 02_Geometry_of_Vectors_2D.tex ├── 02_Geometry_of_Vectors_2D_BW.tex ├── 02_Matrix_Addition.tex ├── 02_Matrix_Inverse_Properties.tex ├── 02_Matrix_Inverses.tex ├── 02_Matrix_Multiplication.tex ├── 02_Solving_AXB.tex ├── 02_Vector_Solutions.tex ├── 03_CramersRule.tex ├── 03_Determinant.tex ├── 03_Determinant_Properties.tex ├── 03_Trace.tex ├── 03_Transpose.tex ├── 04_Eigen.tex ├── 04_Eigen_Properties.tex ├── 05_Geometry_of_Cartesian_Plane.tex ├── 05_Geometry_of_Cartesian_Plane_BW.tex ├── 05_Geometry_of_Vectors_3D.tex ├── 05_Geometry_of_Vectors_3D_BW.tex ├── 05_Linear_Transformations.tex ├── 05_Linear_Transformations_BW.tex ├── Answer_Pages.tex ├── Questions for Sections.tex ├── by-nc.eps ├── by-nc.eu.eps ├── by-nc.pdf ├── by-nc.png ├── cc_license.png ├── copyright_page.tex ├── preface.tex ├── thanks.tex ├── title_page.tex ├── x02_Geometry_of_Vectors_2D.tex ├── x05_Geometry_of_Cartesian_Plane.tex ├── x05_Geometry_of_Vectors_3D.tex └── x05_Linear_Transformations.tex /Fundamentals_of_Matrix_Algebra_3rd_Ed.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/APEXCalculus/Fundamentals-of-Matrix-Algebra/6f2d536d1259395a5bff80ae95623b0719d8d455/Fundamentals_of_Matrix_Algebra_3rd_Ed.pdf -------------------------------------------------------------------------------- /README.txt: -------------------------------------------------------------------------------- 1 | #README 2 | 3 | This repository contains the source files for 4 | Fundamentals of Matrix Algebra, by Gregory Hartman. 5 | 6 | The copywright is under the Creative Commons 7 | Attribution-Noncommercial 3.0 license. (CC-BY-NC) 8 | 9 | Please forward all requests/comments/suggestions to hartmangn@vmi.edu. 10 | 11 | Enjoy. 12 | -------------------------------------------------------------------------------- /cover/amazon_createspace_cover.aux: -------------------------------------------------------------------------------- 1 | \relax 2 | \pgfsyspdfmark {pgfid1}{11497706}{42065886} 3 | \pgfsyspdfmark {pgfid3}{52167065}{11954752} 4 | \pgfsyspdfmark {pgfid2}{11497706}{41279454} 5 | -------------------------------------------------------------------------------- /cover/amazon_createspace_cover.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/APEXCalculus/Fundamentals-of-Matrix-Algebra/6f2d536d1259395a5bff80ae95623b0719d8d455/cover/amazon_createspace_cover.pdf -------------------------------------------------------------------------------- /cover/back_cover.aux: -------------------------------------------------------------------------------- 1 | \relax 2 | \pgfsyspdfmark {pgfid1}{9782558}{43234099} 3 | \pgfsyspdfmark {pgfid2}{9782558}{42447667} 4 | -------------------------------------------------------------------------------- /cover/back_cover.jpg: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/APEXCalculus/Fundamentals-of-Matrix-Algebra/6f2d536d1259395a5bff80ae95623b0719d8d455/cover/back_cover.jpg -------------------------------------------------------------------------------- /cover/back_cover.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/APEXCalculus/Fundamentals-of-Matrix-Algebra/6f2d536d1259395a5bff80ae95623b0719d8d455/cover/back_cover.pdf -------------------------------------------------------------------------------- /cover/back_cover1.jpg: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/APEXCalculus/Fundamentals-of-Matrix-Algebra/6f2d536d1259395a5bff80ae95623b0719d8d455/cover/back_cover1.jpg -------------------------------------------------------------------------------- /cover/front_cover.aux: -------------------------------------------------------------------------------- 1 | \relax 2 | \pgfsyspdfmark {pgfid1}{9782558}{43234099} 3 | \pgfsyspdfmark {pgfid3}{19723510}{13286832} 4 | \pgfsyspdfmark {pgfid2}{9782558}{42447667} 5 | -------------------------------------------------------------------------------- /cover/front_cover.bbl: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/APEXCalculus/Fundamentals-of-Matrix-Algebra/6f2d536d1259395a5bff80ae95623b0719d8d455/cover/front_cover.bbl -------------------------------------------------------------------------------- /cover/front_cover.blg: -------------------------------------------------------------------------------- 1 | This is BibTeX, Version 0.99cThe top-level auxiliary file: C:\Documents and Settings\hartmangn\My Documents\Text Projects\MA 103\MA 103 Master Folder\cover\front_cover.aux 2 | I found no \citation commands---while reading file C:\Documents and Settings\hartmangn\My Documents\Text Projects\MA 103\MA 103 Master Folder\cover\front_cover.aux 3 | I found no \bibdata command---while reading file C:\Documents and Settings\hartmangn\My Documents\Text Projects\MA 103\MA 103 Master Folder\cover\front_cover.aux 4 | I found no \bibstyle command---while reading file C:\Documents and Settings\hartmangn\My Documents\Text Projects\MA 103\MA 103 Master Folder\cover\front_cover.aux 5 | (There were 3 error messages) 6 | -------------------------------------------------------------------------------- /cover/front_cover.jpg: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/APEXCalculus/Fundamentals-of-Matrix-Algebra/6f2d536d1259395a5bff80ae95623b0719d8d455/cover/front_cover.jpg -------------------------------------------------------------------------------- /cover/front_cover.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/APEXCalculus/Fundamentals-of-Matrix-Algebra/6f2d536d1259395a5bff80ae95623b0719d8d455/cover/front_cover.pdf -------------------------------------------------------------------------------- /cover/front_cover1.jpg: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/APEXCalculus/Fundamentals-of-Matrix-Algebra/6f2d536d1259395a5bff80ae95623b0719d8d455/cover/front_cover1.jpg -------------------------------------------------------------------------------- /exercises/01_01_ex_01.tex: -------------------------------------------------------------------------------- 1 | {State which of the following equations is a linear equation. If it is not, state why. 2 | \begin{enumerate} 3 | \item $x+y+z = 10$ 4 | \item $xy + yz+ xz = 1$ 5 | \item $-3x + 9 = 3y - 5z+ x-7$ 6 | \item $\sqrt{5}y + \pi x =-1$ 7 | \item $(x-1)(x+1) = 0$ 8 | \end{enumerate} 9 | } 10 | {(a), (c), (d)} -------------------------------------------------------------------------------- /exercises/01_01_ex_02.tex: -------------------------------------------------------------------------------- 1 | {State which of the following equations is a linear equation. If it is not, state why. 2 | \begin{enumerate} 3 | \item $\sqrt{x_1^2+x_2^2} = 25$ 4 | \item $x_1 + y + t = 1$ 5 | \item $\frac{1}{x} + 9 = 3\cos(y) - 5z$ 6 | \item $\cos(15)y + \frac{x}{4} =-1$ 7 | \item $2^x + 2^y = 16$ 8 | \end{enumerate} 9 | } 10 | {(b), (d)} -------------------------------------------------------------------------------- /exercises/01_01_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$x+y+z = 10$} 2 | {y} 3 | -------------------------------------------------------------------------------- /exercises/01_01_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$xy + yz+ xz = 1$} 2 | {n} 3 | -------------------------------------------------------------------------------- /exercises/01_01_ex_05.tex: -------------------------------------------------------------------------------- 1 | {$-3x + 9 = 3y - 5z+ x-7$} 2 | {y} -------------------------------------------------------------------------------- /exercises/01_01_ex_06.tex: -------------------------------------------------------------------------------- 1 | {$\sqrt{5}y + \pi x =-1$} 2 | {y} -------------------------------------------------------------------------------- /exercises/01_01_ex_07.tex: -------------------------------------------------------------------------------- 1 | {$(x-1)(x+1) = 0$} 2 | {n} -------------------------------------------------------------------------------- /exercises/01_01_ex_08.tex: -------------------------------------------------------------------------------- 1 | {$\sqrt{x_1^2+x_2^2} = 25$} 2 | {n} -------------------------------------------------------------------------------- /exercises/01_01_ex_09.tex: -------------------------------------------------------------------------------- 1 | {$x_1 + y + t = 1$} 2 | {y} 3 | -------------------------------------------------------------------------------- /exercises/01_01_ex_10.tex: -------------------------------------------------------------------------------- 1 | {$\frac{1}{x} + 9 = 3\cos(y) - 5z$} 2 | {n} -------------------------------------------------------------------------------- /exercises/01_01_ex_11.tex: -------------------------------------------------------------------------------- 1 | {$\cos(15)y + \frac{x}{4} =-1$} 2 | {y} -------------------------------------------------------------------------------- /exercises/01_01_ex_12.tex: -------------------------------------------------------------------------------- 1 | {$2^x + 2^y = 16$} 2 | {n} -------------------------------------------------------------------------------- /exercises/01_01_ex_13.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccc} 2 | x&+&y&=&-1\\ 3 | 2x&-&3y&=&8\\ 4 | \end{array}$} 5 | {$x = 1, y=-2$} -------------------------------------------------------------------------------- /exercises/01_01_ex_14.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccc} 2 | 2x&-&3y&=&3\\ 3 | 3x&+&6y&=&8\\ 4 | \end{array}$} 5 | {$x = 2, y=\frac13$} -------------------------------------------------------------------------------- /exercises/01_01_ex_15.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccccc} 2 | x&-&y&+&z&=&1\\ 3 | 2x&+&6y&-&z&=&-4\\ 4 | 4x&-&5y&+&2z&=&0\\ 5 | \end{array}$} 6 | {$x = -1, y=0,z=2$} -------------------------------------------------------------------------------- /exercises/01_01_ex_16.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccccc} 2 | x&+&y&-&z&=&1\\ 3 | 2x&+&y&&&=&2\\ 4 | &&y&+&2z&=&0\\ 5 | \end{array}$} 6 | {$x =1,\ y=0,\ z=0$} -------------------------------------------------------------------------------- /exercises/01_01_ex_17.tex: -------------------------------------------------------------------------------- 1 | {A carpenter can make two sizes of table, grande and venti. The grande table requires 4 table legs and 1 table top; the venti requires 6 table legs and 2 table tops. After doing work, he counts up spare parts in his warehouse and realizes that he has 86 table tops left over, and 300 legs. How many tables of each kind can he build and use up exactly all of his materials?} 2 | {42 grande tables, 22 venti tables} -------------------------------------------------------------------------------- /exercises/01_01_ex_18.tex: -------------------------------------------------------------------------------- 1 | {A jar contains 100 marbles. We know there are twice as many green marbles as red; that the number of blue and yellow marbles together is the same as the number of green; and that three times the number of yellow marbles together with the red marbles gives the same numbers as the blue marbles. How many of each color of marble are in the jar?} 2 | {35 blue, 40 green, 20 red, 5 yellow} -------------------------------------------------------------------------------- /exercises/01_01_ex_19.tex: -------------------------------------------------------------------------------- 1 | {A farmer looks out his window at his chickens and pigs. He tells his daughter that he sees 62 heads and 190 legs. How many chickens and pigs does the farmer have?} 2 | {29 chickens and 33 pigs} -------------------------------------------------------------------------------- /exercises/01_01_ex_20.tex: -------------------------------------------------------------------------------- 1 | {A rescue mission has 85 sandwiches, 65 bags of chips and 210 cookies. They know from experience that men will eat 2 sandwiches, 1 bag of chips and 4 cookies; women will eat 1 sandwich, a bag of chips and 2 cookies; kids will eat half a sandwhich, a bag of chips and 3 cookies. If they want to use all their food up, how many men, women and kids can they feed?} 2 | {30 men, 15 women, 20 kids} -------------------------------------------------------------------------------- /exercises/01_01_ex_21.tex: -------------------------------------------------------------------------------- 1 | {A lady buys 20 trinkets at a yard sale. The cost of each trinket is either \$0.30 or \$0.65. If she spends \$8.80, how many of each type of trinket does she buy?} 2 | {12 \$0.30 trinkets, 8 \$0.65 trinkets} -------------------------------------------------------------------------------- /exercises/01_01_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/01_01_exset_01} 2 | \exsetinput{exercises/01_01_exset_02} 3 | \exinput{exercises/01_01_ex_19} 4 | \exinput{exercises/01_01_ex_21} -------------------------------------------------------------------------------- /exercises/01_01_exercisestest.tex: -------------------------------------------------------------------------------- 1 | exercises/01_01_ex_19 2 | exercises/01_01_ex_21 -------------------------------------------------------------------------------- /exercises/01_01_exset_01.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, state whether or not the given equation is linear.} 3 | \exinput{exercises/01_01_ex_03} 4 | \exinput{exercises/01_01_ex_04} 5 | \exinput{exercises/01_01_ex_05} 6 | \exinput{exercises/01_01_ex_06} 7 | \exinput{exercises/01_01_ex_07} 8 | \exinput{exercises/01_01_ex_08} 9 | \exinput{exercises/01_01_ex_09} 10 | \exinput{exercises/01_01_ex_10} 11 | \exinput{exercises/01_01_ex_11} 12 | \exinput{exercises/01_01_ex_12} -------------------------------------------------------------------------------- /exercises/01_01_exset_02.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, solve the system of linear equations.} 3 | \exinput{exercises/01_01_ex_13} 4 | \exinput{exercises/01_01_ex_14} 5 | \exinput{exercises/01_01_ex_15} 6 | \exinput{exercises/01_01_ex_16} -------------------------------------------------------------------------------- /exercises/01_02_ex_01.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccc} 2 | x&+&y&=&3\\ 3 | 2x&-&3y&=&1\\ 4 | \end{array}$\vs{.2}} 5 | {$x=2,y=1$} -------------------------------------------------------------------------------- /exercises/01_02_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccc} 2 | 2x&+&4y&=&10\\ 3 | -x&+&y&=&4\\ 4 | \end{array}$\vs{.2}} 5 | {$x=-1,y=3$} -------------------------------------------------------------------------------- /exercises/01_02_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccc} 2 | -2x&+&3y&=&2\\ 3 | -x&+&y&=&1\\ 4 | \end{array}$\vs{.2}} 5 | {$x=-1,y=0$} -------------------------------------------------------------------------------- /exercises/01_02_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccccc} 2 | 2x&+&3y&=&2\\ 3 | -2x&+&6y&=&1\\ 4 | \end{array}$\vs{.2}} 5 | {$x=\frac12,y=\frac13$} -------------------------------------------------------------------------------- /exercises/01_02_ex_05.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccccc} 2 | -5x_1&&&+&2x_3&=&14\\ 3 | &&x_2&&&=&1\\ 4 | -3x_1&&&+&x_3&=&8\\ 5 | \end{array}$\vs{.2}} 6 | {$x_1=-2,x_2=1,x_3=2$} -------------------------------------------------------------------------------- /exercises/01_02_ex_06.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccccc} 2 | &-&5x_2&+&2x_3&=&-11\\ 3 | x_1&&&+&2x_3&=&15\\ 4 | &-&3x_2&+&x_3&=&-8\\ 5 | \end{array}$\vs{.2}} 6 | {$x_1=1,x_2=5,x_3=7$} -------------------------------------------------------------------------------- /exercises/01_02_ex_07.tex: -------------------------------------------------------------------------------- 1 | {$\ttb = \bmx{ccc}1&1&1\\2&0&2\\1&2&3\\ \emx$} 2 | {$2R_2\rightarrow R_2$} -------------------------------------------------------------------------------- /exercises/01_02_ex_08.tex: -------------------------------------------------------------------------------- 1 | {$\ttb = \bmx{ccc}1&1&1\\2&1&2\\1&2&3\\ \emx$} 2 | {$R_1+R_2\rightarrow R_2$} -------------------------------------------------------------------------------- /exercises/01_02_ex_09.tex: -------------------------------------------------------------------------------- 1 | {$\ttb = \bmx{ccc}3&5&7\\1&0&1\\1&2&3\\ \emx$} 2 | {$2R_3+R_1\rightarrow R_1$} -------------------------------------------------------------------------------- /exercises/01_02_ex_10.tex: -------------------------------------------------------------------------------- 1 | {$\ttb = \bmx{ccc}1&0&1\\1&1&1\\1&2&3\\ \emx$} 2 | {$R_1\leftrightarrow R_2$} -------------------------------------------------------------------------------- /exercises/01_02_ex_11.tex: -------------------------------------------------------------------------------- 1 | {$\ttb = \bmx{ccc}1&1&1\\1&0&1\\0&2&2\\ \emx$} 2 | {$-R_2+R_3\leftrightarrow R_3$} -------------------------------------------------------------------------------- /exercises/01_02_ex_12.tex: -------------------------------------------------------------------------------- 1 | {$-1R_1\rightarrow R_1$} 2 | {$\bmx{ccc} -2&1&-7\\0&4&-2\\5&0&3\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_02_ex_13.tex: -------------------------------------------------------------------------------- 1 | {$R_2\leftrightarrow R_3$} 2 | {$\bmx{ccc} 2&-1&7\\5&0&3\\0&4&-2\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_02_ex_14.tex: -------------------------------------------------------------------------------- 1 | {$R_1+R_2\rightarrow R_2$} 2 | {$\bmx{ccc} 2&-1&7\\2&3&5\\5&0&3\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_02_ex_15.tex: -------------------------------------------------------------------------------- 1 | {$2R_2+R_3\rightarrow R_3$} 2 | {$\bmx{ccc} 2&-1&7\\0&4&-2\\5&8&-1\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_02_ex_16.tex: -------------------------------------------------------------------------------- 1 | {$\frac12R_2\rightarrow R_2$} 2 | {$\bmx{ccc} 2&-1&7\\0&2&-1\\5&0&3\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_02_ex_17.tex: -------------------------------------------------------------------------------- 1 | {$-\frac52R_1+R_3\rightarrow R_3$} 2 | {$\bmx{ccc} 2&-1&7\\0&4&-2\\0&5/2&-29/2\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_02_ex_18.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccccc} 2 | 3x&+&4y&+&5z&=&7\\ 3 | -x&+&y&-&3z&=&1\\ 4 | 2x&-&2y&+&3z&=&5\\ 5 | \end{array}$} 6 | {$\bmx{cccc} 3&4&5&7\\-1&1&-3&1\\2&-2&3&5\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_02_ex_19.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccccc} 2 | 2x&+&5y&-&6z&=&2\\ 3 | 9x&&&-&8z&=&10\\ 4 | -2x&+&4y&+&z&=&-7\\ 5 | \end{array}$} 6 | {$\bmx{cccc} 2&5&-6&2\\9&0&-8&10\\-2&4&1&-7\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_02_ex_20.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{rl} 2 | x_1 +3x_2-4x_3 + 5x_4 =&17 \\ 3 | -x_1+4x_3+8x_4 =&1\\ 4 | 2x_1+3x_2+4x_3+5x_4=&6 5 | \end{array}$} 6 | {$\bmx{ccccc} 1&3&-4&5&17\\-1&0&4&8&1\\ 2&3&4&5&6 \emx$} -------------------------------------------------------------------------------- /exercises/01_02_ex_21.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{rl} 2 | 3x_1 -2x_2=&4 \\ 3 | 2x_1 =&3\\ 4 | -x_1+9x_2=&8\\ 5 | 5x_1-7x_2=&13\\ 6 | \end{array}$} 7 | {$\bmx{ccc} 3&-2&4\\ 2&0&3\\-1&9&8\\5&-7&13\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_02_ex_22.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc} 2 | 1&2&3\\ 3 | -1&3&9\\ \emx$} 4 | {$\begin{array}{rl} 5 | x_1+2x_2=&3\\ 6 | -x_1+3x_2=&9\\ \end{array}$} -------------------------------------------------------------------------------- /exercises/01_02_ex_23.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc} 2 | -3&4&7\\ 3 | 0&1&-2\\ \emx$} 4 | {$\begin{array}{rl} 5 | -3x_1+4x_2=&7\\ 6 | x_2=&-2\\ \end{array}$} -------------------------------------------------------------------------------- /exercises/01_02_ex_24.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccccc} 2 | 1&1&-1&-1&2\\ 3 | 2&1&3&5&7\\ \emx$} 4 | {$\begin{array}{rl} 5 | x_1+x_2-x_3-x_4=&2\\ 6 | 2x_1+x_2+3x_3+5x_4=&7\\ \end{array}$} -------------------------------------------------------------------------------- /exercises/01_02_ex_25.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccccc} 2 | 1&0&0&0&2\\ 3 | 0&1&0&0&-1\\ 4 | 0&0&1&0&5\\ 5 | 0&0&0&1&3 \emx$} 6 | {$\begin{array}{rl} 7 | x_1=&2\\ 8 | x_2=&-1\\ 9 | x_3=&5\\ 10 | x_4=&3\\ \end{array}$} -------------------------------------------------------------------------------- /exercises/01_02_ex_26.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cccccc} 2 | 1&0&1&0&7&2\\ 3 | 0&1&3&2&0&5\\ 4 | \emx$} 5 | {$\begin{array}{rl} 6 | x_1+x_3+7x_5=&2\\ 7 | x_2+3x_3+2x_4=&5\\ 8 | \end{array}$} -------------------------------------------------------------------------------- /exercises/01_02_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/01_02_exset_04} 2 | \exsetinput{exercises/01_02_exset_05} 3 | \exsetinput{exercises/01_02_exset_03} 4 | \exsetinput{exercises/01_02_exset_02} 5 | \exsetinput{exercises/01_02_exset_01} -------------------------------------------------------------------------------- /exercises/01_02_exset_01.tex: -------------------------------------------------------------------------------- 1 | {In Exercises} 2 | {, rewrite the system of equations in matrix form. Find the solution to the linear system by simultaneously manipulating the equations and the matrix.} 3 | \exinput{exercises/01_02_ex_01} 4 | \exinput{exercises/01_02_ex_02} 5 | \exinput{exercises/01_02_ex_03} 6 | \exinput{exercises/01_02_ex_04} 7 | \exinput{exercises/01_02_ex_05} 8 | \exinput{exercises/01_02_ex_06} -------------------------------------------------------------------------------- /exercises/01_02_exset_02.tex: -------------------------------------------------------------------------------- 1 | {A matrix \tta\ is given below. In Exercises} 2 | {, a matrix \ttb\ is given. Give the row operation that transforms \tta\ into \ttb. $$\tta = \bmx{ccc}1&1&1\\1&0&1\\1&2&3\emx$$} 3 | \exinput{exercises/01_02_ex_07} 4 | \exinput{exercises/01_02_ex_08} 5 | \exinput{exercises/01_02_ex_09} 6 | \exinput{exercises/01_02_ex_10} 7 | \exinput{exercises/01_02_ex_11} -------------------------------------------------------------------------------- /exercises/01_02_exset_03.tex: -------------------------------------------------------------------------------- 1 | {In Exercises} 2 | {, perform the given row operations on \tta, where $$\tta = \bmx{ccc}2&-1&7\\0&4&-2\\5&0&3\emx.$$} 3 | \exinput{exercises/01_02_ex_12} 4 | \exinput{exercises/01_02_ex_13} 5 | \exinput{exercises/01_02_ex_14} 6 | \exinput{exercises/01_02_ex_15} 7 | \exinput{exercises/01_02_ex_16} 8 | \exinput{exercises/01_02_ex_17} -------------------------------------------------------------------------------- /exercises/01_02_exset_04.tex: -------------------------------------------------------------------------------- 1 | {In Exercises} 2 | {, convert the given system of linear equations into an augmented matrix.} 3 | \exinput{exercises/01_02_ex_18} 4 | \exinput{exercises/01_02_ex_19} 5 | \exinput{exercises/01_02_ex_20} 6 | \exinput{exercises/01_02_ex_21} -------------------------------------------------------------------------------- /exercises/01_02_exset_05.tex: -------------------------------------------------------------------------------- 1 | {In Exercises} 2 | {, convert the given augmented matrix into a system of linear equations. Use the variables $x_1$, $x_2$, etc.} 3 | \exinput{exercises/01_02_ex_22} 4 | \exinput{exercises/01_02_ex_23} 5 | \exinput{exercises/01_02_ex_24} 6 | \exinput{exercises/01_02_ex_25} 7 | \exinput{exercises/01_02_ex_26} -------------------------------------------------------------------------------- /exercises/01_03_ex_01.tex: -------------------------------------------------------------------------------- 1 | {\begin{multicols}{2} 2 | \begin{enumerate} 3 | \item $\bmx{cc}1&0\\0&1\\ \emx$ 4 | \item $\bmx{cc}0&1\\1&0\\ \emx$ 5 | \item $\bmx{cc}1&1\\1&1\\ \emx$ 6 | \item $\bmx{ccc}1&0&1\\0&1&2\\ \emx$ 7 | \end{enumerate} 8 | \end{multicols}} 9 | {\begin{multicols}{2} 10 | \begin{enumerate} 11 | \item yes 12 | \item no 13 | \item no 14 | \item yes 15 | \end{enumerate} 16 | \end{multicols} 17 | } -------------------------------------------------------------------------------- /exercises/01_03_ex_02.tex: -------------------------------------------------------------------------------- 1 | {\begin{multicols}{2} 2 | \begin{enumerate} 3 | \item $\bmx{ccc}1&0&0\\0&0&1\\ \emx$ 4 | \item $\bmx{ccc}1&0&1\\0&1&1\\ \emx$ 5 | \item $\bmx{ccc}0&0&0\\1&0&0\\ \emx$ 6 | \item $\bmx{ccc}0&0&0\\0&0&0\\ \emx$ 7 | \end{enumerate} 8 | \end{multicols}} 9 | {\begin{multicols}{2}\begin{enumerate} 10 | \item yes 11 | \item yes 12 | \item no 13 | \item yes 14 | \end{enumerate}\end{multicols}} -------------------------------------------------------------------------------- /exercises/01_03_ex_03.tex: -------------------------------------------------------------------------------- 1 | {\begin{enumerate} 2 | \item $\bmx{ccc}1&1&1\\0&1&1\\0&0&1\\ \emx$ 3 | \item $\bmx{ccc}1&0&0\\0&1&0\\0&0&0 \emx$ 4 | \item $\bmx{ccc}1&0&0\\0&0&1\\0&0&0 \emx$ 5 | \item $\bmx{cccc}1&0&0&-5\\0&1&0&7\\0&0&1&3 \emx$ 6 | \end{enumerate}} 7 | {\begin{multicols}{2}\begin{enumerate} 8 | \item no 9 | \item yes 10 | \item yes 11 | \item yes 12 | \end{enumerate}\end{multicols}} -------------------------------------------------------------------------------- /exercises/01_03_ex_04.tex: -------------------------------------------------------------------------------- 1 | {\begin{enumerate} 2 | \item $\bmx{cccc}2&0&0&2\\0&2&0&2\\0&0&2&2\\ \emx$ 3 | \item $\bmx{cccc}0&1&0&0\\0&0&1&0\\0&0&0&0 \emx$ 4 | \item $\bmx{cccc}0&0&1&-5\\0&0&0&0\\0&0&0&0 \emx$ 5 | \item $\bmx{cccccc}1&1&0&0&1&1\\0&0&1&0&1&1\\0&0&0&1&0&0 \emx$ 6 | \end{enumerate}} 7 | {\begin{multicols}{2}\begin{enumerate} 8 | \item no 9 | \item yes 10 | \item yes 11 | \item yes 12 | \end{enumerate}\end{multicols}} -------------------------------------------------------------------------------- /exercises/01_03_ex_05.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cc} 1&2\\-3&-5\\ \emx$} 2 | {$\bmx{cc} 1&0\\0&1\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_03_ex_06.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cc} 2&-2\\3&-2\\ \emx$} 2 | {$\bmx{cc} 1&0\\0&1\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_03_ex_07.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cc} 4&12\\-2&-6\\ \emx$} 2 | {$\bmx{cc} 1&3\\0&0\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_03_ex_08.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cc} -5&7\\10&14\\ \emx$} 2 | {$\bmx{cc} 1&-7/5\\0&0\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_03_ex_09.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc} -1&1&4\\-2&1&1 \emx$} 2 | {$\bmx{ccc} 1&0&3\\0&1&7\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_03_ex_10.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc} 7&2&3\\3&1&2 \emx$} 2 | {$\bmx{ccc} 1&0&-1\\0&1&5\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_03_ex_11.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc} 3&-3&6\\-1&1&-2 \emx$} 2 | {$\bmx{ccc} 1&-1&2\\0&0&0\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_03_ex_12.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc} 4&5&-6\\-12&-15&18 \emx$} 2 | {$\bmx{ccc} 1&\frac54&-\frac32\\0&0&0\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_03_ex_13.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc} -2&-4&-8\\-2&-3&-5\\ 2&3&6\\ \emx$} 2 | {$\bmx{ccc} 1&0&0\\0&1&0\\0&0&1\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_03_ex_14.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc} 2&1&1\\1&1&1\\2&1&2\\ \emx$} 2 | {$\bmx{ccc} 1&0&0\\0&1&0\\0&0&1\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_03_ex_15.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc} 1&2&1\\1&3&1\\-1&-3&0\\ \emx$} 2 | {$\bmx{ccc} 1&0&0\\0&1&0\\0&0&1\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_03_ex_16.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc} 1&2&3\\0&4&5\\1&6&9\\ \emx$} 2 | {$\bmx{ccc} 1&0&0\\0&1&0\\0&0&1\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_03_ex_17.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cccc} 1&1&1&2\\2&-1&-1&1\\-1&1&1&0\\ \emx$} 2 | {$\bmx{cccc} 1&0&0&1\\0&1&1&1\\0&0&0&0\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_03_ex_18.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cccc} 2&-1&1&5\\3&1&6&-1\\3&0&5&0\\ \emx$} 2 | {$\bmx{cccc} 1&0&0&5\\0&1&0&2\\0&0&1&-3\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_03_ex_19.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cccc} 1&1&-1&7\\2&1&0&10\\3&2&-1&17\\ \emx$} 2 | {$\bmx{cccc} 1&0&1&3\\0&1&-2&4\\0&0&0&0\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_03_ex_20.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cccc} 4&1&8&15\\1&1&2&7\\3&1&5&11\\ \emx$} 2 | {$\bmx{cccc} 1&0&3&4\\0&1&-1&3\\0&0&0&0\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_03_ex_21.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cccccc} 2&2&1&3&1&4\\1&1&1&3&1&4\\ \emx$} 2 | {$\bmx{cccccc} 1&1&0&0&0&0\\0&0&1&3&1&4\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_03_ex_22.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cccccc} 1&-1&3&1&-2&9\\2&-2&6&1&-2&13 \emx$} 2 | {$\bmx{cccccc} 1&-1&3&0&0&4\\0&0&0&1&-2&5\\ \emx$} -------------------------------------------------------------------------------- /exercises/01_03_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/01_03_exset_02} 2 | \exsetinput{exercises/01_03_exset_03} 3 | 4 | -------------------------------------------------------------------------------- /exercises/01_03_exset_01.tex: -------------------------------------------------------------------------------- 1 | {In Exercises} 2 | {, find the solution of the given problem by: 3 | \begin{description} 4 | \item [(a)]creating an appropriate system of linear equations 5 | \item [(b)]forming the augmented matrix that corresponds to this system 6 | \item [(c)]putting the augmented matrix into \rref 7 | \item [(d)]interpreting the \rref\ of the matrix as a solution 8 | \end{description}} 9 | \exinput{exercises/01_01_ex_19} 10 | \exinput{exercises/01_01_ex_21} 11 | \exinput{exercises/01_01_ex_17} 12 | \exinput{exercises/01_01_ex_18} 13 | \exinput{exercises/01_01_ex_20} -------------------------------------------------------------------------------- /exercises/01_03_exset_02.tex: -------------------------------------------------------------------------------- 1 | {In Exercises} 2 | {, state whether or not the given matrices are in \rref. If it is not, state why.} 3 | \exinput{exercises/01_03_ex_01} 4 | \exinput{exercises/01_03_ex_02} 5 | \exinput{exercises/01_03_ex_03} 6 | \exinput{exercises/01_03_ex_04} -------------------------------------------------------------------------------- /exercises/01_03_exset_03.tex: -------------------------------------------------------------------------------- 1 | {In Exercises} 2 | {, use Gaussian Elimination to put the given matrix into \rref.} 3 | \exinput{exercises/01_03_ex_05} 4 | \exinput{exercises/01_03_ex_06} 5 | \exinput{exercises/01_03_ex_07} 6 | \exinput{exercises/01_03_ex_08} 7 | \exinput{exercises/01_03_ex_09} 8 | \exinput{exercises/01_03_ex_10} 9 | \exinput{exercises/01_03_ex_11} 10 | \exinput{exercises/01_03_ex_12} 11 | \exinput{exercises/01_03_ex_13} 12 | \exinput{exercises/01_03_ex_14} 13 | \exinput{exercises/01_03_ex_15} 14 | \exinput{exercises/01_03_ex_16} 15 | \exinput{exercises/01_03_ex_17} 16 | \exinput{exercises/01_03_ex_18} 17 | \exinput{exercises/01_03_ex_19} 18 | \exinput{exercises/01_03_ex_20} 19 | \exinput{exercises/01_03_ex_21} 20 | \exinput{exercises/01_03_ex_22} -------------------------------------------------------------------------------- /exercises/01_04_ex_01.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccc} 2 | 2x_1&+&4x_2&=&2\\ 3 | x_1&+&2x_2&=&1\\ 4 | \end{array}$} 5 | {$x_1=1-2x_2$; $x_2$ is free. Possible solutions: $x_1=1$, $x_2=0$ and $x_1=-1$, $x_2=1$.} -------------------------------------------------------------------------------- /exercises/01_04_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccc} 2 | -x_1&+&5x_2&=&3\\ 3 | 2x_1&-&10x_2&=&-6\\ 4 | \end{array}$} 5 | {$x_1=-3+5x_2$; $x_2$ is free. Possible solutions: $x_1 = 3$, $x_2=0$ and $x_1 = -8$, $x_2 = -1$} -------------------------------------------------------------------------------- /exercises/01_04_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccc} 2 | x_1&+&x_2&=&3\\ 3 | 2x_1&+&x_2&=&4\\ 4 | \end{array}$} 5 | {$x_1=1$; $x_2=2$} -------------------------------------------------------------------------------- /exercises/01_04_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccc} 2 | -3x_1&+&7x_2&=&-7\\ 3 | 2x_1&-&8x_2&=&8\\ 4 | \end{array}$} 5 | {$x_1=0$; $x_2=-1$} -------------------------------------------------------------------------------- /exercises/01_04_ex_05.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccccc} 2 | -2x_1&+&4x_2&+&4x_3&=&6\\ 3 | x_1&-&3x_2&+&2x_3&=&1\\ 4 | \end{array}$} 5 | {$x_1=-11+10x_3$; $x_2=-4+4x_3$; $x_3$ is free. Possible solutions: $x_1=-11$, $x_2 = -4$, $x_3=0$ and $x_1 = -1$, $x_2 = 0$ and $x_3 = 1$.} -------------------------------------------------------------------------------- /exercises/01_04_ex_06.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccccc} 2 | -x_1&+&2x_2&+&2x_3&=&2\\ 3 | 2x_1&+&5x_2&+&x_3&=&2\\ 4 | \end{array}$} 5 | {$x_1=-\frac23+\frac89x_3$; $x_2=\frac23-\frac59x_3$; $x_3$ is free. Possible solutions: $x_1 = -\frac23$, $x_2 = \frac23$, $x_3 = 0$ and $x_1 = \frac49$, $x_2 = -\frac19$, $x_3 = 1$} -------------------------------------------------------------------------------- /exercises/01_04_ex_07.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{rcl} 2 | -x_1-x_2+x_3+x_4&=&0\\ 3 | -2x_1-2x_2+x_3&=&-1\\ 4 | \end{array}$} 5 | {$x_1=1-x_2-x_4$; $x_2$ is free; $x_3=1-2x_4$; $x_4$ is free. Possible solutions: $x_1 = 1$, $x_2 = 0$, $x_3 = 1$, $x_4 = 0$ and $x_1 = -2$, $x_2 = 1$, $x_3 = -3$, $x_4=2$ } -------------------------------------------------------------------------------- /exercises/01_04_ex_08.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{rcl} 2 | x_1+x_2+6x_3+9x_4&=&0\\ 3 | -x_1-x_3-2x_4&=&-3\\ 4 | \end{array}$} 5 | {$x_1=3-x_3-2x_4$; $x_2=-3-5x_3-7x_4$; $x_3$ is free; $x_4$ is free. Possible solutions: $x_1 =3$, $x_2 = -3$, $x_3=0$, $x_4=0$ and $x_1 = 0$, $x_2 = -5$, $x_3 =-1$, $x_4=1$ } -------------------------------------------------------------------------------- /exercises/01_04_ex_09.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccccc} 2 | x_1&+&2x_2&+&2x_3&=&1\\ 3 | 2x_1&+&x_2&+&3x_3&=&1\\ 4 | 3x_1&+&3x_2&+&5x_3&=&2\\ 5 | \end{array}$} 6 | {$x_1=\frac13-\frac43x_3$; $x_2=\frac13-\frac13x_3$; $x_3$ is free. Possible solutions: $x_1 = \frac13$, $x_2=\frac13$, $x_3=0$ and $x_1 = -1$, $x_2 = 0$, $x_3=1$} -------------------------------------------------------------------------------- /exercises/01_04_ex_10.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccccc} 2 | 2x_1&+&4x_2&+&6x_3&=&2\\ 3 | 1x_1&+&2x_2&+&3x_3&=&1\\ 4 | -3x_1&-&6x_2&-&9x_3&=&-3\\ 5 | \end{array}$} 6 | {$x_1=1-2x_2-3x_3$; $x_2$ is free; $x_3$ is free. 7 | Possible solutions: $x_1=1$, $x_2=0$, $x_3=0$ and $x_1=8$, $x_2=1$, $x_3 = -3$} -------------------------------------------------------------------------------- /exercises/01_04_ex_11.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccc} 2 | 2x_1&+&3x_2&=&1\\ 3 | -2x_1&-&3x_2&=&1\\ 4 | \end{array}$} 5 | {No solution; the system is inconsistent.} -------------------------------------------------------------------------------- /exercises/01_04_ex_12.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccc} 2 | x_1&+&2x_2&=&1\\ 3 | -x_1&-&2x_2&=&5\\ 4 | \end{array}$} 5 | {No solution; the system is inconsistent.} -------------------------------------------------------------------------------- /exercises/01_04_ex_13.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccccc} 2 | 2x_1&+&x_2&+&2x_3&=&0\\ 3 | x_1&+&x_2&+&3x_3&=&1\\ 4 | 3x_1&+&2x_2&+&5x_3&=&3\\ 5 | \end{array}$} 6 | {No solution; the system is inconsistent.} -------------------------------------------------------------------------------- /exercises/01_04_ex_14.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccccc} 2 | x_1&+&3x_2&+&3x_3&=&1\\ 3 | 2x_1&-&x_2&+&2x_3&=&-1\\ 4 | 4x_1&+&5x_2&+&8x_3&=&2\\ 5 | \end{array}$} 6 | {No solution; the system is inconsistent.} -------------------------------------------------------------------------------- /exercises/01_04_ex_15.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccc}x_1&+&2x_2&=&1\\2x_1&+&4x_2&=&k\end{array}$} 2 | {Never exactly 1 solution; infinite solutions if $k=2$; no solution if $k\neq 2$.} -------------------------------------------------------------------------------- /exercises/01_04_ex_16.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccc}x_1&+&2x_2&=&1\\x_1&+&kx_2&=&1\end{array}$ } 2 | {Exactly 1 solution if $k\neq 2$; infinite solutions if $k=2$; never no solution.} -------------------------------------------------------------------------------- /exercises/01_04_ex_17.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccc}x_1&+&2x_2&=&1\\x_1&+&kx_2&=&2\end{array}$} 2 | {Exactly 1 solution if $k\neq 2$; no solution if $k=2$; never infinite solutions.} -------------------------------------------------------------------------------- /exercises/01_04_ex_18.tex: -------------------------------------------------------------------------------- 1 | {$\begin{array}{ccccc}x_1&+&2x_2&=&1\\x_1&+&3x_2&=&k\end{array}$} 2 | {Exactly 1 solution for all $k$.} -------------------------------------------------------------------------------- /exercises/01_04_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/01_04_exset_01} 2 | \exsetinput{exercises/01_04_exset_02} -------------------------------------------------------------------------------- /exercises/01_04_exset_01.tex: -------------------------------------------------------------------------------- 1 | {In Exercises} 2 | {, find the solution to the given linear system. If the system has infinite solutions, give 2 particular solutions.} 3 | \exinput{exercises/01_04_ex_01} 4 | \exinput{exercises/01_04_ex_02} 5 | \exinput{exercises/01_04_ex_03} 6 | \exinput{exercises/01_04_ex_04} 7 | \exinput{exercises/01_04_ex_11} 8 | \exinput{exercises/01_04_ex_12} 9 | \exinput{exercises/01_04_ex_05} 10 | \exinput{exercises/01_04_ex_06} 11 | \exinput{exercises/01_04_ex_07} 12 | \exinput{exercises/01_04_ex_08} 13 | \exinput{exercises/01_04_ex_13} 14 | \exinput{exercises/01_04_ex_14} 15 | \exinput{exercises/01_04_ex_09} 16 | \exinput{exercises/01_04_ex_10} -------------------------------------------------------------------------------- /exercises/01_04_exset_02.tex: -------------------------------------------------------------------------------- 1 | {In Exercises } 2 | {, state for which values of $k$ the given system will have exactly 1 solution, infinite solutions, or no solution.} 3 | \exinput{exercises/01_04_ex_15} 4 | \exinput{exercises/01_04_ex_16} 5 | \exinput{exercises/01_04_ex_17} 6 | \exinput{exercises/01_04_ex_18} -------------------------------------------------------------------------------- /exercises/01_05_ex_01.tex: -------------------------------------------------------------------------------- 1 | {$(1,3)$ and $(3,15)$} 2 | {$f(x) = 6x-3$} -------------------------------------------------------------------------------- /exercises/01_05_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$(-2,14)$ and $(3,4)$} 2 | {$f(x) = -2x+10$} -------------------------------------------------------------------------------- /exercises/01_05_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$(1,5)$, $(-1,3)$ and $(3,-1)$} 2 | {$f(x) = -x^2+x+5$} -------------------------------------------------------------------------------- /exercises/01_05_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$(-4,-3)$, $(0,1)$ and $(1,4.5)$} 2 | {$f(x) = \frac12x^2+3x+1$} -------------------------------------------------------------------------------- /exercises/01_05_ex_05.tex: -------------------------------------------------------------------------------- 1 | {$(-2,15)$, $(-1,4)$, $(1,0)$ and $(2,-5)$} 2 | {$f(x) = -x^3+x^2-x+1$} -------------------------------------------------------------------------------- /exercises/01_05_ex_06.tex: -------------------------------------------------------------------------------- 1 | {$(-2,-7)$, $(1,2)$, $(2,9)$ and $(3,28)$} 2 | {$f(x) = x^3+1$} -------------------------------------------------------------------------------- /exercises/01_05_ex_07.tex: -------------------------------------------------------------------------------- 1 | {$(-1,-8)$, $(1,-2)$ and $(3,4)$} 2 | {$f(x) = 3x-5$} -------------------------------------------------------------------------------- /exercises/01_05_ex_08.tex: -------------------------------------------------------------------------------- 1 | {$(-3,3)$, $(1,3)$ and $(2,3)$} 2 | {$f(x) = 3$} -------------------------------------------------------------------------------- /exercises/01_05_ex_09.tex: -------------------------------------------------------------------------------- 1 | {$(-3,10)$, $(-1,2)$, $(1,2)$ and $(2,5)$} 2 | {$f(x) = x^2+1$} -------------------------------------------------------------------------------- /exercises/01_05_ex_10.tex: -------------------------------------------------------------------------------- 1 | {$(0,1)$, $(-3,-3.5)$, $(-2,-2)$ and $(4,7)$} 2 | {$f(x) = \frac32x+1$} -------------------------------------------------------------------------------- /exercises/01_05_ex_13.tex: -------------------------------------------------------------------------------- 1 | {In a football game, 29 points are scored from 8 scoring occasions. There are 2 more successful extra point kicks than successful two point conversions. Find all ways in which the points may have been scored in this game.} 2 | {The augmented matrix from this system is $\bmx{ccccc}1&1&1&1&8\\6&1&2&3&29\\0&1&-1&0&2\\ \emx$. From this we find the solution \begin{align*} t&=4-\frac13f\\ x&=3-\frac13f\\ w&=1-\frac13f.\end{align*} The only time each of these variables are nonnegative integers is when $f=0$ or $f=3$. If $f=0$, then we have 4 touchdowns, 3 extra points and 1 two point conversions (no field goals). If $f=3$, then we have 3 touchdowns, 2 extra points and no two point conversions (and 3 field goals). } -------------------------------------------------------------------------------- /exercises/01_05_ex_14.tex: -------------------------------------------------------------------------------- 1 | {In a basketball game, where points are scored either by a 3 point shot, a 2 point shot or a 1 point free throw, 80 points were scored from 30 successful shots. Find all ways in which the points may have been scored in this game.} 2 | {Let $x_1$, $x_2$ and $x_3$ represent the number of free throws, 2 point and 3 point shots taken. The augmented matrix from this system is $\bmx{cccc}1&1&1&30\\1&2&3&80 \emx$. From this we find the solution \begin{align*} x_1&=-20+x_3\\ x_2&=50-2x_3.\end{align*} In order for $x_1$ and $x_2$ to be nonnegative, we need $20\leq x_3\leq 25$. Thus there are 6 different scenerios: the ``first'' is where 20 three point shots are taken, no free throws, and 10 two point shots; the ``last'' is where 25 three point shots are taken, 5 free throws, and no two point shots. 3 | } -------------------------------------------------------------------------------- /exercises/01_05_ex_15.tex: -------------------------------------------------------------------------------- 1 | {In a basketball game, where points are scored either by a 3 point shot, a 2 point shot or a 1 point free throw, 110 points were scored from 70 successful shots. Find all ways in which the points may have been scored in this game.} 2 | {Let $x_1$, $x_2$ and $x_3$ represent the number of free throws, 2 point and 3 point shots taken. The augmented matrix from this system is $\bmx{cccc}1&1&1&70\\1&2&3&110 \emx$. From this we find the solution \begin{align*} x_1&=30+x_3\\ x_2&=40-2x_3.\end{align*} In order for $x_2$ to be nonnegative, we need $x_3\leq 20$. Thus there are 21 different scenerios: the ``first'' is where 0 three point shots are taken ($x_3=0$, 30 free throws and 40 two point shots; the ``last'' is where 20 three point shots are taken, 50 free throws, and no two point shots. 3 | } -------------------------------------------------------------------------------- /exercises/01_05_ex_16.tex: -------------------------------------------------------------------------------- 1 | {Describe the equations of the linear functions that go through the point (1,3). Give 2 examples.} 2 | {Let $y = ax+b$; all linear functions through (1,3) come in the form $y = (3-b)x+b$. Examples: $b=0$ yields $y = 3x$; $b=2$ yields $y=x+2$. 3 | } -------------------------------------------------------------------------------- /exercises/01_05_ex_17.tex: -------------------------------------------------------------------------------- 1 | {Describe the equations of the linear functions that go through the point (2,5). Give 2 examples.} 2 | {Let $y = ax+b$; all linear functions through (2,5) come in the form $y = (2.5-\frac12b)x+b$. Examples: $b=1$ yields $y = 2x+1$; $b=-1$ yields $y=3x-1$. 3 | } -------------------------------------------------------------------------------- /exercises/01_05_ex_18.tex: -------------------------------------------------------------------------------- 1 | {Describe the equations of the quadratic functions that go through the points $(2,-1)$ and (1,0). Give 2 examples.} 2 | {Let $y = ax^2+bx+c$; we find that $a = -\frac12+\frac12 c$ and $b = \frac12-\frac32c$. Examples: $c=1$ yields $y = -x+1$; $c=3$ yields $y=x^2-4x+3$. 3 | } -------------------------------------------------------------------------------- /exercises/01_05_ex_19.tex: -------------------------------------------------------------------------------- 1 | {Describe the equations of the quadratic functions that go through the points $(-1,3)$ and (2,6). Give 2 examples.} 2 | {Let $y = ax^2+bx+c$; we find that $a = 2-\frac12 c$ and $b = -1+\frac12c$. Examples: $c=0$ yields $y = 2x^2-x$; $c=-2$ yields $y=3x^2-2x-2$. 3 | } -------------------------------------------------------------------------------- /exercises/01_05_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/01_03_exset_01} 2 | \exsetinput{exercises/01_05_exset_01} 3 | \exinput{exercises/01_05_ex_11} 4 | \exinput{exercises/01_05_ex_12} 5 | \exinput{exercises/01_05_ex_13} 6 | \exinput{exercises/01_05_ex_14} 7 | \exinput{exercises/01_05_ex_15} 8 | \exinput{exercises/01_05_ex_16} 9 | \exinput{exercises/01_05_ex_17} 10 | \exinput{exercises/01_05_ex_18} 11 | \exinput{exercises/01_05_ex_19} -------------------------------------------------------------------------------- /exercises/01_05_exset_01.tex: -------------------------------------------------------------------------------- 1 | {In Exercises} 2 | {, find the polynomial with the smallest degree that goes through the given points.} 3 | \exinput{exercises/01_05_ex_01} 4 | \exinput{exercises/01_05_ex_02} 5 | \exinput{exercises/01_05_ex_03} 6 | \exinput{exercises/01_05_ex_04} 7 | \exinput{exercises/01_05_ex_07} 8 | \exinput{exercises/01_05_ex_08} 9 | \exinput{exercises/01_05_ex_05} 10 | \exinput{exercises/01_05_ex_06} 11 | \exinput{exercises/01_05_ex_09} 12 | \exinput{exercises/01_05_ex_10} -------------------------------------------------------------------------------- /exercises/02_01_ex_01.tex: -------------------------------------------------------------------------------- 1 | {$\tta + \ttb$} 2 | {$\bmx{cc} -2& -1\\12 & 13\emx$} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$2\tta - 3\ttb$} 2 | {$\bmx{cc} 11& -8\\-1 & -19\emx$} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$3\tta - \tta$} 2 | {$\bmx{cc} 2 & -2\\14 & 8\emx$} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$4\ttb - 2\tta$} 2 | {$\bmx{c} -14\\-5 \\ -9\emx$} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_05.tex: -------------------------------------------------------------------------------- 1 | {$4\ttb - 2\tta$} 2 | {$\bmx{c} -14\\6 \emx$} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_06.tex: -------------------------------------------------------------------------------- 1 | {$-2\tta + 3\tta$} 2 | {$\bmx{c} -12\\2 \emx$} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_07.tex: -------------------------------------------------------------------------------- 1 | {$-2\tta - 3\tta$} 2 | {$\bmx{c} -15\\-25 \emx$} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_08.tex: -------------------------------------------------------------------------------- 1 | {$-\ttb + 3\ttb-2\ttb$} 2 | {$\bmx{c} 0\\0 \emx$} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_09.tex: -------------------------------------------------------------------------------- 1 | {$2\tta + \ttx = \ttb$} 2 | {$\ttx = \bmx{cc} -5 & 9 \\-1 & -14 \emx$} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_10.tex: -------------------------------------------------------------------------------- 1 | {$\tta - \ttx = 3\ttb$} 2 | {$\ttx = \bmx{cc} 0 & -22 \\-7 & 17 \emx$} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_11.tex: -------------------------------------------------------------------------------- 1 | {$3\tta + 2\ttx = -1\ttb$} 2 | {$\ttx = \bmx{cc} -5 & -2 \\-9/2 & -19/2 \emx$} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_12.tex: -------------------------------------------------------------------------------- 1 | {$\tta - \frac12\ttx = -\ttb$} 2 | {$\ttx = \bmx{cc} 8 & 12 \\10 & 2 \emx$} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_13.tex: -------------------------------------------------------------------------------- 1 | {$a\bmx{c} 1\\2 \emx + b\bmx{c} -1\\5\emx = \bmx{c} 1\\9 \emx$} 2 | {$a = 2$, $b = 1$} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_14.tex: -------------------------------------------------------------------------------- 1 | {$a\bmx{c} -3\\1 \emx + b\bmx{c} 8\\4\emx = \bmx{c} 7\\1 \emx$} 2 | {$a = -1$, $b = 1/2$} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_15.tex: -------------------------------------------------------------------------------- 1 | {$a\bmx{c} 4\\-2 \emx + b\bmx{c} -6\\3\emx = \bmx{c} 10\\-5 \emx$} 2 | {$a = 5/2 + 3/2b$} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_16.tex: -------------------------------------------------------------------------------- 1 | {$a\bmx{c} 1\\1 \emx + b\bmx{c} -1\\3\emx = \bmx{c} 5\\5 \emx$} 2 | {$a = 5$, $b = 0$} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_17.tex: -------------------------------------------------------------------------------- 1 | {$a\bmx{c} 1\\3 \emx + b\bmx{c} -3\\-9\emx = \bmx{c} 4\\-12 \emx$} 2 | {No solution.} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_18.tex: -------------------------------------------------------------------------------- 1 | {$a\bmx{c} 1\\2\\3 \emx + b\bmx{c} 1\\1\\2\emx = \bmx{c} 0\\ -1\\-1 \emx$} 2 | {$a=-1$, $b=1$} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_19.tex: -------------------------------------------------------------------------------- 1 | {$a\bmx{c} 1\\0\\1 \emx + b\bmx{c} 5\\1\\2\emx = \bmx{c} 3\\ 4\\7 \emx$} 2 | {No solution.} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_20.tex: -------------------------------------------------------------------------------- 1 | {$3(\tta - \ttb)+\ttb$} 2 | {$\bmx{cc} 9& -7\\11 & -6\emx$} 3 | -------------------------------------------------------------------------------- /exercises/02_01_ex_21.tex: -------------------------------------------------------------------------------- 1 | {$2(\tta - \ttb)-(\tta - 3\ttb)$} 2 | {$\bmx{cc} -2&1\\12&13\emx$} 3 | -------------------------------------------------------------------------------- /exercises/02_01_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/02_01_exset_01} 2 | \exsetinput{exercises/02_01_exset_02} 3 | \exsetinput{exercises/02_01_exset_03} 4 | \exsetinput{exercises/02_01_exset_04} -------------------------------------------------------------------------------- /exercises/02_01_exset_01.tex: -------------------------------------------------------------------------------- 1 | {\noin Matrices \tta\ and \ttb\ are given below. In Exercises} 2 | {, simplify the given expression. 3 | $$\tta = \bmx{cc} 1 & -1\\ 7 & 4\emx \quad \ttb = \bmx{cc} -3 & 2\\5 & 9\emx$$} 4 | \exinput{exercises/02_01_ex_01} 5 | \exinput{exercises/02_01_ex_02} 6 | \exinput{exercises/02_01_ex_03} 7 | \exinput{exercises/02_01_ex_04} 8 | \exinput{exercises/02_01_ex_20} 9 | \exinput{exercises/02_01_ex_21} -------------------------------------------------------------------------------- /exercises/02_01_exset_02.tex: -------------------------------------------------------------------------------- 1 | {\noin Matrices \tta\ and \ttb\ are given below. In Exercises} 2 | {, simplify the given expression. 3 | $$\tta = \bmx{c} 3\\ 5\emx \quad \ttb = \bmx{c} -2\\4\emx$$} 4 | \exinput{exercises/02_01_ex_05} 5 | \exinput{exercises/02_01_ex_06} 6 | \exinput{exercises/02_01_ex_07} 7 | \exinput{exercises/02_01_ex_08} -------------------------------------------------------------------------------- /exercises/02_01_exset_03.tex: -------------------------------------------------------------------------------- 1 | {\noin Matrices \tta\ and \ttb\ are given below. In Exercises} 2 | {, find \ttx\ that satisfies the equation. 3 | $$\tta = \bmx{cc} 3 & -1\\ 2 & 5\emx \quad \ttb = \bmx{cc} 1 & 7\\3 & -4\emx$$} 4 | \exinput{exercises/02_01_ex_09} 5 | \exinput{exercises/02_01_ex_10} 6 | \exinput{exercises/02_01_ex_11} 7 | \exinput{exercises/02_01_ex_12} -------------------------------------------------------------------------------- /exercises/02_01_exset_04.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, find values for the scalars $a$ and $b$ that satisfy the given equation.} 3 | \exinput{exercises/02_01_ex_13} 4 | \exinput{exercises/02_01_ex_14} 5 | \exinput{exercises/02_01_ex_15} 6 | \exinput{exercises/02_01_ex_16} 7 | \exinput{exercises/02_01_ex_17} 8 | \exinput{exercises/02_01_ex_18} 9 | \exinput{exercises/02_01_ex_19} -------------------------------------------------------------------------------- /exercises/02_02_ex_01.tex: -------------------------------------------------------------------------------- 1 | {$\vu = \bmx{cc} 1&-4\emx \quad \vv = \bmx{c} -2\\5\emx$} 2 | {$-22$} -------------------------------------------------------------------------------- /exercises/02_02_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$\vu = \bmx{cc} 2&3\emx \quad \vv = \bmx{c} 7\\-4\emx$} 2 | {$2$} -------------------------------------------------------------------------------- /exercises/02_02_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$\vu = \bmx{cc} 1&-1\emx \quad \vv = \bmx{c} 3\\3\emx$} 2 | {$0$} -------------------------------------------------------------------------------- /exercises/02_02_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$\vu = \bmx{cc} 0.6&0.8\emx \quad \vv = \bmx{c} 0.6\\0.8\emx$} 2 | {$1$} -------------------------------------------------------------------------------- /exercises/02_02_ex_05.tex: -------------------------------------------------------------------------------- 1 | {$\vu = \bmx{ccc} 1&2&-1\emx \ \vv = \bmx{c} 2\\1\\-1\emx$} 2 | {$5$} -------------------------------------------------------------------------------- /exercises/02_02_ex_06.tex: -------------------------------------------------------------------------------- 1 | {$\vu = \bmx{ccc} 3&2&-2\emx \ \vv = \bmx{c} -1\\0\\9\emx$} 2 | {$-21$} -------------------------------------------------------------------------------- /exercises/02_02_ex_07.tex: -------------------------------------------------------------------------------- 1 | {$\vu = \bmx{ccc} 8&-4&3\emx \ \vv = \bmx{c} 2\\4\\5\emx$} 2 | {$15$} -------------------------------------------------------------------------------- /exercises/02_02_ex_08.tex: -------------------------------------------------------------------------------- 1 | {$\vu = \bmx{ccc} -3&6&1\emx$ \ $\vv = \bmx{c} 1\\-1\\1\emx$} 2 | {$-8$} -------------------------------------------------------------------------------- /exercises/02_02_ex_09.tex: -------------------------------------------------------------------------------- 1 | {$\vu = \bmx{cccc} 1&2&3&4\emx$ 2 | 3 | $\vv = \bmx{c} 1\\-1\\1\\-1\emx$} 4 | {$-2$} -------------------------------------------------------------------------------- /exercises/02_02_ex_10.tex: -------------------------------------------------------------------------------- 1 | {$\vu = \bmx{cccc} 6&2&-1&2\emx$ 2 | 3 | $\vv = \bmx{c} 3\\2\\9\\5\emx$} 4 | {$23$} -------------------------------------------------------------------------------- /exercises/02_02_ex_11.tex: -------------------------------------------------------------------------------- 1 | {$\vu = \bmx{ccc} 1&2&3\emx \quad \vv = \bmx{c} 3\\2\emx$} 2 | {Not possible.} -------------------------------------------------------------------------------- /exercises/02_02_ex_12.tex: -------------------------------------------------------------------------------- 1 | {$\vu = \bmx{cc} 2&-5\emx \quad \vv = \bmx{c} 1\\1\\1\emx$} 2 | {Not possible.} -------------------------------------------------------------------------------- /exercises/02_02_ex_13.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{cc} 1&2\\-1&4\emx$\ $\ttb = \bmx{cc}2&5\\3&-1\emx$} 2 | {$\tta\ttb=\bmx{cc} 8&3\\10&-9\emx$ 3 | 4 | $\ttb\tta=\bmx{cc} -3&24\\4&2\emx$} -------------------------------------------------------------------------------- /exercises/02_02_ex_14.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{cc} 3&7\\2&5\emx$\ $\ttb = \bmx{cc}1&-1\\3&-3\emx$} 2 | {$\tta\ttb=\bmx{cc} 24&-24\\17&-17\emx$ 3 | 4 | $\ttb\tta=\bmx{cc} 1&2\\3&6\emx$} -------------------------------------------------------------------------------- /exercises/02_02_ex_15.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{cc} 3&-1\\2&2\emx$ 2 | 3 | $\ttb = \bmx{ccc}1&0&7\\4&2&9\emx$} 4 | {$\tta\ttb=\bmx{ccc} -1&-2&12\\10&4&32\emx$ 5 | 6 | $\ttb\tta$ is not possible.} -------------------------------------------------------------------------------- /exercises/02_02_ex_16.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{cc} 0&1\\1&-1\\-2&-4\emx$ 2 | 3 | $\ttb = \bmx{cc}-2&0\\3&8\emx$} 4 | {$\tta\ttb=\bmx{cc} 3&8\\-5&-8\\-8&-32\emx$ 5 | 6 | $\ttb\tta$ is not possible.} -------------------------------------------------------------------------------- /exercises/02_02_ex_17.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{ccc} 9&4&3\\9&-5&9\emx$ 2 | 3 | $\ttb = \bmx{cc}-2&5\\-2&-1\emx$} 4 | {$\tta\ttb$ is not possible. 5 | 6 | $\ttb\tta = \bmx{ccc} 27&-33&39\\-27&-3&-15\emx$ } -------------------------------------------------------------------------------- /exercises/02_02_ex_18.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{cc} -2&-1\\9&-5\\3&-1\emx$ 2 | 3 | $\ttb = \bmx{ccc}-5&6&-4\\0&6&-3\emx$} 4 | {$\tta\ttb =\bmx{ccc}10&-18&11\\-45&24&-21\\-15&12&-9\emx$ 5 | 6 | $\ttb\tta = \bmx{cc} 52 & -21\\45&-27\emx$ } -------------------------------------------------------------------------------- /exercises/02_02_ex_19.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{cc} 2&6\\6&2\\5&-1\emx$ 2 | 3 | $\ttb = \bmx{ccc}-4&5&0\\-4&4&-4\emx$} 4 | {$\tta\ttb =\bmx{ccc}-32&34&-24\\-32&38&-8\\-16&21&4\emx$ 5 | 6 | $\ttb\tta = \bmx{cc} 22&-14\\-4&-12\emx$ } -------------------------------------------------------------------------------- /exercises/02_02_ex_20.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} -5&2\\ -5&-2 2 | \\ -5&-4\end{array}\hskip -3pt \right]$ 3 | 4 | $\ttb = \left[\hskip -3pt \begin{array}{ccc} 0&-5&6\\ -5&-3&-1 5 | \end{array}\hskip -3pt \right] $} 6 | {$\tta\ttb = \left[\hskip -3pt \begin{array}{ccc} -10&19&-32\\ 10&31&-28 7 | \\ 20&37&-26\end{array}\hskip -3pt \right] $ 8 | 9 | $\ttb\tta = \left[\hskip -3pt \begin{array}{cc} -5&-14\\ 45&0\end{array}\hskip -3pt \right] $} 10 | 11 | -------------------------------------------------------------------------------- /exercises/02_02_ex_21.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 8&-2\\ 4&5 2 | \\ 2&-5\end{array}\hskip -3pt \right] $ 3 | 4 | $\ttb = \left[\hskip -3pt \begin{array}{ccc} -5&1&-5\\ 8&3&-2 5 | \end{array}\hskip -3pt \right] $} 6 | {$\tta\ttb = \left[\hskip -3pt \begin{array}{ccc} -56&2&-36\\ 20&19&-30 7 | \\ -50&-13&0\end{array}\hskip -3pt \right] $ 8 | 9 | $\ttb\tta = \left[\hskip -3pt \begin{array}{cc} -46&40\\ 72&9\end{array}\hskip -3pt \right] $} 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | -------------------------------------------------------------------------------- /exercises/02_02_ex_22.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 1&4\\ 7&6\end {array} \hskip -3pt 2 | \right] $ 3 | 4 | $\ttb = \left[\hskip -3pt \begin{array}{cccc} 1&-1&-5&5\\ -2&1&3&-5 5 | \end{array}\hskip -3pt \right] $} 6 | {$\tta\ttb = \left[\hskip -3pt \begin{array}{cccc} -7&3&7&-15\\ -5&-1&-17&5 7 | \end{array}\hskip -3pt \right] $ 8 | 9 | $\ttb\tta$ is not possible.} 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | -------------------------------------------------------------------------------- /exercises/02_02_ex_23.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} -1&5\\ 6&7\end {array} \hskip -3pt 2 | \right] $ 3 | 4 | $\ttb = \left[\hskip -3pt \begin{array}{cccc} 5&-3&-4&-4\\ -2&-5&-5&-1 5 | \end{array}\hskip -3pt \right] $} 6 | {$\tta\ttb = \left[\hskip -3pt \begin{array}{cccc} -15&-22&-21&-1\\ 16&-53& 7 | -59&-31\end{array}\hskip -3pt \right] $ 8 | 9 | $\ttb\tta$ is not possible.} 10 | 11 | 12 | 13 | 14 | 15 | -------------------------------------------------------------------------------- /exercises/02_02_ex_24.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} -1&2&1\\ -1&2&-1 2 | \\ 0&0&-2\end{array}\hskip -3pt \right] $ 3 | 4 | $\ttb = \left[\hskip -3pt \begin{array}{ccc} 0&0&-2\\ 1&2&-1 5 | \\ 1&0&0\end{array}\hskip -3pt \right] 6 | $} 7 | {$\tta\ttb = \left[\hskip -3pt \begin{array}{ccc} 3&4&0\\ 1&4&0 8 | \\ -2&0&0\end{array}\hskip -3pt \right] $ 9 | 10 | $\ttb\tta = \left[\hskip -3pt \begin{array}{ccc} 0&0&4\\ -3&6&1 11 | \\ -1&2&1\end{array}\hskip -3pt \right]$ } 12 | -------------------------------------------------------------------------------- /exercises/02_02_ex_25.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} -1&1&1\\ -1&-1&-2 2 | \\ 1&1&-2\end{array}\hskip -3pt \right] $ 3 | 4 | $\ttb = \left[\hskip -3pt \begin{array}{ccc} -2&-2&-2\\ 0&-2&0 5 | \\ -2&0&2\end{array}\hskip -3pt \right] 6 | $} 7 | {$\tta\ttb = \left[\hskip -3pt \begin{array}{ccc} 0&0&4\\ 6&4&-2 8 | \\ 2&-4&-6\end{array}\hskip -3pt \right] $ 9 | 10 | $\ttb\tta = \left[\hskip -3pt \begin{array}{ccc} 2&-2&6\\ 2&2&4 11 | \\ 4&0&-6\end{array}\hskip -3pt \right] $ } 12 | 13 | -------------------------------------------------------------------------------- /exercises/02_02_ex_26.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} -4&3&3\\ -5&-1&-5 2 | \\ -5&0&-1\end{array}\hskip -3pt \right] $ 3 | 4 | $\ttb = \left[\hskip -3pt \begin{array}{ccc} 0&5&0\\ -5&-4&3 5 | \\ 5&-4&3\end{array}\hskip -3pt \right] $} 6 | {$\tta\ttb = \left[\hskip -3pt \begin{array}{ccc} 0&-44&18\\ -20&-1&-18 7 | \\ -5&-21&-3\end{array}\hskip -3pt \right]$ 8 | 9 | $\ttb\tta = \left[\hskip -3pt \begin{array}{ccc} -25&-5&-25\\ 25&-11&2 10 | \\ -15&19&32\end{array}\hskip -3pt \right]$ } 11 | -------------------------------------------------------------------------------- /exercises/02_02_ex_27.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} -4&-1&3\\ 2&-3&5 2 | \\ 1&5&3\end{array}\hskip -3pt \right]$ 3 | 4 | $\ttb = \left[\hskip -3pt \begin{array}{ccc} -2&4&3\\ -1&1&-1 5 | \\ 4&0&2\end{array}\hskip -3pt \right]$} 6 | {$\tta\ttb = \left[\hskip -3pt \begin{array}{ccc} 21&-17&-5\\ 19&5&19 7 | \\ 5&9&4\end{array}\hskip -3pt \right]$ 8 | 9 | $\ttb\tta = \left[\hskip -3pt \begin{array}{ccc} 19&5&23\\ 5&-7&-1 10 | \\ -14&6&18\end{array}\hskip -3pt \right]$ } 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | -------------------------------------------------------------------------------- /exercises/02_02_ex_28.tex: -------------------------------------------------------------------------------- 1 | {$\ttd = \left[\hskip -3pt \begin{array}{ccc} -1&0&0\\ 0&2&0 2 | \\ 0&0&3\end{array}\hskip -3pt \right]$ 3 | 4 | $\tta = \left[\hskip -3pt \begin{array}{ccc} 1&2&3\\ 4&5&6 5 | \\ 7&8&9\end{array}\hskip -3pt \right] $} 6 | {$\ttd\tta= \left[\hskip -3pt \begin{array}{ccc} -1&4&9\\ -4&10&18 7 | \\ -7&16&27\end{array}\hskip -3pt \right]$\ 8 | $\tta\ttd = \left[\hskip -3pt \begin{array}{ccc} -1&-2&-3\\ 8&10&12 9 | \\ 21&24&27\end{array}\hskip -3pt \right] $ 10 | } 11 | 12 | 13 | 14 | -------------------------------------------------------------------------------- /exercises/02_02_ex_29.tex: -------------------------------------------------------------------------------- 1 | {$\ttd = \left[\hskip -3pt \begin{array}{ccc} 1&1&1\\ 2&2&2 2 | \\ -3&-3&-3\end{array}\hskip -3pt \right] $ 3 | 4 | $\tta = \left[\hskip -3pt \begin{array}{ccc} 2&0&0\\ 0&-3&0 5 | \\ 0&0&5\end{array}\hskip -3pt \right] $} 6 | {$\ttd\tta= \left[\hskip -3pt \begin{array}{ccc} 2&2&2\\ -6&-6&-6 7 | \\ -15&-15&-15\end{array}\hskip -3pt \right]$\ 8 | $\tta\ttd = \left[\hskip -3pt \begin{array}{ccc} 2&-3&5\\ 4&-6&10 9 | \\ -6&9&-15\end{array}\hskip -3pt \right] $ 10 | } 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | -------------------------------------------------------------------------------- /exercises/02_02_ex_30.tex: -------------------------------------------------------------------------------- 1 | {$\ttd = \left[\hskip -3pt \begin{array}{cc} 3&0\\ 0&-1\end {array} \hskip -3pt 2 | \right]$ 3 | 4 | $\tta = \left[\hskip -3pt \begin{array}{cc} 2&4\\ 6&8\end {array} \hskip -3pt 5 | \right] $} 6 | {$\ttd\tta= \left[\hskip -3pt \begin{array}{cc} 6&-4\\ 18&-8\end{array}\hskip -3pt \right] $ 7 | 8 | $\tta\ttd = \left[\hskip -3pt \begin{array}{cc} 6&12\\ -6&-8\end{array}\hskip -3pt \right]$ 9 | } 10 | -------------------------------------------------------------------------------- /exercises/02_02_ex_31.tex: -------------------------------------------------------------------------------- 1 | {$\ttd = \left[\hskip -3pt \begin{array}{cc} 4&0\\ 0&-3\end{array}\hskip -3pt 2 | \right]$ 3 | 4 | $\tta = \left[\hskip -3pt \begin{array}{cc} 1&2\\ 1&2\end{array}\hskip -3pt 5 | \right] $} 6 | {$\ttd\tta= \left[\hskip -3pt \begin{array}{cc} 4&-6\\ 4&-6\end{array}\hskip -3pt 7 | \right] $ 8 | 9 | $\tta\ttd = \left[\hskip -3pt \begin{array}{cc} 4&8\\ -3&-6\end{array}\hskip -3pt \right]$ 10 | } 11 | 12 | -------------------------------------------------------------------------------- /exercises/02_02_ex_32.tex: -------------------------------------------------------------------------------- 1 | {$\ttd = \left[\hskip -3pt \begin{array}{cc} d_1&0\\ 0&d_2\end{array}\hskip -3pt 2 | \right]$ 3 | 4 | $\tta = \left[\hskip -3pt \begin{array}{cc} a&b\\ c&d\end{array}\hskip -3pt 5 | \right] $} 6 | {$\ttd\tta= \left[\hskip -3pt \begin{array}{cc} d_1a&d_1b\\ d_2c&d_2d\end{array}\hskip -3pt 7 | \right] $ 8 | 9 | $\tta\ttd = \left[\hskip -3pt \begin{array}{cc} d_1a&d_2b\\ d_1c&d_2d\end{array}\hskip -3pt \right]$ 10 | } 11 | 12 | -------------------------------------------------------------------------------- /exercises/02_02_ex_33.tex: -------------------------------------------------------------------------------- 1 | {$\ttd = \left[\hskip -3pt \begin{array}{ccc} d_1&0&0\\ 0&d_2&0 2 | \\ 0&0&d_3\end{array}\hskip -3pt \right]$ 3 | 4 | $\tta = \left[\hskip -3pt \begin{array}{ccc} a&b&c\\ d&e&f 5 | \\ g&h&i\end{array}\hskip -3pt \right] $} 6 | {$\ttd\tta= \left[\hskip -3pt \begin{array}{ccc} d_1a&d_1b&d_1c\\ d_2d&d_2e&d_2f 7 | \\ d_3g&d_3h&d_3i\end{array}\hskip -3pt \right] $\ 8 | $\tta\ttd = \left[\hskip -3pt \begin{array}{ccc} d_1a&d_2b&d_3c\\ d_1d&d_2e&d_3f 9 | \\ d_1g&d_2h&d_3i\end{array}\hskip -3pt \right]$ 10 | } 11 | 12 | -------------------------------------------------------------------------------- /exercises/02_02_ex_34.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 2&3\\ 1&-1\end {array} \hskip -3pt 2 | \right]$,\quad 3 | $\vx = \left[\hskip -3pt \begin{array}{c} 4\\ 9\end{array}\hskip -3pt \right] $} 4 | {$\tta\vx= \left[\hskip -3pt \begin{array}{c} 35\\ -5\end {array} \hskip -3pt 5 | \right]$ 6 | } 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | -------------------------------------------------------------------------------- /exercises/02_02_ex_35.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} -1&4\\ 7&3\end {array} \hskip -3pt 2 | \right] $,\quad 3 | $\vx = \left[\hskip -3pt \begin{array}{c} 2\\ -1\end{array}\hskip -3pt \right]$} 4 | {$\tta\vx= \left[\hskip -3pt \begin{array}{c} -6\\ 11\end {array} \hskip -3pt 5 | \right]$ 6 | } 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | -------------------------------------------------------------------------------- /exercises/02_02_ex_36.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} 2&0&3\\ 1&1&1 2 | \\ 3&-1&2\end{array}\hskip -3pt \right] $,\quad 3 | $\vx = \left[\hskip -3pt \begin{array}{c} 1\\ 4\\ 2 4 | \end{array}\hskip -3pt \right]$} 5 | {$\tta\vx= \left[\hskip -3pt \begin{array}{c} 8\\ 7\\ 3 6 | \end{array}\hskip -3pt \right]$ 7 | } 8 | 9 | -------------------------------------------------------------------------------- /exercises/02_02_ex_37.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} -2&0&3\\ 1&1&-2 2 | \\ 4&2&-1\end{array}\hskip -3pt \right]$,\quad 3 | $\vx = \left[\hskip -3pt \begin{array}{c} 4\\ 3\\ 4 | 1\end{array}\hskip -3pt \right] $} 5 | {$\tta\vx= \left[\hskip -3pt \begin{array}{c} -5\\ 5\\ 6 | 21\end{array}\hskip -3pt \right] $ 7 | } 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | -------------------------------------------------------------------------------- /exercises/02_02_ex_38.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} 1&2&3\\ 1&0&2 2 | \\ 2&3&1\end{array}\hskip -3pt \right]$,\quad 3 | $\vx = \left[\hskip -3pt \begin{array}{c} x_1\\ x_2\\ 4 | x_3\end{array}\hskip -3pt \right] $} 5 | {$\tta\vx= \left[\hskip -3pt \begin{array}{c} x_1+2x_2+3x_3\\ x_1+2x_3\\ 2x_1+3x_2+x_3\end{array}\hskip -3pt \right] $ 6 | } 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | -------------------------------------------------------------------------------- /exercises/02_02_ex_39.tex: -------------------------------------------------------------------------------- 1 | {Let $\tta = \bmx{cc} 0&1\\1&0\emx$. Find $\tta^2$ and $\tta^3$.} 2 | {$\tta^2 = \bmx{cc} 1&0\\0&1\emx$; $\tta^3 = \bmx{cc} 0&1\\1&0\emx$} 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | -------------------------------------------------------------------------------- /exercises/02_02_ex_40.tex: -------------------------------------------------------------------------------- 1 | {Let $\tta = \bmx{cc} 2&0\\0&3\emx$. Find $\tta^2$ and $\tta^3$.} 2 | {$\tta^2 = \bmx{cc} 4&0\\0&9\emx$; $\tta^3 = \bmx{cc} 8&0\\0&27\emx$} 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | -------------------------------------------------------------------------------- /exercises/02_02_ex_41.tex: -------------------------------------------------------------------------------- 1 | {Let $\tta = \bmx{ccc} -1&0&0\\0&3&0\\0&0&5\emx$. Find $\tta^2$ and $\tta^3$.} 2 | {$\tta^2 = \bmx{ccc} 1&0&0\\0&9&0\\0&0&25\emx$; $\tta^3 = \bmx{ccc} -1&0&0\\0&27&0\\0&0&125\emx$} 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | -------------------------------------------------------------------------------- /exercises/02_02_ex_42.tex: -------------------------------------------------------------------------------- 1 | {Let $\tta = \bmx{ccc} 0&1&0\\0&0&1\\1&0&0\emx$. Find $\tta^2$ and $\tta^3$.} 2 | {$\tta^2 = \bmx{ccc} 0&0&1\\1&0&0\\0&1&0\emx$; $\tta^3 = \bmx{ccc} 1&0&0\\0&1&0\\0&0&1\emx$} 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | -------------------------------------------------------------------------------- /exercises/02_02_ex_43.tex: -------------------------------------------------------------------------------- 1 | {Let $\tta = \bmx{ccc} 0&0&1\\0&0&0\\0&1&0\emx$. Find $\tta^2$ and $\tta^3$.} 2 | {$\tta^2 = \bmx{ccc} 0&1&0\\0&0&0\\0&0&0\emx$; $\tta^3 = \bmx{ccc} 0&0&0\\0&0&0\\0&0&0\emx$} 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | -------------------------------------------------------------------------------- /exercises/02_02_ex_45.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 2&-1\\ 4&3\end{array}\hskip -3pt \right]$ 2 | ,\quad $\vx = \left[\hskip -3pt \begin{array}{c} x_1\\ x_2\end{array}\hskip -3pt \right] $} 3 | {$\tta\vx= \left[\hskip -3pt \begin{array}{c}2x_1-x_2\\ 4x_1+3x_2\end{array}\hskip -3pt \right] $ 4 | } 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | -------------------------------------------------------------------------------- /exercises/02_02_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/02_02_exset_01} 2 | \exsetinput{exercises/02_02_exset_02} 3 | \exsetinput{exercises/02_02_exset_03} 4 | \exsetinput{exercises/02_02_exset_04} 5 | \exinput{exercises/02_02_ex_39} 6 | \exinput{exercises/02_02_ex_40} 7 | \exinput{exercises/02_02_ex_41} 8 | \exinput{exercises/02_02_ex_42} 9 | \exinput{exercises/02_02_ex_43} 10 | \exinput{exercises/02_02_ex_44} 11 | -------------------------------------------------------------------------------- /exercises/02_02_exset_01.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, row and column vectors \vu\ and \vv\ are defined. Find the product \vu\vv, where possible.} 3 | \exinput{exercises/02_02_ex_01} 4 | \exinput{exercises/02_02_ex_02} 5 | \exinput{exercises/02_02_ex_03} 6 | \exinput{exercises/02_02_ex_04} 7 | \exinput{exercises/02_02_ex_05} 8 | \exinput{exercises/02_02_ex_06} 9 | \exinput{exercises/02_02_ex_07} 10 | \exinput{exercises/02_02_ex_08} 11 | \exinput{exercises/02_02_ex_09} 12 | \exinput{exercises/02_02_ex_10} 13 | \exinput{exercises/02_02_ex_11} 14 | \exinput{exercises/02_02_ex_12} -------------------------------------------------------------------------------- /exercises/02_02_exset_03.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, a \textit{diagonal} matrix \ttd\ and a matrix \tta\ are given. Find the products \ttd\tta\ and \tta\ttd, where possible.} 3 | \exinput{exercises/02_02_ex_30} 4 | \exinput{exercises/02_02_ex_31} 5 | \exinput{exercises/02_02_ex_28} 6 | \exinput{exercises/02_02_ex_29} 7 | \exinput{exercises/02_02_ex_32} 8 | \exinput{exercises/02_02_ex_33} -------------------------------------------------------------------------------- /exercises/02_02_exset_04.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, a matrix \tta\ and a vector \vx\ are given. Find the product \tta\vx.} 3 | \exinput{exercises/02_02_ex_34} 4 | \exinput{exercises/02_02_ex_35} 5 | \exinput{exercises/02_02_ex_36} 6 | \exinput{exercises/02_02_ex_37} 7 | \exinput{exercises/02_02_ex_45} 8 | \exinput{exercises/02_02_ex_38} -------------------------------------------------------------------------------- /exercises/02_02_exset_05.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, a matrix \tta\ and a vector \vx\ are given. Find the product \tta\vx.} 3 | \exinput{exercises/02_02_ex_34} 4 | \exinput{exercises/02_02_ex_35} 5 | \exinput{exercises/02_02_ex_36} 6 | \exinput{exercises/02_02_ex_37} 7 | \exinput{exercises/02_02_ex_39} 8 | \exinput{exercises/02_02_ex_38} -------------------------------------------------------------------------------- /exercises/02_03_ex_01.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 0&2\\ -1&3\end {array} \hskip -3pt 2 | \right]$, $\vb=\left[\hskip -3pt \begin{array}{c} -2\\ -1\end {array} \hskip -3pt 3 | \right]$} 4 | {\begin{enumerate} 5 | \item $\vx=\bmx{c}0\\0\emx$ 6 | \item $\vx=\left[\hskip -3pt \begin{array}{c} -2\\ -1\end {array} \hskip -3pt 7 | \right]$ 8 | \end{enumerate} 9 | } 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | -------------------------------------------------------------------------------- /exercises/02_03_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} -4&-1\\ -3&-2\end {array} \hskip -3pt 2 | \right]$, $\vb=\left[\hskip -3pt \begin{array}{c} 1\\ 4\end{array}\hskip -3pt \right]$} 3 | {\begin{enumerate} 4 | \item $\vx=\bmx{c}0\\0\emx$ 5 | \item $\vx=\left[\hskip -3pt \begin{array}{c} 2/5\\ -13/5 6 | \end{array}\hskip -3pt \right]$ 7 | \end{enumerate} 8 | } 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | -------------------------------------------------------------------------------- /exercises/02_03_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 1&-2\\ 0&1\end {array} \hskip -3pt 2 | \right]$, $\vb=\left[\hskip -3pt \begin{array}{c} 0\\ -5\end{array}\hskip -3pt \right]$} 3 | {\begin{enumerate} 4 | \item $\vx=\bmx{c}0\\0\emx$ 5 | \item $\vx=\left[\hskip -3pt \begin{array}{c} -10\\ -5\end {array} \hskip -3pt 6 | \right]$ 7 | \end{enumerate} 8 | } 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | -------------------------------------------------------------------------------- /exercises/02_03_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 1&0\\ 5&-4\end {array} \hskip -3pt 2 | \right]$, $\vb=\left[\hskip -3pt \begin{array}{c} -2\\ -1\end {array} \hskip -3pt 3 | \right]$} 4 | {\begin{enumerate} 5 | \item $\vx=\bmx{c}0\\0\emx$ 6 | \item $\vx=\left[\hskip -3pt \begin{array}{c} -2\\ -9/4\end {array} \hskip -3pt 7 | \right]$ 8 | \end{enumerate} 9 | } 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | -------------------------------------------------------------------------------- /exercises/02_03_ex_05.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 2&-3\\ -4&6\end {array} \hskip -3pt 2 | \right]$, $\vb=\left[\hskip -3pt \begin{array}{c} 1\\ -1\end {array} \hskip -3pt 3 | \right]$} 4 | {\begin{enumerate} 5 | \item $\vx=x_2\bmx{c}3/2\\1\emx$ 6 | \item No solution. 7 | \end{enumerate} 8 | } 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | -------------------------------------------------------------------------------- /exercises/02_03_ex_06.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} -4&3&2\\ -4&5&0 2 | \end{array}\hskip -3pt \right]$, $\vb=\left[\hskip -3pt \begin{array}{c} -4\\ -4\end{array}\hskip -3pt \right]$} 3 | {\begin{enumerate} 4 | \item $\vx=x_3\left[\hskip -3pt \begin{array}{c} 5/4\\ 1\\ 1\end{array}\hskip -3pt 5 | \right]$ 6 | \item $\vx=\left[\hskip -3pt \begin{array}{c} 1\\ 0\\ 0\end{array}\hskip -3pt \right]+x_3\left[\hskip -3pt \begin{array}{c} 5/4\\ 1\\ 1\end{array}\hskip -3pt 7 | \right]$ 8 | \end{enumerate} 9 | } 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | -------------------------------------------------------------------------------- /exercises/02_03_ex_07.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} 1&5&-2\\ 1&4&5 2 | \end{array}\hskip -3pt \right]$, $\vb=\left[\hskip -3pt \begin{array}{c} 0\\ 1\end{array}\hskip -3pt \right]$} 3 | {\begin{enumerate} 4 | \item $\vx=x_3\left[\hskip -3pt \begin{array}{c} -33\\ 7\\ 1\end {array} \hskip -3pt 5 | \right]$ 6 | \item $\vx=\left[\hskip -3pt \begin{array}{c} 5\\ -1\\ 0\end{array}\hskip -3pt \right]+x_3\left[\hskip -3pt \begin{array}{c} -33\\ 7\\ 1\end {array} \hskip -3pt 7 | \right]$ 8 | \end{enumerate} 9 | } 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | -------------------------------------------------------------------------------- /exercises/02_03_ex_08.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} -1&-2&-2\\ 3&4&-2 2 | \end{array}\hskip -3pt \right] $, $\vb=\left[\hskip -3pt \begin{array}{c} -4\\ -4\end{array}\hskip -3pt \right]$} 3 | {\begin{enumerate} 4 | \item $\vx=x_3\left[\hskip -3pt \begin{array}{c} 14\\ -10\\ 0\end{array}\hskip -3pt \right]$ 5 | \item $\vx=\left[\hskip -3pt \begin{array}{c} -4\\ 2\end{array}\hskip -3pt 6 | \right]+x_3\left[\hskip -3pt \begin{array}{c} 14\\ -10\\ 0\end{array}\hskip -3pt \right]$ 7 | \end{enumerate} 8 | } 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | -------------------------------------------------------------------------------- /exercises/02_03_ex_09.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} 2&2&2\\ 5&5&-3 2 | \end{array}\hskip -3pt \right] $, $\vb=\left[\hskip -3pt \begin{array}{c} 3\\ -3\end{array}\hskip -3pt \right] $} 3 | {\begin{enumerate} 4 | \item $\vx=x_2\left[\hskip -3pt \begin{array}{c} -1\\ 1\\ 0\end{array}\hskip -3pt \right]$ 5 | \item $\vx=\left[\hskip -3pt \begin{array}{c} 3/16\\ 0\\ {21/16}\end {array} \hskip -3pt 6 | \right]+x_2\left[\hskip -3pt \begin{array}{c} -1\\ 1\\ 0\end{array}\hskip -3pt \right]$ 7 | \end{enumerate}} 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | -------------------------------------------------------------------------------- /exercises/02_03_ex_10.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cccc} 1&5&-4&-1\\ 1&0&-2&1 2 | \end{array}\hskip -3pt \right] $, 3 | 4 | $\vb=\left[\hskip -3pt \begin{array}{c} 0\\ -2\end{array}\hskip -3pt \right] $} 5 | {\begin{enumerate} 6 | \item $\vx=x_3\bmx{c}2\\2/5\\1\\0\emx+x_4\bmx{c}-1\\2/5\\0\\1\emx$ 7 | 8 | \item $\vx=\bmx{c}-2\\2/5\\0\\0\emx + x_3\bmx{c}2\\2/5\\1\\0\emx +x_4\bmx{c}-1\\2/5\\0\\1\emx$ 9 | \end{enumerate}} 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | -------------------------------------------------------------------------------- /exercises/02_03_ex_11.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cccc} -4&2&-5&4\\ 0&1&-1&5 2 | \end{array}\hskip -3pt \right] $, 3 | 4 | $\vb=\left[\hskip -3pt \begin{array}{c} -3\\ -2\end {array}\hskip -3pt \right] $} 5 | {\begin{enumerate} 6 | \item $\vx=x_3\bmx{c}-3/4\\1\\1\\0\emx+x_4\bmx{c}-3/2\\-5\\0\\1\emx$ 7 | 8 | \item $\vx=\bmx{c}-1/4\\-2\\0\\0\emx + x_3\bmx{c}-3/4\\1\\1\\0\emx+x_4\bmx{c}-3/2\\-5\\0\\1\emx$ 9 | \end{enumerate}} 10 | 11 | -------------------------------------------------------------------------------- /exercises/02_03_ex_12.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccccc} 0&0&2&1&4\\ -2&-1&-4&-1 2 | &5\end{array}\hskip -3pt \right]$, 3 | 4 | $\vb=\left[\hskip -3pt \begin{array}{c} 3\\ 4\end{array}\hskip -3pt \right]$} 5 | {\begin{enumerate} 6 | \item $\vx=x_2\bmx{c}-1/2\\1\\0\\0\\0\emx+x_4\bmx{c}1/2\\0\\-1/2\\1\\0\emx+x_5\bmx{c}13/2\\0\\-2\\0\\1\emx$ 7 | 8 | \item $\vx=\bmx{c}-5\\0\\ 3/2 \\0\\0\emx + x_2\bmx{c}-1/2\\1\\0\\0\\0\emx+x_4\bmx{c}1/2 \\0\\-1/2\\1\\0\emx+x_5\bmx{c}13/2\\0\\-2\\0\\1\emx$ 9 | \end{enumerate}} 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | -------------------------------------------------------------------------------- /exercises/02_03_ex_13.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccccc} 3&0&-2&-4&5\\ 2&3&2&0&2 2 | \\ -5&0&4&0&5\end{array}\hskip -3pt \right] $, 3 | 4 | $\vb=\left[\hskip -3pt \begin{array}{c} -1\\ -5\\ 5 | 4\end{array}\hskip -3pt \right]$} 6 | {\begin{enumerate} 7 | \item $\vx=x_4\bmx{c}8\\-12\\10\\1\\0\emx+x_5\bmx{c} -15\\ 68/3 \\-20\\0\\1 \emx$ 8 | 9 | \item $\vx=\bmx{c} 2\\ -16/3\\ 7/2 \\ 0\\ 0 \emx + x_4\bmx{c}8\\-12\\10\\1\\0\emx+x_5\bmx{c} -15\\ 68/3 \\-20\\0\\1 \emx$ 10 | \end{enumerate}} 11 | 12 | 13 | -------------------------------------------------------------------------------- /exercises/02_03_ex_14.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccccc} -1&3&1&-3&4\\ 3&-3&-1&1 2 | &-4\\ -2&3&-2&-3&1\end{array}\hskip -3pt \right] $, 3 | 4 | $\vb=\left[\hskip -3pt \begin{array}{c} 1\\ 1\\ -5 5 | \end{array}\hskip -3pt \right] $} 6 | {\begin{enumerate} 7 | \item $\vx=x_4\bmx{c}1\\ 13/9 \\-1/3 \\1\\0\emx+x_5\bmx{c} 0\\ -1 \\-1\\0\\1 \emx$ 8 | 9 | \item $\vx=\bmx{c} 1\\ 1/9 \\ 5/3 \\ 0\\ 0 \emx + x_4\bmx{c}1\\ 13/9 \\ -1/3 \\1\\0\emx+x_5\bmx{c} 0\\ -1 \\-1\\0\\1 \emx$ 10 | \end{enumerate}} 11 | 12 | 13 | 14 | 15 | -------------------------------------------------------------------------------- /exercises/02_03_ex_15.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccccc} -4&-2&-1&4&0\\ 5&-4&3&-1 2 | &1\\ 4&-5&3&1&-4\end{array}\hskip -3pt \right] $, 3 | 4 | $\vb=\left[\hskip -3pt \begin{array}{c} 3\\ 2\\ 1 5 | \end{array}\hskip -3pt \right] $} 6 | {\begin{enumerate} 7 | \item $\vx=x_4\bmx{c}3\\ -1 \\-6 \\1\\0\emx + x_5\bmx{c} -17\\ 12 \\ 44 \\0\\1 \emx$ 8 | 9 | \item $\vx=\bmx{c} 7\\ -6 \\ -19 \\ 0\\ 0 \emx + x_4\bmx{c}3\\ -1 \\-6 \\1\\0\emx + x_5\bmx{c} -17\\ 12 \\ 44 \\0\\1 \emx$ 10 | \end{enumerate}} 11 | 12 | 13 | 14 | 15 | -------------------------------------------------------------------------------- /exercises/02_03_ex_16.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 1&-2\\ -3&6 \end{array}\hskip -3pt \right] $, 2 | 3 | $\vb=\left[\hskip -3pt \begin{array}{c} 0\\ 0 \end{array}\hskip -3pt \right] $, 4 | $\vu = \left[\hskip -3pt \begin{array}{c} 2\\ 1 \end{array}\hskip -3pt \right]$, $\vv = \left[\hskip -3pt \begin{array}{c} -10\\ -5 \end{array}\hskip -3pt \right]$ } 5 | {Multiply $\tta\vu$ and $\tta\vv$ to verify.} 6 | 7 | 8 | 9 | 10 | -------------------------------------------------------------------------------- /exercises/02_03_ex_17.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 1&0\\ 2&0 \end{array}\hskip -3pt \right] $, 2 | 3 | $\vb=\left[\hskip -3pt \begin{array}{c} 0\\ 0 \end{array}\hskip -3pt \right] $, 4 | $\vu = \left[\hskip -3pt \begin{array}{c} 0\\ -1 \end{array}\hskip -3pt \right]$, $\vv = \left[\hskip -3pt \begin{array}{c} 0\\ 59 \end{array}\hskip -3pt \right] $} 5 | {Multiply $\tta\vu$ and $\tta\vv$ to verify.} 6 | 7 | 8 | 9 | 10 | -------------------------------------------------------------------------------- /exercises/02_03_ex_18.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 1&0\\ 2&0 \end{array}\hskip -3pt \right] $, 2 | 3 | $\vb=\left[\hskip -3pt \begin{array}{c} -3\\ -6 \end{array}\hskip -3pt \right] $, 4 | $\vu = \left[\hskip -3pt \begin{array}{c} -3\\ -1 \end{array}\hskip -3pt \right]$, $\vv = \left[\hskip -3pt \begin{array}{c} -3\\ 59 \end{array}\hskip -3pt \right] $} 5 | {Multiply $\tta\vu$ and $\tta\vv$ to verify.} 6 | 7 | 8 | 9 | 10 | -------------------------------------------------------------------------------- /exercises/02_03_ex_19.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 1&-2\\ -3&6 \end{array}\hskip -3pt \right] $, 2 | 3 | $\vb=\left[\hskip -3pt \begin{array}{c} 2\\ -6 \end{array}\hskip -3pt \right] $, 4 | $\vu = \left[\hskip -3pt \begin{array}{c} 0\\ -1 \end{array}\hskip -3pt \right]$, $\vv = \left[\hskip -3pt \begin{array}{c} 2\\ 0 \end{array}\hskip -3pt \right]$ } 5 | {Multiply $\tta\vu$ and $\tta\vv$ to verify.} 6 | 7 | 8 | 9 | 10 | -------------------------------------------------------------------------------- /exercises/02_03_ex_20.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cccc} 0 & -3 & -1 & -3 \\ -4 & 2 & -3 & 5 \end{array}\hskip -3pt \right] $, 2 | 3 | $\vb=\left[\hskip -3pt \begin{array}{c} 0\\ 0 \end{array}\hskip -3pt \right] $, 4 | $\vu = \left[\hskip -3pt \begin{array}{c} 11 \\ 4 \\ -12 \\ 0 \end{array}\hskip -3pt \right]$, 5 | 6 | $\vv = \left[\hskip -3pt \begin{array}{c} 9\\ -12\\ 0 \\ 12 \end{array}\hskip -3pt \right] $} 7 | {Multiply $\tta\vu$ and $\tta\vv$ to verify.} 8 | 9 | 10 | 11 | 12 | -------------------------------------------------------------------------------- /exercises/02_03_ex_21.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cccc} 0 & -3 & -1 & -3 \\ -4 & 2 & -3 & 5 \end{array}\hskip -3pt \right] $, 2 | 3 | $\vb=\left[\hskip -3pt \begin{array}{c} 48\\ 36 \end{array}\hskip -3pt \right] $, 4 | $\vu = \left[\hskip -3pt \begin{array}{c} -17 \\ -16 \\ 0 \\ 0 \end{array}\hskip -3pt \right]$, 5 | 6 | $\vv = \left[\hskip -3pt \begin{array}{c} -8\\ -28\\ 0 \\ 12 \end{array}\hskip -3pt \right] $} 7 | {Multiply $\tta\vu$ and $\tta\vv$ to verify.} 8 | 9 | 10 | 11 | 12 | -------------------------------------------------------------------------------- /exercises/02_03_ex_22.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} 2&-2&-1 \\ -1&1&-1\\ -2&2&-1 \end{array}\hskip -3pt \right] $, 2 | 3 | $\vb=\left[\hskip -3pt \begin{array}{c} 1\\ 1 \\ 1 \end{array}\hskip -3pt \right] $, 4 | $\vu = \left[\hskip -3pt \begin{array}{c} 1 \\ 1 \\ 0 \end{array}\hskip -3pt \right]$, 5 | $\vv = \left[\hskip -3pt \begin{array}{c} 1\\ 1 \\ -1 \end{array}\hskip -3pt \right] $} 6 | {Multiply $\tta\vu$, $\tta\vv$ and $\tta(\vu+\vv)$ to verify.} 7 | -------------------------------------------------------------------------------- /exercises/02_03_ex_23.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} 1 & -1 & 3 \\ 3&-3&-3\\ -1&1&1 \end{array}\hskip -3pt \right] $, 2 | 3 | $\vb=\left[\hskip -3pt \begin{array}{c} -1\\ -3 \\ 1 \end{array}\hskip -3pt \right] $, 4 | $\vu = \left[\hskip -3pt \begin{array}{c} 2 \\ 2 \\ 0 \end{array}\hskip -3pt \right]$, 5 | $\vv = \left[\hskip -3pt \begin{array}{c} 2\\ 3 \\ 0 \end{array}\hskip -3pt \right] $} 6 | {Multiply $\tta\vu$, $\tta\vv$ and $\tta(\vu+\vv)$ to verify.} 7 | -------------------------------------------------------------------------------- /exercises/02_03_ex_24.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{ccc} 2 & 0 & 0 \\ 0&1&-3\\ 3&1&-3 \emx $, 2 | 3 | $\vb=\left[\hskip -3pt \begin{array}{c} 2\\ -4 \\ -1 \end{array}\hskip -3pt \right] $, 4 | $\vu = \left[\hskip -3pt \begin{array}{c} 0 \\ 6 \\ 2 \end{array}\hskip -3pt \right]$, 5 | $\vv = \left[\hskip -3pt \begin{array}{c} 1\\ -1 \\ 1 \end{array}\hskip -3pt \right] $} 6 | {Multiply $\tta\vu$, $\tta\vv$ and $\tta(\vu+\vv)$ to verify.} 7 | -------------------------------------------------------------------------------- /exercises/02_03_ex_25.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 2&4\\ -1&-2\end {array} \hskip -3pt 2 | \right]$, $\vb=\left[\hskip -3pt \begin{array}{c} 0\\ 0\end {array} \hskip -3pt 3 | \right]$} 4 | {$\vx = x_2\bmx{c} -2\\1\emx = x_2\vv$ 5 | 6 | \begin{tikzpicture}[x={(.4cm,0)},y={(0,.4cm)}, >=latex] 7 | \drawxlines{-4.5}{4.5}{-4,...,4}; 8 | \drawylines{-2.5}{2.5}{-2,...,2}; 9 | \draw [<->](-4,2)--(4,-2); 10 | \draw [->,thick] (0,0)--(-2,1) node [above] {\vv}; 11 | \end{tikzpicture} 12 | } 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | -------------------------------------------------------------------------------- /exercises/02_03_ex_26.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 2&-5\\ -4&-10\end {array} \hskip -3pt 2 | \right]$, $\vb=\left[\hskip -3pt \begin{array}{c} 1\\ 2\end {array} \hskip -3pt 3 | \right]$} 4 | {$\vx = \bmx{c} 0.5\\0\emx + x_2\bmx{c} 2.5\\1\emx = \vect{x_p} + x_2\vv$ 5 | 6 | \begin{tikzpicture}[x={(.4cm,0)},y={(0,.4cm)}, >=latex] 7 | \drawxlines{-4.5}{4.5}{-4,...,4}; 8 | \drawylines{-2.5}{2.5}{-2,...,2}; 9 | \draw [<->](-4,-1.8)--(4,1.4); 10 | \draw [->,thick] (0,0)--(0.5,0) node [below] {\vect{x_p}}; 11 | \draw [->,thick] (0.5,0)--(3,1) node [above] {\vv}; 12 | \end{tikzpicture} } 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | -------------------------------------------------------------------------------- /exercises/02_03_ex_27.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 2&4\\ -1&-2\end {array} \hskip -3pt 2 | \right]$, $\vb=\left[\hskip -3pt \begin{array}{c} -6\\ 3\end {array} \hskip -3pt 3 | \right]$} 4 | {$\vx = \bmx{c} -3\\0\emx + x_2\bmx{c} -2\\1\emx = \vect{x_p} + x_2\vv$ 5 | 6 | \begin{tikzpicture}[x={(.4cm,0)},y={(0,.4cm)}, >=latex] 7 | \drawxlines{-4.5}{4.5}{-4,...,4}; 8 | \drawylines{-4.5}{2.5}{-4,...,2}; 9 | \draw [<->](-4,.5)--(4,-3.5); 10 | \draw [->,thick] (0,0)--(-3,0) node [below] {\vect{x_p}}; 11 | \draw [->,thick] (-3,0)--(-5,1) node [above] {\vv}; 12 | \end{tikzpicture} 13 | } 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | -------------------------------------------------------------------------------- /exercises/02_03_ex_28.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 2&-5\\ -4&-10\end {array} \hskip -3pt 2 | \right]$, $\vb=\left[\hskip -3pt \begin{array}{c} 0\\ 0\end {array} \hskip -3pt 3 | \right]$} 4 | {$\vx = x_2\bmx{c} 2.5\\1\emx = x_2\vv$ 5 | 6 | \begin{tikzpicture}[x={(.4cm,0)},y={(0,.4cm)}, >=latex] 7 | \drawxlines{-4.5}{4.5}{-4,...,4}; 8 | \drawylines{-2.5}{2.5}{-2,...,2}; 9 | \draw [<->](-4,-1.6)--(4,1.6); 10 | \draw [->,thick] (0,0)--(2.5,1) node [above] {\vv}; 11 | \end{tikzpicture} } 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | -------------------------------------------------------------------------------- /exercises/02_03_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/02_03_exset_02} 2 | \exsetinput{exercises/02_03_exset_03} 3 | \exsetinput{exercises/02_03_exset_01} 4 | \exsetinput{exercises/02_03_exset_04} 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/02_03_exset_02.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, a matrix \tta\ and vectors \vb, \vu\ and \vv\ are given. Verify that \vu\ and \vv\ are both solutions to the equation \ttaxb; that is, show that $\tta\vu=\tta\vv=\vb$.} 3 | \exinput{exercises/02_03_ex_16} 4 | \exinput{exercises/02_03_ex_19} 5 | \exinput{exercises/02_03_ex_17} 6 | \exinput{exercises/02_03_ex_18} 7 | \exinput{exercises/02_03_ex_20} 8 | \exinput{exercises/02_03_ex_21} -------------------------------------------------------------------------------- /exercises/02_03_exset_03.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, a matrix \tta\ and vectors \vb, \vu\ and \vv\ are given. Verify that $\tta\vu=\zero$, $\tta\vv=\vb$ and $\tta(\vu+\vv)=\vb$.} 3 | \exinput{exercises/02_03_ex_22} 4 | \exinput{exercises/02_03_ex_23} 5 | \exinput{exercises/02_03_ex_24} -------------------------------------------------------------------------------- /exercises/02_03_exset_04.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, a matrix \tta\ and vector \vb\ are given. 3 | Solve the equation \ttaxb, write the solution in vector format, and sketch the solution as the appropriate line on the Cartesian plane.} 4 | \exinput{exercises/02_03_ex_25} 5 | \exinput{exercises/02_03_ex_27} 6 | \exinput{exercises/02_03_ex_26} 7 | \exinput{exercises/02_03_ex_28} -------------------------------------------------------------------------------- /exercises/02_04_ex_01.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 4&-1 \\ -7&5 \end{array}\hskip -3pt \right] $, 2 | 3 | $\ttb=\left[\hskip -3pt \begin{array}{cc} 8 & -31\\ -27 & 38 \end{array}\hskip -3pt \right] $} 4 | {$\ttx=\left[\hskip -3pt \begin{array}{cc} 1 & -9\\ -4 & -5 \end{array}\hskip -3pt \right] $} 5 | -------------------------------------------------------------------------------- /exercises/02_04_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 1&-3 \\ -3&6 \end{array}\hskip -3pt \right] $, 2 | 3 | $\ttb=\left[\hskip -3pt \begin{array}{cc} 12&-10\\ -27 & 27 \end{array}\hskip -3pt \right] $} 4 | {$\ttx=\left[\hskip -3pt \begin{array}{cc} 3&-7\\ -3&1 \end{array}\hskip -3pt \right] $} 5 | -------------------------------------------------------------------------------- /exercises/02_04_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 3&3 \\ 6&4 \end{array}\hskip -3pt \right] $, 2 | 3 | $\ttb=\left[\hskip -3pt \begin{array}{cc} 15&-39\\ 16&-66 \end{array}\hskip -3pt \right] $} 4 | {$\ttx=\left[\hskip -3pt \begin{array}{cc} -2&-7\\ 7&-6 \end{array}\hskip -3pt \right] $} 5 | -------------------------------------------------------------------------------- /exercises/02_04_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} -3&-6 \\ 4&0 \end{array}\hskip -3pt \right] $, 2 | 3 | $\ttb=\left[\hskip -3pt \begin{array}{cc} 48&-30\\ 0&-8 \end{array}\hskip -3pt \right] $} 4 | {$\ttx=\left[\hskip -3pt \begin{array}{cc} 0&-2\\ -8&6 \end{array}\hskip -3pt \right] $} 5 | -------------------------------------------------------------------------------- /exercises/02_04_ex_05.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} -1&-2 \\ -2&-3 \end{array}\hskip -3pt \right] $, 2 | 3 | $\ttb=\left[\hskip -3pt \begin{array}{ccc} 13&4&7\\ 22&5&12 \end{array}\hskip -3pt \right] $} 4 | {$\ttx=\left[\hskip -3pt \begin{array}{ccc} -5&2&-3\\ -4&-3&-2 \end{array}\hskip -3pt \right] $} 5 | -------------------------------------------------------------------------------- /exercises/02_04_ex_06.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} -4&1 \\ -1&-2 \end{array}\hskip -3pt \right] $, 2 | 3 | $\ttb=\left[\hskip -3pt \begin{array}{ccc} -2&-10&19\\ 13&2&-2 \end{array}\hskip -3pt \right] $} 4 | {$\ttx=\left[\hskip -3pt \begin{array}{ccc} -1&2&-4\\ -6&-2&3 \end{array}\hskip -3pt \right] $} 5 | -------------------------------------------------------------------------------- /exercises/02_04_ex_07.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} -5&-4&-1 \\ 8&-2&-3 \\ 6&1&-8\end{array}\hskip -3pt \right] $, 2 | 3 | $\ttb=\left[\hskip -3pt \begin{array}{ccc} -21&-8&-19\\ 65&-11&-10 \\ 75 & -51&33 \end{array}\hskip -3pt \right] $} 4 | {$\ttx=\left[\hskip -3pt \begin{array}{ccc} 6&1&-1\\ -1&-1&7 \\ -5&7&-4 \end{array}\hskip -3pt \right] $} 5 | -------------------------------------------------------------------------------- /exercises/02_04_ex_08.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} -2&0&4 \\ -5&-4&5 \\ -3&5&-3\end{array}\hskip -3pt \right] $, 2 | 3 | $\ttb=\left[\hskip -3pt \begin{array}{ccc} -18&2&-14\\ -38&18&-13 \\ 10&2&-18 \end{array}\hskip -3pt \right] $} 4 | {$\ttx=\left[\hskip -3pt \begin{array}{ccc} 3&-3&3\\ 2&-2&-3 \\ -3&-1&-2 \end{array}\hskip -3pt \right] $} 5 | -------------------------------------------------------------------------------- /exercises/02_04_ex_09.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} 0&-2&1 \\ 0&2&2 \\ 1&2&-3\end{array}\hskip -3pt \right] $, 2 | \quad 3 | $\ttb=\tti_3 $} 4 | {$\ttx=\left[\hskip -3pt \begin{array}{ccc} 5/3&2/3&1\\ -1/3 &1/6&0\\ 1/3&1/3&0 \end{array}\hskip -3pt \right] $} 5 | -------------------------------------------------------------------------------- /exercises/02_04_ex_10.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} -3&3&-2 \\ 1&-3&2 \\ -1&-1&2\end{array}\hskip -3pt \right] $, 2 | \quad 3 | $\ttb=\tti_3 $} 4 | {$\ttx=\left[\hskip -3pt \begin{array}{ccc} -1/2 & -1/2 & 0\\ -1/2 & -1 & 1/2 \\ -1/2 & -3/4 & 3/4 \end{array}\hskip -3pt \right] $} 5 | -------------------------------------------------------------------------------- /exercises/02_04_ex_11.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 1&0 \\ 3&-1 \end{array}\hskip -3pt \right] $, 2 | \quad 3 | $\ttb=\tti_2 $} 4 | {$\ttx=\left[\hskip -3pt \begin{array}{cc} 1&0 \\ 3&-1 \end{array}\hskip -3pt \right] $} 5 | -------------------------------------------------------------------------------- /exercises/02_04_ex_12.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 2&2 \\ 3&1 \end{array}\hskip -3pt \right] $, \quad 2 | $\ttb=\tti_2 $} 3 | {$\ttx=\left[\hskip -3pt \begin{array}{cc} -1/4 & 1/2 \\ 3/4 & -1/2 \end{array}\hskip -3pt \right] $} 4 | -------------------------------------------------------------------------------- /exercises/02_04_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/02_04_exset_01} 2 | 3 | 4 | 5 | 6 | -------------------------------------------------------------------------------- /exercises/02_04_exset_01.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, matrices \tta\ and \ttb\ are given. Solve the matrix equation $\tta\ttx=\ttb$.} 3 | \exinput{exercises/02_04_ex_01} 4 | \exinput{exercises/02_04_ex_02} 5 | \exinput{exercises/02_04_ex_03} 6 | \exinput{exercises/02_04_ex_04} 7 | \exinput{exercises/02_04_ex_05} 8 | \exinput{exercises/02_04_ex_06} 9 | \exinput{exercises/02_04_ex_11} 10 | \exinput{exercises/02_04_ex_12} 11 | \exinput{exercises/02_04_ex_08} 12 | \exinput{exercises/02_04_ex_07} 13 | \exinput{exercises/02_04_ex_09} 14 | \exinput{exercises/02_04_ex_10} -------------------------------------------------------------------------------- /exercises/02_05_ex_01.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 1&5\\ -5&-24\end {array} \hskip -3pt 2 | \right]$ 3 | } 4 | {$\left[\hskip -3pt \begin{array}{cc} -24&-5\\ 5&1\end {array} \hskip -3pt 5 | \right]$ } 6 | -------------------------------------------------------------------------------- /exercises/02_05_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 1&-4\\ 1&-3\end {array} \hskip -3pt 2 | \right]$ 3 | } 4 | {$\left[\hskip -3pt \begin{array}{cc} -3&4\\ -1&1\end {array} \hskip -3pt 5 | \right]$ } 6 | -------------------------------------------------------------------------------- /exercises/02_05_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 3&0\\ 0&7\end {array} \hskip -3pt 2 | \right] $ 3 | } 4 | {$\left[\hskip -3pt \begin{array}{cc} 1/3&0\\ 0&1/7\end {array} \hskip -3pt 5 | \right]$ } 6 | -------------------------------------------------------------------------------- /exercises/02_05_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 2&5\\ 3&4\end {array} \hskip -3pt 2 | \right] $ 3 | } 4 | {$\left[\hskip -3pt \begin{array}{cc} -4/7&5/7\\ 3/7&-2/7\end {array} \hskip -3pt 5 | \right]$ } 6 | -------------------------------------------------------------------------------- /exercises/02_05_ex_05.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 1&-3\\ -2&6\end {array} \hskip -3pt 2 | \right] $ 3 | } 4 | {\ttai\ does not exist.} 5 | -------------------------------------------------------------------------------- /exercises/02_05_ex_06.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 3&7\\ 2&4\end {array} \hskip -3pt 2 | \right] $ 3 | } 4 | {$\left[\hskip -3pt \begin{array}{cc} -2&7/2\\ 1&-3/2\end {array} \hskip -3pt 5 | \right] $} 6 | -------------------------------------------------------------------------------- /exercises/02_05_ex_07.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 1&0\\ 0&1\end {array} \hskip -3pt 2 | \right] $ 3 | } 4 | {$\left[\hskip -3pt \begin{array}{cc} 1&0\\ 0&1\end {array} \hskip -3pt 5 | \right] $} 6 | -------------------------------------------------------------------------------- /exercises/02_05_ex_08.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 0&1\\ 1&0\end {array} \hskip -3pt 2 | \right] $ 3 | } 4 | {$\left[\hskip -3pt \begin{array}{cc} 0&1\\ 1&0\end {array} \hskip -3pt 5 | \right] $} 6 | -------------------------------------------------------------------------------- /exercises/02_05_ex_09.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} -2&3\\ 1&5\end {array} \hskip -3pt 2 | \right] $ 3 | } 4 | {$\left[\hskip -3pt \begin{array}{cc} -5/13&3/13\\ 1/13&2/13\end {array} \hskip -3pt 5 | \right] $} 6 | -------------------------------------------------------------------------------- /exercises/02_05_ex_10.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} -5&-2\\ 9&2\end {array} \hskip -3pt 2 | \right] $ 3 | } 4 | {$\left[\hskip -3pt \begin{array}{cc} 1/4&1/4\\ -9/8&-5/8\end {array} \hskip -3pt 5 | \right] $} 6 | -------------------------------------------------------------------------------- /exercises/02_05_ex_11.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 1&2\\ 3&4\end {array} \hskip -3pt 2 | \right] $ 3 | } 4 | {$\left[\hskip -3pt \begin{array}{cc} -2&1\\ 3/2&-1/2\end {array} \hskip -3pt 5 | \right] $} 6 | -------------------------------------------------------------------------------- /exercises/02_05_ex_12.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 5&7\\ 5/3& 7/3\end {array} \hskip -3pt 2 | \right] $ 3 | } 4 | {\ttai\ does not exist.} 5 | -------------------------------------------------------------------------------- /exercises/02_05_ex_13.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 25&-10&-4\\ -18&7&3 2 | \\ -6&2&1\end{array}\hskip -3pt \right]$ 3 | } 4 | {$\left[\hskip -3pt \begin{array}{ccc} 1&2&-2\\ 0&1&-3 5 | \\ 6&10&-5\end{array}\hskip -3pt \right] $} 6 | -------------------------------------------------------------------------------- /exercises/02_05_ex_14.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 1&0&0\\ 4&1&-7 2 | \\ 20&7&-48\end{array}\hskip -3pt \right] $ 3 | } 4 | {$ \left[\hskip -3pt \begin{array}{ccc} 1&0&0\\ 52&-48&7 5 | \\ 8&-7&1\end{array}\hskip -3pt \right]$} 6 | -------------------------------------------------------------------------------- /exercises/02_05_ex_15.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} -4&1&5\\ -5&1&9 2 | \\ -10&2&19\end{array}\hskip -3pt \right]$ 3 | } 4 | {$\left[\hskip -3pt \begin{array}{ccc} 1&-9&4\\ 5&-26&11 5 | \\ 0&-2&1\end{array}\hskip -3pt \right]$} 6 | -------------------------------------------------------------------------------- /exercises/02_05_ex_16.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 1&-5&0\\ -2&15&4 2 | \\ 4&-19&1\end{array}\hskip -3pt \right]$ 3 | } 4 | {$ \left[\hskip -3pt \begin{array}{ccc} 91&5&-20\\ 18&1&-4 5 | \\ -22&-1&5\end{array}\hskip -3pt \right]$} 6 | -------------------------------------------------------------------------------- /exercises/02_05_ex_17.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 25&-8&0\\ -78&25&0 2 | \\ 48&-15&1\end{array}\hskip -3pt \right]$ 3 | } 4 | {$ \left[\hskip -3pt \begin{array}{ccc} 25&8&0\\ 78&25&0 5 | \\ -30&-9&1\end{array}\hskip -3pt \right]$} 6 | -------------------------------------------------------------------------------- /exercises/02_05_ex_18.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 1&0&0\\ 7&5&8 2 | \\ -2&-2&-3\end{array}\hskip -3pt \right]$ 3 | } 4 | {$ \left[\hskip -3pt \begin{array}{ccc} 1&0&0\\ 5&-3&-8 5 | \\ -4&2&5\end{array}\hskip -3pt \right]$} 6 | -------------------------------------------------------------------------------- /exercises/02_05_ex_19.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 0&0&1\\ 1&0&0 2 | \\ 0&1&0\end{array}\hskip -3pt \right]$ 3 | } 4 | {$ \left[\hskip -3pt \begin{array}{ccc} 0&1&0\\ 0&0&1 5 | \\ 1&0&0\end{array}\hskip -3pt \right]$} 6 | -------------------------------------------------------------------------------- /exercises/02_05_ex_20.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 0&1&0\\ 1&0&0 2 | \\ 0&0&1\end{array}\hskip -3pt \right]$ 3 | } 4 | {$ \left[\hskip -3pt \begin{array}{ccc} 0&1&0\\ 1&0&0 5 | \\ 0&0&1\end{array}\hskip -3pt \right]$} 6 | -------------------------------------------------------------------------------- /exercises/02_05_ex_21.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 2&3&4\\ -3&6&9 2 | \\ -1&9&13\end{array}\hskip -3pt \right]$ 3 | } 4 | {\ttai\ does not exist.} 5 | -------------------------------------------------------------------------------- /exercises/02_05_ex_22.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 5&-1&0\\ 7&7&1 2 | \\ -2&-8&-1\end{array}\hskip -3pt \right]$ 3 | } 4 | {\ttai\ does not exist.} 5 | -------------------------------------------------------------------------------- /exercises/02_05_ex_23.tex: -------------------------------------------------------------------------------- 1 | {$ \left[\hskip -3pt \begin{array}{cccc} 1&0&0&0\\ -19&-9&0&4 2 | \\ 33&4&1&-7\\ 4&2&0&-1\end {array} \hskip -3pt 3 | \right] $ 4 | } 5 | {$ \left[\hskip -3pt \begin{array}{cccc} 1&0&0&0\\ -3&-1&0&-4 6 | \\ -35&-10&1&-47\\ -2&-2&0&-9 7 | \end{array}\hskip -3pt \right] $} 8 | -------------------------------------------------------------------------------- /exercises/02_05_ex_24.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} 1&0&0&0\\ 27&1&0&4 2 | \\ 18&0&1&4\\ 4&0&0&1\end {array} \hskip -3pt 3 | \right]$ 4 | } 5 | {$ \left[\hskip -3pt \begin{array}{cccc} 1&0&0&0\\ -11&1&0&-4 6 | \\ -2&0&1&-4\\ -4&0&0&1\end {array} \hskip -3pt 7 | \right] $} 8 | -------------------------------------------------------------------------------- /exercises/02_05_ex_25.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} -15&45&-3&4\\ 55&-164&15 2 | &-15\\ -215&640&-62&59\\ -4&12&0&1 3 | \end{array}\hskip -3pt \right] $ 4 | } 5 | {$ \left[\hskip -3pt \begin{array}{cccc} 28&18&3&-19\\ 5&1&0&-5 6 | \\ 4&5&1&0\\ 52&60&12&-15 7 | \end{array}\hskip -3pt \right]$} 8 | -------------------------------------------------------------------------------- /exercises/02_05_ex_26.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} 1&0&2&8\\ 0&1&0&0 2 | \\ 0&-4&-29&-110\\ 0&-3&-5&-19 3 | \end{array}\hskip -3pt \right] $ 4 | } 5 | {$ \left[\hskip -3pt \begin{array}{cccc} 1&28&-2&12\\ 0&1&0&0 6 | \\ 0&254&-19&110\\ 0&-67&5&-29 7 | \end{array}\hskip -3pt \right]$} 8 | -------------------------------------------------------------------------------- /exercises/02_05_ex_27.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} 0&0&1&0\\ 0&0&0&1 2 | \\ 1&0&0&0\\ 0&1&0&0 3 | \end{array}\hskip -3pt \right] $ 4 | } 5 | {$ \left[\hskip -3pt \begin{array}{cccc} 0&0&1&0\\ 0&0&0&1 6 | \\ 1&0&0&0\\ 0&1&0&0 7 | \end{array}\hskip -3pt \right]$} 8 | -------------------------------------------------------------------------------- /exercises/02_05_ex_28.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} 1&0&0&0\\ 0&2&0&0 2 | \\ 0&0&3&0\\ 0&0&0&-4 3 | \end{array}\hskip -3pt \right] $ 4 | } 5 | {$ \left[\hskip -3pt \begin{array}{cccc} 1&0&0&0\\ 0&1/2&0&0 6 | \\ 0&0&1/3&0\\ 0&0&0&-1/4 7 | \end{array}\hskip -3pt \right]$} 8 | -------------------------------------------------------------------------------- /exercises/02_05_ex_29.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} 1&2&12\\ 0&1&6 2 | \\ -3&0&1\end{array}\hskip -3pt \right],$ 3 | 4 | $\vb = \left[\hskip -3pt \begin{array}{c} -17\\ -5 5 | \\ 20\end{array}\hskip -3pt \right]$ 6 | } 7 | {$\vx = \left[\hskip -3pt \begin{array}{c} -7\\ 1\\ - 8 | 1\end{array}\hskip -3pt \right]$} 9 | -------------------------------------------------------------------------------- /exercises/02_05_ex_30.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} 1&0&-3\\ 8&-2&-13 2 | \\ 12&-3&-20\end{array}\hskip -3pt \right],$ 3 | 4 | $\vb = \left[\hskip -3pt \begin{array}{c} -34\\ -159 5 | \\ -243\end{array}\hskip -3pt \right]$ 6 | } 7 | {$\vx = \left[\hskip -3pt \begin{array}{c} -7\\ -7\\ 8 | 9\end{array}\hskip -3pt \right] $} 9 | -------------------------------------------------------------------------------- /exercises/02_05_ex_31.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} 5&0&-2\\ -8&1&5 2 | \\ -2&0&1\end{array}\hskip -3pt \right] ,$ 3 | 4 | $\vb = \left[\hskip -3pt \begin{array}{c} 33\\ -70 5 | \\ -15\end{array}\hskip -3pt \right]$ 6 | } 7 | {$\vx = \left[\hskip -3pt \begin{array}{c} 3\\ -1\\ - 8 | 9\end{array}\hskip -3pt \right]$} 9 | -------------------------------------------------------------------------------- /exercises/02_05_ex_32.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} 1&-6&0\\ 0&1&0 2 | \\ 2&-8&1\end{array}\hskip -3pt \right] ,$ 3 | 4 | $\vb = \left[\hskip -3pt \begin{array}{c} -69\\ 10 5 | \\ -102\end{array}\hskip -3pt \right]$ 6 | } 7 | {$\vx = \left[\hskip -3pt \begin{array}{c} -9\\ 10\\ 8 | -4\end{array}\hskip -3pt \right]$} 9 | -------------------------------------------------------------------------------- /exercises/02_05_ex_33.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 3&5\\ 2&3 2 | \end{array}\hskip -3pt \right],$ 3 | \quad 4 | $\vb = \left[\hskip -3pt \begin{array}{c} 21\\ 13\end {array} \hskip -3pt 5 | \right]$ 6 | } 7 | {$\vx = \left[\hskip -3pt \begin{array}{c} 2\\ 3\end{array}\hskip -3pt \right] $} 8 | -------------------------------------------------------------------------------- /exercises/02_05_ex_34.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 1&-4\\ 4&-15\end {array} \hskip -3pt 2 | \right] ,$ 3 | \quad 4 | $\vb =\left[\hskip -3pt \begin{array}{c} 21\\ 77\end {array} \hskip -3pt 5 | \right]$ 6 | } 7 | {$\vx = \left[\hskip -3pt \begin{array}{c} -7\\ -7\end {array} \hskip -3pt 8 | \right]$} 9 | -------------------------------------------------------------------------------- /exercises/02_05_ex_35.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 9&70\\ -4&-31\end {array} \hskip -3pt 2 | \right],$ 3 | \quad 4 | $\vb =\left[\hskip -3pt \begin{array}{c} -2\\ 1\end {array} \hskip -3pt 5 | \right]$ 6 | } 7 | {$\vx = \left[\hskip -3pt \begin{array}{c} -8\\ 1\end {array} \hskip -3pt 8 | \right]$} 9 | -------------------------------------------------------------------------------- /exercises/02_05_ex_36.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 10&-57\\ 3&-17\end {array} \hskip -3pt 2 | \right],$ 3 | \quad 4 | $\vb =\left[\hskip -3pt \begin{array}{c} -14\\ -4\end {array} \hskip -3pt 5 | \right]$ 6 | } 7 | {$\vx = \left[\hskip -3pt \begin{array}{c} 10\\ 2\end{array}\hskip -3pt \right]$} 8 | -------------------------------------------------------------------------------- /exercises/02_05_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/02_05_exset_01} 2 | \exsetinput{exercises/02_05_exset_02} 3 | \exsetinput{exercises/02_05_exset_03} 4 | 5 | 6 | 7 | -------------------------------------------------------------------------------- /exercises/02_05_exset_01.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, A matrix \tta\ is given. Find \ttai\ using Theorem \ref{thm:2by2}, if it exists.} 3 | \exinput{exercises/02_05_ex_01} 4 | \exinput{exercises/02_05_ex_02} 5 | \exinput{exercises/02_05_ex_03} 6 | \exinput{exercises/02_05_ex_04} 7 | \exinput{exercises/02_05_ex_05} 8 | \exinput{exercises/02_05_ex_06} 9 | \exinput{exercises/02_05_ex_07} 10 | \exinput{exercises/02_05_ex_08} -------------------------------------------------------------------------------- /exercises/02_05_exset_03.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, a matrix \tta\ and a vector \vb\ are given. Solve the equation \ttaxb\ using Theorem \ref{thm:inverse_solution}.} 3 | \exinput{exercises/02_05_ex_33} 4 | \exinput{exercises/02_05_ex_34} 5 | \exinput{exercises/02_05_ex_35} 6 | \exinput{exercises/02_05_ex_36} 7 | \exinput{exercises/02_05_ex_29} 8 | \exinput{exercises/02_05_ex_30} 9 | \exinput{exercises/02_05_ex_31} 10 | \exinput{exercises/02_05_ex_32} -------------------------------------------------------------------------------- /exercises/02_06_ex_01.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[ 2 | \begin{array}{cc} 3 | 1 & 2 \\ 4 | 1 & 1 5 | \end{array} \hskip -3pt 6 | \right]$, \quad 7 | $\ttb = \left[ 8 | \begin{array}{cc} 9 | 3 & 5 \\ 10 | 2 & 5 11 | \end{array} \hskip -3pt 12 | \right]$} 13 | {$(\tta\ttb)^{-1} = \left[ 14 | \begin{array}{cc} 15 | -2 & 3 \\ 16 | 1 & -1.4 17 | \end{array} \hskip -3pt 18 | \right]$} 19 | -------------------------------------------------------------------------------- /exercises/02_06_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[ 2 | \begin{array}{cc} 3 | 1 & 2 \\ 4 | 3 & 4 5 | \end{array}\hskip -3pt \right]$, \quad 6 | $\ttb = \left[ 7 | \begin{array}{cc} 8 | 7 & 1 \\ 9 | 2 & 1 10 | \end{array} \hskip -3pt 11 | \right]$} 12 | {$(\tta\ttb)^{-1} = \left[ 13 | \begin{array}{cc} 14 | -7/10 & 3/10 \\ 15 | 29/10 & -11/10 16 | \end{array} \hskip -3pt 17 | \right]$} 18 | -------------------------------------------------------------------------------- /exercises/02_06_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[ 2 | \begin{array}{cc} 3 | 2 & 5 \\ 4 | 3 & 8 5 | \end{array}\hskip -3pt \right]$, \quad 6 | $\ttb = \left[ 7 | \begin{array}{cc} 8 | 1 & -1 \\ 9 | 1 & 4 10 | \end{array} \hskip -3pt 11 | \right]$} 12 | {$(\tta\ttb)^{-1} = \left[ 13 | \begin{array}{cc} 14 | 29/5 & -18/5 \\ 15 | -11/5 & 7/5 16 | \end{array} \hskip -3pt 17 | \right]$} 18 | -------------------------------------------------------------------------------- /exercises/02_06_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[ 2 | \begin{array}{cc} 3 | 2 & 4 \\ 4 | 2 & 5 5 | \end{array}\hskip -3pt \right]$, \quad 6 | $\ttb = \left[ 7 | \begin{array}{cc} 8 | 2 & 2 \\ 9 | 6 & 5 10 | \end{array} \hskip -3pt 11 | \right]$} 12 | {$(\tta\ttb)^{-1} = \left[ 13 | \begin{array}{cc} 14 | -29/4 & 6\\ 15 | 17/2 & -7 16 | \end{array} \hskip -3pt 17 | \right]$} 18 | -------------------------------------------------------------------------------- /exercises/02_06_ex_05.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[ 2 | \begin{array}{cc} 3 | -3 & 5 \\ 4 | 1 & -2 5 | \end{array}\hskip -3pt \right]$ 6 | } 7 | {$\ttai = \left[ 8 | \begin{array}{cc} 9 | -2 & -5\\ 10 | -1 & -3 11 | \end{array} \hskip -3pt 12 | \right]$, 13 | 14 | $(\ttai)^{-1} = \left[ 15 | \begin{array}{cc} 16 | -3 & 5 \\ 17 | 1 & -2 18 | \end{array}\hskip -3pt \right]$ 19 | } 20 | -------------------------------------------------------------------------------- /exercises/02_06_ex_06.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[ 2 | \begin{array}{cc} 3 | 3 & 5\\ 4 | 2 & 4 5 | \end{array}\hskip -3pt \right]$ 6 | } 7 | {$\ttai = \left[ 8 | \begin{array}{cc} 9 | 2 & -5/2\\ 10 | -1 & 3/2 11 | \end{array} \hskip -3pt 12 | \right]$, 13 | 14 | $(\ttai)^{-1} = \left[ 15 | \begin{array}{cc} 16 | 3 & 5\\ 17 | 2 & 4 18 | \end{array}\hskip -3pt \right]$ 19 | } 20 | -------------------------------------------------------------------------------- /exercises/02_06_ex_07.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[ 2 | \begin{array}{cc} 3 | 2 & 7\\ 4 | 1 & 3 5 | \end{array}\hskip -3pt \right]$ 6 | } 7 | {$\ttai = \left[ 8 | \begin{array}{cc} 9 | -3 & 7\\ 10 | 1 & -2 \end{array} \hskip -3pt 11 | \right]$, 12 | 13 | $(\ttai)^{-1} = \left[ 14 | \begin{array}{cc} 15 | 2 & 7\\ 16 | 1 & 3 17 | \end{array}\hskip -3pt \right]$ 18 | } 19 | -------------------------------------------------------------------------------- /exercises/02_06_ex_08.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[ 2 | \begin{array}{cc} 3 | 9 & 0 \\ 4 | 7 & 9 5 | \end{array}\hskip -3pt \right]$ 6 | } 7 | {$\ttai = \left[ 8 | \begin{array}{cc} 9 | 1/9 & 0 \\ 10 | -7/81 & 1/9 11 | \end{array} \hskip -3pt 12 | \right]$, 13 | 14 | $(\ttai)^{-1} = \left[ 15 | \begin{array}{cc} 16 | 9 & 0 \\ 17 | 7 & 9 18 | \end{array}\hskip -3pt \right]$ 19 | } 20 | -------------------------------------------------------------------------------- /exercises/02_06_ex_09.tex: -------------------------------------------------------------------------------- 1 | {Find $2\times 2$ matrices \tta\ and \ttb\ that are each invertible, but $\tta+\ttb$ is not.} 2 | {Solutions will vary.} -------------------------------------------------------------------------------- /exercises/02_06_ex_10.tex: -------------------------------------------------------------------------------- 1 | {Create a random $6\times 6$ matrix \tta, then have a calculator or computer compute $\tta\ttai$. Was the identity matrix returned exactly? Comment on your results.} 2 | {Likely some entries that should be 0 will not be exaclty 0, but rather very small values.} -------------------------------------------------------------------------------- /exercises/02_06_ex_11.tex: -------------------------------------------------------------------------------- 1 | {Use a calculator or computer to compute $\tta\ttai$, where $$\tta = \bmx{cccc} 1&2&3&4\\1&4&9&16\\1&8&27&64\\1&16&81&256\emx.$$ Was the identity matrix returned exactly? Comment on your results. 2 | } 3 | {Likely some entries that should be 0 will not be exactly 0, but rather very small values.} -------------------------------------------------------------------------------- /exercises/02_06_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/02_06_exset_01} 2 | \exsetinput{exercises/02_06_exset_02} 3 | \exinput{exercises/02_06_ex_09} 4 | \exinput{exercises/02_06_ex_10} 5 | \exinput{exercises/02_06_ex_11} 6 | 7 | 8 | -------------------------------------------------------------------------------- /exercises/02_06_exset_01.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, matrices \tta\ and \ttb\ are given. Compute $(\tta\ttb)^{-1}$ and $\ttb^{-1}\tta^{-1}$.} 3 | \exinput{exercises/02_06_ex_01} 4 | \exinput{exercises/02_06_ex_02} 5 | \exinput{exercises/02_06_ex_03} 6 | \exinput{exercises/02_06_ex_04} -------------------------------------------------------------------------------- /exercises/02_06_exset_02.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, a $2\times 2$ matrix \tta\ is given. Compute \ttai\ and $(\ttai)^{-1}$ using Theorem \ref{thm:2by2}.} 3 | \exinput{exercises/02_06_ex_05} 4 | \exinput{exercises/02_06_ex_06} 5 | \exinput{exercises/02_06_ex_07} 6 | \exinput{exercises/02_06_ex_08} -------------------------------------------------------------------------------- /exercises/03_01_ex_01.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} -9&4&10\\ 6&-3&-7 2 | \\ -8&1&-1\end{array}\hskip -3pt \right]$} 3 | {$ \left[\hskip -3pt \begin{array}{ccc} -9&6&-8\\ 4&-3&1 4 | \\ 10&-7&-1\end{array}\hskip -3pt \right]$} 5 | 6 | -------------------------------------------------------------------------------- /exercises/03_01_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 4&-5&2\\ 1&5&9 2 | \\ 9&2&3\end{array}\hskip -3pt \right] $} 3 | {$\left[\hskip -3pt \begin{array}{ccc} 4&1&9\\ -5&5&2 4 | \\ 2&9&3\end{array}\hskip -3pt \right] $} 5 | 6 | -------------------------------------------------------------------------------- /exercises/03_01_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 2&-5&-3\\ 5&5&-6 2 | \\ 7&-4&-10\end{array}\hskip -3pt \right] $} 3 | {$\left[\hskip -3pt \begin{array}{ccc} 2&5&7\\ -5&5&-4 4 | \\ -3&-6&-10\end{array}\hskip -3pt \right]$} 5 | 6 | -------------------------------------------------------------------------------- /exercises/03_01_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 4&2&-9\\ 5&-4&-10 2 | \\ -6&6&9\end{array}\hskip -3pt \right] $} 3 | {$\left[\hskip -3pt \begin{array}{ccc} 4&5&-6\\ 2&-4&6 4 | \\ -9&-10&9\end{array}\hskip -3pt \right]$ 5 | } 6 | -------------------------------------------------------------------------------- /exercises/03_01_ex_05.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} -5&-9\\ 3&1 2 | \\ -10&-8\end{array}\hskip -3pt \right] $} 3 | {$\left[\hskip -3pt \begin{array}{ccc} -5&3&-10\\ -9&1&-8 4 | \end{array}\hskip -3pt \right]$} 5 | 6 | -------------------------------------------------------------------------------- /exercises/03_01_ex_06.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} -2&10\\ 1&-7 2 | \\ 9&-2\end{array}\hskip -3pt \right] $} 3 | {$\left[\hskip -3pt \begin{array}{ccc} -2&1&9\\ 10&-7&-2 4 | \end{array}\hskip -3pt \right]$} 5 | 6 | -------------------------------------------------------------------------------- /exercises/03_01_ex_07.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} 4&-7&-4&-9\\ -9&6&3&-9 2 | \end{array}\hskip -3pt \right] $} 3 | {$\left[\hskip -3pt \begin{array}{cc} 4&-9\\ -7&6 4 | \\ -4&3\\ -9&-9\end{array}\hskip -3pt \right]$} 5 | 6 | -------------------------------------------------------------------------------- /exercises/03_01_ex_08.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} 3&-10&0&6\\ -10&-2&-3&1 2 | \end{array}\hskip -3pt \right] $} 3 | {$\left[\hskip -3pt \begin{array}{cc} 3&-10\\ -10&-2 4 | \\ 0&-3\\ 6&1\end{array}\hskip -3pt \right]$} 5 | 6 | -------------------------------------------------------------------------------- /exercises/03_01_ex_09.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} -7&4\\ 4&-6\end {array} \hskip -3pt 2 | \right]$ } 3 | {\tta\ is symmetric. $\left[\hskip -3pt \begin{array}{cc} -7&4\\ 4&-6\end {array} \hskip -3pt 4 | \right]$ } 5 | 6 | 7 | 8 | -------------------------------------------------------------------------------- /exercises/03_01_ex_10.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 3&1\\ -7&8\end {array} \hskip -3pt 2 | \right] $} 3 | {$\left[\hskip -3pt \begin{array}{cc} 3&-7\\ 1&8\end {array} \hskip -3pt 4 | \right] $ 5 | } 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_01_ex_11.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} -7&-8&2&-3\end{array}\hskip -3pt \right]$ } 2 | {$ \left[\hskip -3pt \begin{array}{c} -7\\ -8\\ 3 | 2\\ -3\end{array}\hskip -3pt \right] $} 4 | 5 | 6 | 7 | -------------------------------------------------------------------------------- /exercises/03_01_ex_12.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} -9&8&2&-7\end{array}\hskip -3pt \right] $} 2 | {$ \left[\hskip -3pt \begin{array}{c} -9\\ 8\\ 2 3 | \\ -7\end{array}\hskip -3pt \right]$} 4 | 5 | 6 | 7 | -------------------------------------------------------------------------------- /exercises/03_01_ex_13.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 4&0&0\\ -2&-7&0 2 | \\ 4&-2&5\end{array}\hskip -3pt \right] $} 3 | {\tta\ is lower triangular and \ttat\ is upper triangular; $\left[\hskip -3pt \begin{array}{ccc} 4&-2&4\\ 0&-7&-2 4 | \\ 0&0&5\end{array}\hskip -3pt \right] $} 5 | 6 | 7 | 8 | -------------------------------------------------------------------------------- /exercises/03_01_ex_14.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} -3&-4&-5\\ 0&-3&5 2 | \\ 0&0&-3\end{array}\hskip -3pt \right] $} 3 | {\tta\ is upper triangular; \ttat\ is lower triangular. $\left[\hskip -3pt \begin{array}{ccc} -3&0&0\\ -4&-3&0 4 | \\ -5&5&-3\end{array}\hskip -3pt \right]$} 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_01_ex_15.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} 6&-7&2&6\\ 0&-8&-1&0 2 | \\ 0&0&1&-7\end{array}\hskip -3pt \right]$} 3 | {\tta\ is upper triangular; \ttat\ is lower triangular.$\left[\hskip -3pt \begin{array}{ccc} 6&0&0\\ -7&-8&0 4 | \\ 2&-1&1\\ 6&0&-7\end {array} \hskip -3pt 5 | \right] $} 6 | 7 | 8 | 9 | 10 | -------------------------------------------------------------------------------- /exercises/03_01_ex_16.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 1&0&0\\ 0&2&0 2 | \\ 0&0&-1\end{array}\hskip -3pt \right] $} 3 | {\tta\ is diagonal, as is \ttat.$\left[\hskip -3pt \begin{array}{ccc} 1&0&0\\ 0&2&0 4 | \\ 0&0&-1\end{array}\hskip -3pt \right] 5 | $} 6 | 7 | 8 | 9 | 10 | -------------------------------------------------------------------------------- /exercises/03_01_ex_17.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 1&0\\ 0&9\end {array} \hskip -3pt 2 | \right] $} 3 | {\tta\ is diagonal, as is \ttat.$\left[\hskip -3pt \begin{array}{cc} 1&0\\ 0&9\end{array}\hskip -3pt \right]$} 4 | 5 | 6 | 7 | 8 | -------------------------------------------------------------------------------- /exercises/03_01_ex_18.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 13&-3\\ -3&1\end {array} \hskip -3pt 2 | \right]$} 3 | {\tta\ is symmetric. $\left[\hskip -3pt \begin{array}{cc} 13&-3\\ -3&1\end {array} \hskip -3pt 4 | \right]$} 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_01_ex_19.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 4&0&-2\\ 0&2&3 2 | \\ -2&3&6\end{array}\hskip -3pt \right]$} 3 | {\tta\ is symmetric. $\left[\hskip -3pt \begin{array}{ccc} 4&0&-2\\ 0&2&3\\ -2&3&6\end{array}\hskip -3pt \right] 4 | $} 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_01_ex_20.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 0&3&-2\\ 3&-4&1 2 | \\ -2&1&0\end{array}\hskip -3pt \right] $} 3 | {\tta\ is symmetric. $\left[\hskip -3pt \begin{array}{ccc} 0&3&-2\\ 3&-4&1\\ -2&1&0\end{array}\hskip -3pt \right] $} 4 | 5 | 6 | 7 | 8 | -------------------------------------------------------------------------------- /exercises/03_01_ex_21.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 6&-4&-5\\ -4&0&2 2 | \\ -5&2&-2\end{array}\hskip -3pt \right] 3 | $} 4 | {\tta\ is symmetric. $\left[\hskip -3pt \begin{array}{ccc} 6&-4&-5\\ -4&0&2 5 | \\ -5&2&-2\end{array}\hskip -3pt \right] 6 | $} 7 | 8 | 9 | 10 | 11 | -------------------------------------------------------------------------------- /exercises/03_01_ex_22.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 0&1&-2\\ -1&0&4 2 | \\ 2&-4&0\end{array}\hskip -3pt \right]$} 3 | {\tta\ is skew symmetric. $\left[\hskip -3pt \begin{array}{ccc} 0&-1&2\\ 1&0&-4\\ -2&4&0\end{array}\hskip -3pt \right] 4 | $} 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_01_ex_23.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 0&-6&1\\ 6&0&4 2 | \\ -1&-4&0\end{array}\hskip -3pt \right]$} 3 | {\tta\ is skew symmetric. $\left[\hskip -3pt \begin{array}{ccc} 0&-6&1\\ 6&0&4\\ -1&-4&0\end{array}\hskip -3pt \right] 4 | $} 5 | 6 | 7 | 8 | 9 | 10 | -------------------------------------------------------------------------------- /exercises/03_01_ex_24.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 0&0&0\\ 0&0&0 2 | \\ 0&0&0\end{array}\hskip -3pt \right]$} 3 | {\tta\ is upper and lower triangular; it is diagonal; it is both symmetric and skew symmetric. It's got it all. $\left[\hskip -3pt \begin{array}{ccc} 0&0&0\\ 0&0&0\\ 0&0&0\end{array}\hskip -3pt \right] 4 | $} 5 | 6 | 7 | 8 | 9 | 10 | -------------------------------------------------------------------------------- /exercises/03_01_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/03_01_exset_01} -------------------------------------------------------------------------------- /exercises/03_02_ex_01.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} -4&1&1\\ -2&0&0 2 | \\ -1&-2&-5\end{array}\hskip -3pt \right] 3 | $} 4 | {$-9$} 5 | 6 | 7 | 8 | 9 | 10 | -------------------------------------------------------------------------------- /exercises/03_02_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 0&-3&1\\ 5&-5&5 2 | \\ -4&1&0\end{array}\hskip -3pt \right] 3 | $} 4 | {$-5$} 5 | 6 | 7 | 8 | 9 | 10 | -------------------------------------------------------------------------------- /exercises/03_02_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} -2&-3&5\\ 5&2&0 2 | \\ -1&-3&1\end{array}\hskip -3pt \right] $} 3 | {$1$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_02_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 4&2&-1\\ -4&1&4 2 | \\ 0&-5&5\end{array}\hskip -3pt \right]$} 3 | {$10$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_02_ex_05.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 1&-5\\ 9&5\end {array} \hskip -3pt 2 | \right]$} 3 | {$6$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_02_ex_06.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} -3&-10\\ -6&4\end {array} \hskip -3pt 2 | \right]$} 3 | {$1$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_02_ex_07.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 7&5\\ -5&-4\end {array} \hskip -3pt 2 | \right] $} 3 | {$3$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_02_ex_08.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} -6&0\\ -10&9\end {array} \hskip -3pt 2 | \right] $} 3 | {$3$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_02_ex_09.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} -10&6&-7&-9\\ -2&1&6&-9 2 | \\ 0&4&-4&0\\ -3&-9&3&-10 3 | \end{array}\hskip -3pt \right] $} 4 | {$-23$} 5 | 6 | 7 | 8 | 9 | 10 | -------------------------------------------------------------------------------- /exercises/03_02_ex_10.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} 5&2&2&2\\ -7&4&-7&-3 2 | \\ 9&-9&-7&2\\ -4&8&-8&-2 3 | \end{array}\hskip -3pt \right]$} 4 | {$0$} 5 | 6 | 7 | 8 | 9 | 10 | -------------------------------------------------------------------------------- /exercises/03_02_ex_11.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 2&6&4\\ -1&8&-10 2 | \end{array}\hskip -3pt \right]$} 3 | {Not defined; the matrix must be square.} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_02_ex_12.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 6&5\\ 2&10 2 | \\ 3&3\end{array}\hskip -3pt \right]$} 3 | {Not defined; the matrix must be square.} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_02_ex_13.tex: -------------------------------------------------------------------------------- 1 | {$\tti_4$} 2 | {4} 3 | 4 | 5 | 6 | 7 | 8 | -------------------------------------------------------------------------------- /exercises/03_02_ex_14.tex: -------------------------------------------------------------------------------- 1 | {$\tti_n$} 2 | {$n$} 3 | 4 | 5 | 6 | 7 | 8 | -------------------------------------------------------------------------------- /exercises/03_02_ex_15.tex: -------------------------------------------------------------------------------- 1 | {A matrix \tta\ that is skew symmetric.} 2 | {$0$} 3 | 4 | 5 | 6 | 7 | 8 | -------------------------------------------------------------------------------- /exercises/03_02_ex_16.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} -8&-10&10\\ 10&5&-6 2 | \\ -10&1&3\end{array}\hskip -3pt \right]$ 3 | 4 | $\ttb = \left[\hskip -3pt \begin{array}{ccc} -10&-4&-3\\ -4&-5&4 5 | \\ 3&7&3\end{array}\hskip -3pt \right]$ 6 | } 7 | {\begin{enumerate} 8 | \item tr(\tta)=0; tr(\ttb)=$-12$; tr($\tta+\ttb$)=$-12$ 9 | \item tr(\tta\ttb) = 86 = tr(\ttb\tta) 10 | \end{enumerate} 11 | } 12 | 13 | 14 | 15 | 16 | 17 | -------------------------------------------------------------------------------- /exercises/03_02_ex_17.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} -10&7&5\\ 7&7&-5 2 | \\ 8&-9&2\end{array}\hskip -3pt \right]$ 3 | 4 | $\ttb = \left[\hskip -3pt \begin{array}{ccc} -3&-4&9\\ 4&-1&-9 5 | \\ -7&-8&10\end{array}\hskip -3pt \right]$ 6 | } 7 | {\begin{enumerate} 8 | \item tr(\tta)=$-1$; tr(\ttb)=$6$; tr($\tta+\ttb$)=$5$ 9 | \item tr(\tta\ttb) = 201 = tr(\ttb\tta) 10 | \end{enumerate} 11 | } 12 | 13 | 14 | 15 | 16 | -------------------------------------------------------------------------------- /exercises/03_02_ex_18.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 1&-1\\ 9&-6\end {array} \hskip -3pt 2 | \right]$, 3 | \quad 4 | $\ttb = \left[\hskip -3pt \begin{array}{cc} -1&0\\ -6&3\end {array} \hskip -3pt 5 | \right]$ 6 | } 7 | { 8 | \begin{enumerate} 9 | \item tr(\tta)=$-5$; tr(\ttb)=$-4$; tr($\tta+\ttb$)=$-9$ 10 | \item tr(\tta\ttb) = 23 = tr(\ttb\tta) 11 | \end{enumerate} 12 | } 13 | 14 | 15 | 16 | 17 | 18 | -------------------------------------------------------------------------------- /exercises/03_02_ex_19.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 0&-8\\ 1&8\end {array} \hskip -3pt 2 | \right]$, 3 | \quad 4 | $\ttb = \left[\hskip -3pt \begin{array}{cc} -4&5\\ -4&2\end {array} \hskip -3pt 5 | \right] $ 6 | } 7 | {\begin{enumerate} 8 | \item tr(\tta)=$8$; tr(\ttb)=$-2$; tr($\tta+\ttb$)=$6$ 9 | \item tr(\tta\ttb) = 53 = tr(\ttb\tta) 10 | \end{enumerate} 11 | } 12 | 13 | 14 | 15 | 16 | 17 | -------------------------------------------------------------------------------- /exercises/03_02_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/03_02_exset_01} 2 | \exsetinput{exercises/03_02_exset_02} -------------------------------------------------------------------------------- /exercises/03_02_exset_01.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, find the trace of the given matrix.} 3 | \exinput{exercises/03_02_ex_05} 4 | \exinput{exercises/03_02_ex_06} 5 | \exinput{exercises/03_02_ex_07} 6 | \exinput{exercises/03_02_ex_08} 7 | \exinput{exercises/03_02_ex_01} 8 | \exinput{exercises/03_02_ex_02} 9 | \exinput{exercises/03_02_ex_03} 10 | \exinput{exercises/03_02_ex_04} 11 | \exinput{exercises/03_02_ex_11} 12 | \exinput{exercises/03_02_ex_12} 13 | \exinput{exercises/03_02_ex_09} 14 | \exinput{exercises/03_02_ex_10} 15 | \exinput{exercises/03_02_ex_13} 16 | \exinput{exercises/03_02_ex_14} 17 | \exinput{exercises/03_02_ex_15} -------------------------------------------------------------------------------- /exercises/03_02_exset_02.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, verify Theorem \ref{thm:trace} by: 3 | \begin{enumerate} 4 | \item Showing that tr(\tta)$+$tr(\ttb) = tr($\tta+\ttb$) and 5 | \item Showing that tr(\tta\ttb) = tr(\ttb\tta). 6 | \end{enumerate} 7 | } 8 | \exinput{exercises/03_02_ex_18} 9 | \exinput{exercises/03_02_ex_19} 10 | \exinput{exercises/03_02_ex_16} 11 | \exinput{exercises/03_02_ex_17} -------------------------------------------------------------------------------- /exercises/03_03_ex_01.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 10&7\\ 8&9\end {array} \hskip -3pt 2 | \right]$} 3 | {$34$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_03_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 6&-1\\ -7&8\end {array} \hskip -3pt 2 | \right]$} 3 | {$41$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_03_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} -1&-7\\ -5&9\end {array} \hskip -3pt 2 | \right] $} 3 | {$-44$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_03_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} -10&-1\\ -4&7\end {array} \hskip -3pt 2 | \right]$} 3 | {$-74$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_03_ex_05.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 8&10\\ 2&-3\end {array} \hskip -3pt 2 | \right]$} 3 | {$-44$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_03_ex_06.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 10&-10\\ -10&0\end {array} \hskip -3pt 2 | \right]$} 3 | {$-100$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_03_ex_07.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} 1&-3\\ 7&7\end {array} \hskip -3pt 2 | \right]$} 3 | {$28$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_03_ex_08.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cc} -4&-5\\ -1&-4\end {array} \hskip -3pt 2 | \right] $} 3 | {$11$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_03_ex_09.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} -7&-3&10\\ 3&7&6 2 | \\ 1&6&10\end{array}\hskip -3pt \right] $} 3 | {\begin{enumerate} 4 | \item The submatrices are $\left[\hskip -3pt \begin{array}{cc} 7&6\\ 6&10\end {array} \hskip -3pt 5 | \right] $, $ \left[\hskip -3pt \begin{array}{cc} 3&6\\ 1&10\end{array}\hskip -3pt \right] $, and 6 | $\left[\hskip -3pt \begin{array}{cc} 3&7\\ 1&6\end {array} \hskip -3pt 7 | \right] $, respectively. 8 | \item $C_{1,2}=34$, $C_{1,2}=-24$, $C_{1,3}=11$ 9 | \end{enumerate} 10 | } -------------------------------------------------------------------------------- /exercises/03_03_ex_10.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} -2&-9&6\\ -10&-6&8 2 | \\ 0&-3&-2\end{array}\hskip -3pt \right]$} 3 | {\begin{enumerate} 4 | \item The submatrices are $\left[\hskip -3pt \begin{array}{cc} -6&8\\ -3&-2\end {array} \hskip -3pt 5 | \right] $, $ \left[\hskip -3pt \begin{array}{cc} -10&8\\ 0&-2\end{array}\hskip -3pt \right]$, and 6 | $\left[\hskip -3pt \begin{array}{cc} 10&-6\\ 0&-3\end {array} \hskip -3pt 7 | \right]$, respectively. 8 | \item $C_{1,2}=36$, $C_{1,2}=-20$, $C_{1,3}=-30$ 9 | \end{enumerate} 10 | } 11 | 12 | 13 | 14 | 15 | 16 | -------------------------------------------------------------------------------- /exercises/03_03_ex_11.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} -5&-3&3\\ -3&3&10 2 | \\ -9&3&9\end{array}\hskip -3pt \right] $} 3 | {\begin{enumerate} 4 | \item The submatrices are 5 | $\left[\hskip -3pt \begin{array}{cc} 3&10\\ 3&9\end {array} \hskip -3pt 6 | \right]$, 7 | $\left[\hskip -3pt \begin{array}{cc} -3&10\\ -9&9\end {array} \hskip -3pt 8 | \right]$, and 9 | $\left[\hskip -3pt \begin{array}{cc} -3&3\\ -9&3\end {array} \hskip -3pt 10 | \right]$, respectively. 11 | \item $C_{1,2}=-3$, $C_{1,2}=-63$, $C_{1,3}=18$ 12 | \end{enumerate} 13 | } 14 | 15 | 16 | 17 | 18 | 19 | -------------------------------------------------------------------------------- /exercises/03_03_ex_12.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} -6&-4&6\\ -8&0&0 2 | \\ -10&8&-1\end{array}\hskip -3pt \right] $} 3 | {\begin{enumerate} 4 | \item The submatrices are 5 | $\left[\hskip -3pt \begin{array}{ccc} -6&-4&6\\ -8&0&0 6 | \\ -10&8&-1\end{array}\hskip -3pt \right] $, 7 | $\left[\hskip -3pt \begin{array}{cc} -8&0\\ -10&-1\end {array} \hskip -3pt 8 | \right]$, and 9 | $\left[\hskip -3pt \begin{array}{cc} -8&0\\ -10&8\end {array} \hskip -3pt 10 | \right]$, respectively. 11 | \item $C_{1,2}=0$, $C_{1,2}=-8$, $C_{1,3}=-64$ 12 | \end{enumerate} 13 | } 14 | 15 | 16 | 17 | 18 | 19 | -------------------------------------------------------------------------------- /exercises/03_03_ex_13.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 3&2&3\\ -6&1&-10 2 | \\ -8&-9&-9\end{array}\hskip -3pt \right]$} 3 | {$-59$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_03_ex_14.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 8&-9&-2\\ -9&9&-7 2 | \\ 5&-1&9\end{array}\hskip -3pt \right] $} 3 | {$250$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_03_ex_15.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} -4&3&-4\\ -4&-5&3 2 | \\ 3&-4&5\end{array}\hskip -3pt \right]$} 3 | {$15$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_03_ex_16.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 1&-2&1\\ 5&5&4 2 | \\ 4&0&0\end{array}\hskip -3pt \right]$} 3 | {$-52$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_03_ex_17.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 1&-4&1\\ 0&3&0 2 | \\ 1&2&2\end{array}\hskip -3pt \right] $} 3 | {$3$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_03_ex_18.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 3&-1&0\\ -3&0&-4 2 | \\ 0&-1&-4\end{array}\hskip -3pt \right]$} 3 | {$0$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_03_ex_19.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} -5&0&-4\\ 2&4&-1 2 | \\ -5&0&-4\end{array}\hskip -3pt \right]$} 3 | {$0$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_03_ex_20.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 1&0&0\\ 0&1&0 2 | \\ -1&1&1\end{array}\hskip -3pt \right] $} 3 | {$1$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_03_ex_21.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} 0&0&-1&-1\\ 1&1&0&1 2 | \\ 1&1&-1&0\\ -1&0&1&0\end{array}\hskip -3pt \right] 3 | $} 4 | {$0$} 5 | 6 | 7 | 8 | 9 | 10 | -------------------------------------------------------------------------------- /exercises/03_03_ex_22.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} -1&0&0&-1\\ -1&0&0&1 2 | \\ 1&1&1&0\\ 1&0&-1&-1\end {array}\hskip -3pt \right] 3 | $} 4 | {$2$} 5 | 6 | 7 | 8 | 9 | 10 | -------------------------------------------------------------------------------- /exercises/03_03_ex_23.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} -5&1&0&0\\ -3&-5&2&5 2 | \\ -2&4&-3&4\\ 5&4&-3&3\end {array}\hskip -3pt \right] 3 | $} 4 | {$-113$} 5 | 6 | 7 | 8 | 9 | 10 | -------------------------------------------------------------------------------- /exercises/03_03_ex_24.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} 2&-1&4&4\\ 3&-3&3&2 2 | \\ 0&4&-5&1\\ -2&-5&-2&-5\end{array}\hskip -3pt \right]$} 3 | {$179$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_03_ex_25.tex: -------------------------------------------------------------------------------- 1 | {Let \tta\ be a $2\times 2$ matrix; $$\tta = \bmx{cc}a&b\\c&d\emx.$$ Show why det$(\tta)=ad-bc$ by computing the cofactor expansion of \tta\ along the first row.} 2 | {Hint: $C_{1,1} = d$.} 3 | 4 | 5 | 6 | 7 | 8 | -------------------------------------------------------------------------------- /exercises/03_03_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/03_03_exset_01} 2 | \exsetinput{exercises/03_03_exset_02} 3 | \exsetinput{exercises/03_03_exset_03} 4 | \exinput{exercises/03_03_ex_25} -------------------------------------------------------------------------------- /exercises/03_03_exset_01.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, find the determinant of the $2\times 2$ matrix.} 3 | \exinput{exercises/03_03_ex_01} 4 | \exinput{exercises/03_03_ex_02} 5 | \exinput{exercises/03_03_ex_03} 6 | \exinput{exercises/03_03_ex_04} 7 | \exinput{exercises/03_03_ex_05} 8 | \exinput{exercises/03_03_ex_06} 9 | \exinput{exercises/03_03_ex_07} 10 | \exinput{exercises/03_03_ex_08} -------------------------------------------------------------------------------- /exercises/03_03_exset_02.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, a matrix \tta\ is given. 3 | \begin{list}{(\alph{enumi})}{}\usecounter{enumi} 4 | \item Construct the submatrices used to compute the minors $\tta_{1,1}$, $\tta_{1,2}$ and $\tta_{1,3}$. 5 | \item Find the cofactors $C_{1,1}$, $C_{1,2}$, and $C_{1,3}$. 6 | \end{list}} 7 | \exinput{exercises/03_03_ex_09} 8 | \exinput{exercises/03_03_ex_10} 9 | \exinput{exercises/03_03_ex_11} 10 | \exinput{exercises/03_03_ex_12} -------------------------------------------------------------------------------- /exercises/03_03_exset_03.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, find the determinant of the given matrix using cofactor expansion along the first row. 3 | } 4 | \exinput{exercises/03_03_ex_13} 5 | \exinput{exercises/03_03_ex_14} 6 | \exinput{exercises/03_03_ex_15} 7 | \exinput{exercises/03_03_ex_16} 8 | \exinput{exercises/03_03_ex_17} 9 | \exinput{exercises/03_03_ex_18} 10 | \exinput{exercises/03_03_ex_19} 11 | \exinput{exercises/03_03_ex_20} 12 | \exinput{exercises/03_03_ex_21} 13 | \exinput{exercises/03_03_ex_22} 14 | \exinput{exercises/03_03_ex_23} 15 | \exinput{exercises/03_03_ex_24} -------------------------------------------------------------------------------- /exercises/03_04_ex_01.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 1&2&3\\ -5&0&3 2 | \\ 4&0&6\end{array}\hskip -3pt \right]$} 3 | {$84$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_04_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} -4&4&-4\\ 0&0&-3 2 | \\ -2&-2&-1\end{array}\hskip -3pt \right] $} 3 | {$48$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_04_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} -4&1&1\\ 0&0&0 2 | \\ -1&-2&-5\end{array}\hskip -3pt \right]$} 3 | {$0$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_04_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 0&-3&1\\ 0&0&5 2 | \\ -4&1&0\end{array}\hskip -3pt \right]$} 3 | {$60$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_04_ex_05.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} -2&-3&5\\ 5&2&0 2 | \\ -1&0&0\end{array}\hskip -3pt \right]$} 3 | {$10$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_04_ex_06.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} -2&-2&0\\ 2&-5&-3 2 | \\ -5&1&0\end{array}\hskip -3pt \right]$} 3 | {$-36$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_04_ex_07.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} -3&0&-5\\ -2&-3&3 2 | \\ -1&0&1\end{array}\hskip -3pt \right]$} 3 | {$24$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_04_ex_08.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccc} 0&4&-4\\ 3&1&-3 2 | \\ -3&-4&0\end{array}\hskip -3pt \right]$} 3 | {$72$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_04_ex_09.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} 5&-5&0&1\\ 2&4&-1&-1 2 | \\ 5&0&0&4\\ -1&-2&0&5\end{array}\hskip -3pt \right]$} 3 | {$175$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_04_ex_10.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} -1&3&3&4\\ 0&0&0&0 2 | \\ 4&-5&-2&0\\ 0&0&2&0\end{array}\hskip -3pt \right]$} 3 | {$0$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_04_ex_11.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} -5&-5&0&-2\\ 0&0&5&0 2 | \\ 1&3&3&1\\ -4&-2&-1&-5\end{array}\hskip -3pt \right]$} 3 | {$-200$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_04_ex_12.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{cccc} -1&0&-2&5\\ 3&-5&1&-2 2 | \\ -5&-2&-1&-3\\ -1&0&0&0\end{array}\hskip -3pt \right]$} 3 | {$57$} 4 | 5 | 6 | 7 | 8 | 9 | -------------------------------------------------------------------------------- /exercises/03_04_ex_13.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccccc} 4&0&5&1&0\\ 1&0&3&1&5 2 | \\ 2&2&0&2&2\\ 1&0&0&0&0 3 | \\ 4&4&2&5&3\end{array}\hskip -3pt \right]$} 4 | {$34$} 5 | 6 | 7 | 8 | 9 | 10 | -------------------------------------------------------------------------------- /exercises/03_04_ex_14.tex: -------------------------------------------------------------------------------- 1 | {$\left[\hskip -3pt \begin{array}{ccccc} 2&1&1&1&1\\ 4&1&2&0&2 2 | \\ 0&0&1&0&0\\ 1&3&2&0&3 3 | \\ 5&0&5&0&4\end{array}\hskip -3pt \right] $} 4 | {$29$} 5 | 6 | 7 | 8 | 9 | 10 | -------------------------------------------------------------------------------- /exercises/03_04_ex_15.tex: -------------------------------------------------------------------------------- 1 | {$\ttm=\left[\hskip -3pt \begin{array}{ccc} 9&7&8\\ 1&3&7 2 | \\ 6&3&3\end{array}\hskip -3pt \right] $, 3 | 4 | det$(\ttm)=45$. 5 | \begin{enumerate} 6 | \item $\tta = \left[\hskip -3pt \begin{array}{ccc} 18&14&16\\ 1&3&7 7 | \\ 6&3&3\end{array}\hskip -3pt \right]$ 8 | \item $\ttb = \left[\hskip -3pt \begin{array}{ccc} 9&7&8\\ 1&3&7 9 | \\ 96&73&83\end{array}\hskip -3pt \right]$ 10 | \item $\ttc = \left[\hskip -3pt \begin{array}{ccc} 9&1&6\\ 7&3&3 11 | \\ 8&7&3\end{array}\hskip -3pt \right] $ 12 | \end{enumerate}} 13 | {\begin{enumerate} 14 | \item det$(\tta) = 90$; $2R_1\rightarrow R_1$ 15 | \item det$(\ttb) = 45$; $10R_1+R_3\rightarrow R_3$ 16 | \item det$(\ttc) = 45$; $\ttc = \ttat$ 17 | \end{enumerate}} 18 | 19 | 20 | 21 | 22 | 23 | -------------------------------------------------------------------------------- /exercises/03_04_ex_16.tex: -------------------------------------------------------------------------------- 1 | {$\ttm=\left[\hskip -3pt \begin{array}{ccc} 0&3&5\\ 3&1&0 2 | \\ -2&-4&-1\end{array}\hskip -3pt \right] $, 3 | 4 | det$(\ttm)=-41$. 5 | \begin{enumerate} 6 | \item $\tta = \left[\hskip -3pt \begin{array}{ccc} 0&3&5\\ -2&-4&-1 7 | \\ 3&1&0\end{array}\hskip -3pt \right] $ 8 | \item $\ttb = \left[\hskip -3pt \begin{array}{ccc} 0&3&5\\ 3&1&0 9 | \\ 8&16&4\end{array}\hskip -3pt \right] $ 10 | \item $\ttc = \left[\hskip -3pt \begin{array}{ccc} 3&4&5\\ 3&1&0 11 | \\ -2&-4&-1\end{array}\hskip -3pt \right] $ 12 | \end{enumerate}} 13 | {\begin{enumerate} 14 | \item det$(\tta) = 41$; $R_2\leftrightarrow R_3$ 15 | \item det$(\ttb) = 164$; $-4R_3\rightarrow R_3$ 16 | \item det$(\ttc) = -41$; $R_2+R_1\rightarrow R_1$ 17 | \end{enumerate}} 18 | 19 | 20 | 21 | 22 | 23 | -------------------------------------------------------------------------------- /exercises/03_04_ex_19.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 2&0\\ 1&2\end {array}\hskip -3pt \right] $, 2 | 3 | $\ttb = \left[\hskip -3pt \begin{array}{cc} 0&-4\\ 1&3\end{array}\hskip -3pt \right] $} 4 | {det$(\tta)=4$, det$(\ttb)=4$, det$(\tta\ttb)=16$} 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | -------------------------------------------------------------------------------- /exercises/03_04_ex_20.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 3&-1\\ 4&1\end {array} \hskip -3pt 2 | \right]$, 3 | 4 | $\ttb = \left[\hskip -3pt \begin{array}{cc} -4&-1\\ -5&3\end {array} \hskip -3pt 5 | \right]$} 6 | {det$(\tta)=7$, det$(\ttb)=-17$, det$(\tta\ttb)=-119$} 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | -------------------------------------------------------------------------------- /exercises/03_04_ex_21.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} -4&4\\ 5&-2\end {array} \hskip -3pt 2 | \right]$, 3 | 4 | $\ttb = \left[\hskip -3pt \begin{array}{cc} -3&-4\\ 5&-3\end {array} \hskip -3pt 5 | \right]$} 6 | {det$(\tta)=-12$, det$(\ttb)=29$, det$(\tta\ttb)=-348$} 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | -------------------------------------------------------------------------------- /exercises/03_04_ex_22.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} -3&-1\\ 2&-3\end {array} \hskip -3pt 2 | \right] $, 3 | 4 | $\ttb = \left[\hskip -3pt \begin{array}{cc} 0&0\\ 4&-4\end {array} \hskip -3pt 5 | \right]$} 6 | {det$(\tta)=11$, det$(\ttb)=0$, det$(\tta\ttb)=0$} 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | -------------------------------------------------------------------------------- /exercises/03_04_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/03_04_exset_01} 2 | \exsetinput{exercises/03_04_exset_02} 3 | \exsetinput{exercises/03_04_exset_03} 4 | \exsetinput{exercises/03_04_exset_04} -------------------------------------------------------------------------------- /exercises/03_04_exset_01.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, find the determinant of the given matrix using cofactor expansion along any row or column you choose.} 3 | \exinput{exercises/03_04_ex_01} 4 | \exinput{exercises/03_04_ex_02} 5 | \exinput{exercises/03_04_ex_03} 6 | \exinput{exercises/03_04_ex_04} 7 | \exinput{exercises/03_04_ex_05} 8 | \exinput{exercises/03_04_ex_06} 9 | \exinput{exercises/03_04_ex_07} 10 | \exinput{exercises/03_04_ex_08} 11 | \exinput{exercises/03_04_ex_09} 12 | \exinput{exercises/03_04_ex_10} 13 | \exinput{exercises/03_04_ex_11} 14 | \exinput{exercises/03_04_ex_12} 15 | \exinput{exercises/03_04_ex_13} 16 | \exinput{exercises/03_04_ex_14} -------------------------------------------------------------------------------- /exercises/03_04_exset_02.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, a matrix \ttm\ and det(\ttm) are given. Matrices \tta, \ttb\ and \ttc\ are formed by performing operations on \ttm. Determine the determinants of \tta, \ttb\ and \ttc\ using Theorems 3 | \ref{thm:determinant_row_operations} and \ref{thm:determinant_properties}, 4 | % 10 and 11, 5 | and indicate the operations used to form \tta, \ttb\ and \ttc.} 6 | \exinput{exercises/03_04_ex_16} 7 | \exinput{exercises/03_04_ex_15} 8 | \exinput{exercises/03_04_ex_17} 9 | \exinput{exercises/03_04_ex_18} -------------------------------------------------------------------------------- /exercises/03_04_exset_03.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, matrices \tta\ and \ttb\ are given. Verify part 3 of Theorem \ref{thm:determinant_properties} 3 | %11 4 | by computing det(\tta), det(\ttb) and det(\tta\ttb).} 5 | \exinput{exercises/03_04_ex_19} 6 | \exinput{exercises/03_04_ex_20} 7 | \exinput{exercises/03_04_ex_21} 8 | \exinput{exercises/03_04_ex_22} -------------------------------------------------------------------------------- /exercises/03_04_exset_04.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, find the determinant of the given matrix using Key Idea \ref{idea:3by3shortcut}. 3 | } 4 | \exinput{exercises/03_03_ex_13} 5 | \exinput{exercises/03_03_ex_14} 6 | \exinput{exercises/03_03_ex_15} 7 | \exinput{exercises/03_03_ex_16} 8 | \exinput{exercises/03_03_ex_17} 9 | \exinput{exercises/03_03_ex_18} 10 | \exinput{exercises/03_03_ex_19} 11 | \exinput{exercises/03_03_ex_20} -------------------------------------------------------------------------------- /exercises/03_05_ex_01.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} 3&0&-3\\ 5&4&4 2 | \\ 5&5&-4\end{array}\hskip -3pt \right]$, 3 | \quad 4 | $\vb = \left[\hskip -3pt \begin{array}{c} 24\\ 0\\ 5 | 31\end{array}\hskip -3pt \right] $} 6 | {\begin{enumerate} 7 | \item $\det{\tta} = -123$, $\det{\tta_1} = -492$, $\det{\tta_2} = 123$, $\det{\tta_3} = 492$ 8 | \item $\vx = \left[\hskip -3pt \begin{array}{c} 4\\ -1\\ -4\end{array}\hskip -3pt \right]$ 9 | \end{enumerate} 10 | } 11 | 12 | -------------------------------------------------------------------------------- /exercises/03_05_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} 4&-4&0\\ 5&1&-1 2 | \\ 3&-1&2\end{array}\hskip -3pt \right] $, 3 | \quad 4 | $\vb = \left[\hskip -3pt \begin{array}{c} 16\\ 22\\ 5 | 8\end{array}\hskip -3pt \right] $} 6 | {\begin{enumerate} 7 | \item $\det{\tta} = 56$, $\det{\tta_1} = 224$, $\det{\tta_2} = 0$, $\det{\tta_3} = -112$ 8 | \item $\vx = \left[\hskip -3pt \begin{array}{c} 4\\ 0\\ -2 9 | \end{array}\hskip -3pt \right]$ 10 | \end{enumerate} 11 | } 12 | 13 | 14 | 15 | 16 | 17 | 18 | -------------------------------------------------------------------------------- /exercises/03_05_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} 1&0&-10\\ 4&-3&-10 2 | \\ -9&6&-2\end{array}\hskip -3pt \right] $, 3 | 4 | $\vb = \left[\hskip -3pt \begin{array}{c} -40\\ -94 5 | \\ 132\end{array}\hskip -3pt \right]$} 6 | {\begin{enumerate} 7 | \item $\det{\tta} = 96$, $\det{\tta_1} = -960$, $\det{\tta_2} = 768$, $\det{\tta_3} = 288$ 8 | \item $\vx = \left[\hskip -3pt \begin{array}{c} -10\\ 8\\ 9 | 3\end{array}\hskip -3pt \right]$ 10 | \end{enumerate} 11 | } 12 | 13 | 14 | 15 | 16 | 17 | -------------------------------------------------------------------------------- /exercises/03_05_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} -6&-7&-7\\ 5&4&1 2 | \\ 5&4&8\end{array}\hskip -3pt \right] $, 3 | 4 | $\vb = \left[\hskip -3pt \begin{array}{c} 58\\ -35 5 | \\ -49\end{array}\hskip -3pt \right] $} 6 | {\begin{enumerate} 7 | \item $\det{\tta} = 77$, $\det{\tta_1} = -385$, $\det{\tta_2} = -154$, $\det{\tta_3} = -154$ 8 | \item $\vx = \left[\hskip -3pt \begin{array}{c} -5\\ -2\\ 9 | -2\end{array}\hskip -3pt \right] $ 10 | \end{enumerate} 11 | } 12 | -------------------------------------------------------------------------------- /exercises/03_05_ex_05.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 9&5\\ -4&-7\end {array} \hskip -3pt 2 | \right] $, 3 | \quad 4 | $\vb = \left[\hskip -3pt \begin{array}{c} -45\\ 20\end {array} \hskip -3pt 5 | \right]$} 6 | {\begin{enumerate} 7 | \item $\det{\tta} = -43$, $\det{\tta_1} = 215$, $\det{\tta_2} = 0$ 8 | \item $\vx = \left[\hskip -3pt \begin{array}{c} -5\\ 0\end{array}\hskip -3pt \right]$ 9 | \end{enumerate} 10 | } 11 | -------------------------------------------------------------------------------- /exercises/03_05_ex_06.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 0&-6\\ 9&-10\end {array} \hskip -3pt 2 | \right]$, 3 | \quad 4 | $\vb = \left[\hskip -3pt \begin{array}{c} 6\\ -17\end {array} \hskip -3pt 5 | \right]$} 6 | {\begin{enumerate} 7 | \item $\det{\tta} = 54$, $\det{\tta_1} = -162$, $\det{\tta_2} = -54$ 8 | \item $\vx = \left[\hskip -3pt \begin{array}{c} -3\\ -1\end {array} \hskip -3pt 9 | \right]$ 10 | \end{enumerate} 11 | } 12 | 13 | -------------------------------------------------------------------------------- /exercises/03_05_ex_07.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 2&10\\ -1&3\end {array} \hskip -3pt 2 | \right]$, 3 | \quad 4 | $\vb = \left[\hskip -3pt \begin{array}{c} 42\\ 19\end {array} \hskip -3pt 5 | \right]$} 6 | {\begin{enumerate} 7 | \item $\det{\tta} = 16$, $\det{\tta_1} = -64$, $\det{\tta_2} = 80$ 8 | \item $\vx = \left[\hskip -3pt \begin{array}{c} -4\\ 5\end{array}\hskip -3pt \right]$ 9 | \end{enumerate} 10 | } 11 | 12 | -------------------------------------------------------------------------------- /exercises/03_05_ex_08.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 7&-7\\ -7&9\end {array} \hskip -3pt 2 | \right] $, 3 | \quad 4 | $\vb = \left[\hskip -3pt \begin{array}{c} 28\\ -26\end {array} \hskip -3pt 5 | \right]$} 6 | {\begin{enumerate} 7 | \item $\det{\tta} = 14$, $\det{\tta_1} = 70$, $\det{\tta_2} = 14$ 8 | \item $\vx = \left[\hskip -3pt \begin{array}{c} 5\\ 1\end{array}\hskip -3pt \right]$ 9 | \end{enumerate} 10 | } 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | -------------------------------------------------------------------------------- /exercises/03_05_ex_09.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} -8&16\\ 10&-20\end {array} \hskip -3pt 2 | \right]$, 3 | \quad 4 | $\vb = \left[\hskip -3pt \begin{array}{c} -48\\ 60\end {array} \hskip -3pt 5 | \right]$} 6 | {\begin{enumerate} 7 | \item $\det{\tta} = 0$, $\det{\tta_1} = 0$, $\det{\tta_2} = 0$ 8 | \item Infinite solutions exist. 9 | \end{enumerate} 10 | } 11 | 12 | 13 | 14 | -------------------------------------------------------------------------------- /exercises/03_05_ex_10.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{cc} 7&14\\ -2&-4\end {array} \hskip -3pt 2 | \right] $, 3 | \quad 4 | $\vb = \left[\hskip -3pt \begin{array}{c} -1\\ 4\end{array}\hskip -3pt \right]$} 5 | {\begin{enumerate} 6 | \item $\det{\tta} = 0$, $\det{\tta_1} = -52$, $\det{\tta_2} = 26$ 7 | \item No solution exists. 8 | \end{enumerate} 9 | } 10 | 11 | 12 | 13 | 14 | 15 | -------------------------------------------------------------------------------- /exercises/03_05_ex_11.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} 4&9&3\\ -5&-2&-13 2 | \\ -1&10&-13\end{array}\hskip -3pt \right] $, 3 | 4 | $\vb = \left[\hskip -3pt \begin{array}{c} -28\\ 35 5 | \\ 7\end{array}\hskip -3pt \right]$} 6 | {\begin{enumerate} 7 | \item $\det{\tta} = 0$, $\det{\tta_1} = 0$, $\det{\tta_2} = 0$, $\det{\tta_3} = 0$ 8 | \item Infinite solutions exist. 9 | \end{enumerate} 10 | } 11 | 12 | 13 | 14 | 15 | -------------------------------------------------------------------------------- /exercises/03_05_ex_12.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \left[\hskip -3pt \begin{array}{ccc} 7&-4&25\\ -2&1&-7 2 | \\ 9&-7&34\end{array}\hskip -3pt \right]$, 3 | 4 | $\vb = \left[\hskip -3pt \begin{array}{c} -1\\ -3\\ 5 | 5\end{array}\hskip -3pt \right]$} 6 | {\begin{enumerate} 7 | \item $\det{\tta} = 0$, $\det{\tta_1} = 147$, $\det{\tta_2} = -49$, $\det{\tta_3} = -49$ 8 | \item No solution exists. 9 | \end{enumerate} 10 | } 11 | 12 | 13 | 14 | -------------------------------------------------------------------------------- /exercises/03_05_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/03_05_exset_01} -------------------------------------------------------------------------------- /exercises/03_05_exset_01.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, matrices \tta\ and \vb\ are given. 3 | \begin{list}{(\alph{enumi})}{}\usecounter{enumi} 4 | \item Give \det{\tta} and \det{\tta_i} for all $i$. 5 | \item Use Cramer's Rule to solve \ttaxb. If Cramer's Rule cannot be used to find the solution, then state whether or not a solution exists. 6 | \end{list}} 7 | \exinput{exercises/03_05_ex_08} 8 | \exinput{exercises/03_05_ex_05} 9 | \exinput{exercises/03_05_ex_09} 10 | \exinput{exercises/03_05_ex_06} 11 | \exinput{exercises/03_05_ex_07} 12 | \exinput{exercises/03_05_ex_10} 13 | \exinput{exercises/03_05_ex_01} 14 | \exinput{exercises/03_05_ex_11} 15 | \exinput{exercises/03_05_ex_02} 16 | \exinput{exercises/03_05_ex_03} 17 | \exinput{exercises/03_05_ex_12} 18 | \exinput{exercises/03_05_ex_04} -------------------------------------------------------------------------------- /exercises/04_01_ex_01.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cc}-1 & -4 \\ -3 & -2\emx$} 2 | {$\lda_1 = -5$ with $\vx[1] = \bmx{c}1\\1\emx$; 3 | 4 | $\lda_2 = 2$ with $\vx[2] = \bmx{c}-4\\3\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cc}-4 & 72 \\ -1 & 13\emx$} 2 | {$\lda_1 = 4$ with $\vx[1] = \bmx{c}9\\1\emx$; 3 | 4 | $\lda_2 = 5$ with $\vx[2] = \bmx{c}8\\1\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cc}2 & -12 \\ 2 & -8\emx$} 2 | {$\lda_1 = -4$ with $\vx[1] = \bmx{c}2\\1\emx$; 3 | 4 | $\lda_2 = -2$ with $\vx[2] = \bmx{c}3\\1\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cc}3 & 12 \\ 1 & -1\emx$} 2 | {$\lda_1 = -3$ with $\vx[1] = \bmx{c}-2\\1\emx$; 3 | 4 | $\lda_2 = 5$ with $\vx[2] = \bmx{c}6\\1\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_05.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cc}5 & 9 \\ -1 & -5\emx$} 2 | {$\lda_1 = -4$ with $\vx[1] = \bmx{c}-1\\1\emx$; 3 | 4 | $\lda_2 = 4$ with $\vx[2] = \bmx{c}-9\\1\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_06.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cc}3 & -1 \\ -1 & 3\emx$} 2 | {$\lda_1 = 2$ with $\vx[1] = \bmx{c}1\\1\emx$; 3 | 4 | $\lda_2 = 4$ with $\vx[2] = \bmx{c}-1\\1\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_07.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cc}0 & 1 \\ 25 & 0\emx$} 2 | {$\lda_1 = -5$ with $\vx[1] = \bmx{c}-1\\5\emx$; 3 | 4 | $\lda_2 = 5$ with $\vx[2] = \bmx{c}1\\5\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_08.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cc}-3 & 1 \\ 0 & -1\emx$} 2 | {$\lda_1 = -1$ with $\vx[1] = \bmx{c}1\\2\emx$; 3 | 4 | $\lda_2 = -3$ with $\vx[2] = \bmx{c}1\\0\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_09.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc}1 & -2 & -3 \\ 0 & 3 & 0 \\ 0 & -1 & -1\emx$} 2 | {$\lda_1 = -1$ with $\vx[1] = \bmx{c}3\\0\\2\emx$; 3 | 4 | $\lda_2 = 1$ with $\vx[2] = \bmx{c}1\\0\\0\emx$ 5 | 6 | $\lda_3 = 3$ with $\vx[3] = \bmx{c}5\\-8\\2\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_10.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc}5 & -2 & 3 \\ 0 & 4 & 0 \\ 0 & -1 & 3\emx$} 2 | {$\lda_1 = 3$ with $\vx[1] = \bmx{c}-3\\0\\2\emx$; 3 | 4 | $\lda_2 = 4$ with $\vx[2] = \bmx{c}-5\\-1\\1\emx$ 5 | 6 | $\lda_3 = 5$ with $\vx[3] = \bmx{c}1\\0\\0\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_11.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc}1 & 0 & 12 \\ 2 & -5 & 0 \\ 1 & 0 & 2\emx$} 2 | {$\lda_1 = -5$ with $\vx[1] = \bmx{c}0\\1\\0\emx$; 3 | 4 | $\lda_2 = -2$ with $\vx[2] = \bmx{c}-12\\-8\\3\emx$ 5 | 6 | $\lda_3 = 5$ with $\vx[3] = \bmx{c}15\\3\\5\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_12.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc}1 & 0 & -18 \\ -4 & 3 & -1 \\ 1 & 0 & -8\emx$} 2 | {$\lda_1 = -5$ with $\vx[1] = \bmx{c}24\\13\\8\emx$; 3 | 4 | $\lda_2 = -2$ with $\vx[2] = \bmx{c}6\\5\\1\emx$ 5 | 6 | $\lda_3 = 3$ with $\vx[3] = \bmx{c}0\\1\\0\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_13.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc} -1 & 18 & 0 \\ 1 & 2 & 0 \\ 5 & -3 & -1\emx$} 2 | {$\lda_1 = -4$ with $\vx[1] = \bmx{c}-6\\1\\11\emx$; 3 | 4 | $\lda_2 = -1$ with $\vx[2] = \bmx{c}0\\0\\1\emx$ 5 | 6 | $\lda_3 = 5$ with $\vx[3] = \bmx{c}3\\1\\2\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_14.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc} 5 & 0 & 0 \\ 1 & 1 & 0 \\ -1 & 5 & -2\emx$} 2 | {$\lda_1 = -2$ with $\vx[1] = \bmx{c}0\\0\\1\emx$; 3 | 4 | $\lda_2 = 1$ with $\vx[2] = \bmx{c}0\\3\\5\emx$ 5 | 6 | $\lda_3 = 5$ with $\vx[3] = \bmx{c}28\\7\\1\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_15.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc} 2 & -1 & 1 \\ 0 & 3 & 6 \\ 0 & 0 & 7\emx$} 2 | {$\lda_1 = 2$ with $\vx[1] = \bmx{c}1\\0\\0\emx$; 3 | 4 | $\lda_2 = 3$ with $\vx[2] = \bmx{c}-1\\1\\0\emx$ 5 | 6 | $\lda_3 = 7$ with $\vx[3] = \bmx{c}-1\\15\\10\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_16.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{ccc} -16 & -28 & -19 \\ 42 & 69 & 46 \\ -42 & -72 & -49\emx$ 2 | 3 | $\lambda = 5$} 4 | {$\vx = \bmx{c} 3\\-7\\7\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_17.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{ccc} 7 & -5 & -10 \\ 6 & 2 & -6 \\ 2 & -5 & -5\emx$ 2 | 3 | $\lambda = -3$} 4 | {$\vx = \bmx{c} 1\\0\\1\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_18.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{ccc} 4 & 5 & -3 \\ -7 & -8 & 3 \\ 1 & -5 & 8\emx$ 2 | 3 | $\lambda = 2$} 4 | {$\vx = \bmx{c} -1\\1\\1\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_19.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{cc} 16 & 6 \\ -18 & -5\emx$ 2 | 3 | $\lambda = 4$} 4 | {$\vx = \bmx{c} -1\\2\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_20.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{cc} -2 & 6 \\ -9 & 13\emx$ 2 | 3 | $\lambda = 7$} 4 | {$\vx = \bmx{c} 2\\3\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_21.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{cc} 9 & 8 \\ -6 & -5\emx$ 2 | 3 | $\vx = \bmx{c} -4\\3\emx$} 4 | {$\lambda = 3$} -------------------------------------------------------------------------------- /exercises/04_01_ex_22.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{cc} 19 & -6 \\ 48 & -15\emx$ 2 | 3 | $\vx = \bmx{c} 1\\3\emx$} 4 | {$\lambda = 1$} -------------------------------------------------------------------------------- /exercises/04_01_ex_23.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{ccc} -11 & -19 & 14 \\ -6 & -8 & 6 \\ -12 & -22 & 15\emx$ 2 | 3 | $\vx = \bmx{c} 3\\2\\4\emx$} 4 | {$\lambda = -5$} -------------------------------------------------------------------------------- /exercises/04_01_ex_24.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{ccc} -7 & 1 & 3 \\ 10 & 2 & -3 \\ -20 & -14 & 1\emx$ 2 | 3 | $\vx = \bmx{c} 1\\-2\\4\emx$} 4 | {$\lambda = 3$} -------------------------------------------------------------------------------- /exercises/04_01_ex_25.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{ccc} -12 & -10 & 0 \\ 15 & 13 & 0 \\ 15 & 18 & -5\emx$ 2 | 3 | $\vx = \bmx{c} -1\\1\\1\emx$} 4 | {$\lambda = -2$} -------------------------------------------------------------------------------- /exercises/04_01_ex_26.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{cc} 1& -2\\ -2 & 4 \emx$ 2 | 3 | $\vx = \bmx{c} 2\\1\emx$} 4 | {$\lambda = 0$} -------------------------------------------------------------------------------- /exercises/04_01_ex_27.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc}3&5&-5\\ -2&3&2\\-2&5&0\emx$} 2 | {$\lambda_1=-2$ with $\vx = \bmx{c} 1\\0\\1\emx$; 3 | 4 | $\lambda_2=3$ with $\vx = \bmx{c} 1\\1\\1\emx$; 5 | 6 | $\lambda_3=5$ with $\vx = \bmx{c} 0\\1\\1\emx$} -------------------------------------------------------------------------------- /exercises/04_01_ex_28.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc}1&2&1\\1&2&3\\1&1&1\emx$} 2 | {$\lambda_1=0$ with $\vx = \bmx{c} 1\\3\\1\emx$; 3 | 4 | $\lambda_2=-1$ with $\vx = \bmx{c} 2\\2\\1\emx$; 5 | 6 | $\lambda_3=2$ with $\vx = \bmx{c} 1\\1\\1\emx$} -------------------------------------------------------------------------------- /exercises/04_01_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/04_01_exset_03} 2 | \exsetinput{exercises/04_01_exset_02} 3 | \exsetinput{exercises/04_01_exset_01} 4 | -------------------------------------------------------------------------------- /exercises/04_01_exset_01.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, find the \el s of the given matrix. For each \el, give an \ev.} 3 | \exinput{exercises/04_01_ex_01} 4 | \exinput{exercises/04_01_ex_02} 5 | \exinput{exercises/04_01_ex_03} 6 | \exinput{exercises/04_01_ex_04} 7 | \exinput{exercises/04_01_ex_05} 8 | \exinput{exercises/04_01_ex_06} 9 | \exinput{exercises/04_01_ex_07} 10 | \exinput{exercises/04_01_ex_08} 11 | \exinput{exercises/04_01_ex_09} 12 | \exinput{exercises/04_01_ex_10} 13 | \exinput{exercises/04_01_ex_11} 14 | \exinput{exercises/04_01_ex_12} 15 | \exinput{exercises/04_01_ex_13} 16 | \exinput{exercises/04_01_ex_14} 17 | \exinput{exercises/04_01_ex_15} 18 | \exinput{exercises/04_01_ex_27} 19 | \exinput{exercises/04_01_ex_28} -------------------------------------------------------------------------------- /exercises/04_01_exset_02.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, a matrix \tta\ and one of its \el s are given. Find an \ev\ of \tta\ for the given \el.} 3 | \exinput{exercises/04_01_ex_19} 4 | \exinput{exercises/04_01_ex_20} 5 | \exinput{exercises/04_01_ex_16} 6 | \exinput{exercises/04_01_ex_17} 7 | \exinput{exercises/04_01_ex_18} -------------------------------------------------------------------------------- /exercises/04_01_exset_03.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, a matrix \tta\ and one of its \ev s are given. Find the \el\ of \tta\ for the given \ev.} 3 | \exinput{exercises/04_01_ex_21} 4 | \exinput{exercises/04_01_ex_22} 5 | \exinput{exercises/04_01_ex_26} 6 | \exinput{exercises/04_01_ex_23} 7 | \exinput{exercises/04_01_ex_24} 8 | \exinput{exercises/04_01_ex_25} -------------------------------------------------------------------------------- /exercises/04_02_ex_01.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cc}0&4\\-1&5\emx$} 2 | {\begin{enumerate} 3 | \item $\lda_1 = 1$ with $\vx[1] = \bmx{c}4\\1\emx$; 4 | 5 | $\lda_2 = 4$ with $\vx[2] = \bmx{c}1\\1\emx$ 6 | 7 | \item $\lda_1 = 1$ with $\vx[1] = \bmx{c}-1\\1\emx$; 8 | 9 | $\lda_2 = 4$ with $\vx[2] = \bmx{c}-1\\4\emx$ 10 | 11 | \item $\lda_1 = 1/4$ with $\vx[1] = \bmx{c}1\\1\emx$; 12 | 13 | $\lda_2 = 1$ with $\vx[2] = \bmx{c}4\\1\emx$ 14 | 15 | \item 5 16 | \item 4 17 | \end{enumerate} 18 | } -------------------------------------------------------------------------------- /exercises/04_02_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cc}-2&-14\\-1&3\emx$} 2 | {\begin{enumerate} 3 | \item $\lda_1 = -4$ with $\vx[1] = \bmx{c}7\\1\emx$; 4 | 5 | $\lda_2 = 5$ with $\vx[2] = \bmx{c}-2\\1\emx$ 6 | 7 | \item $\lda_1 = -4$ with $\vx[1] = \bmx{c}1\\2\emx$; 8 | 9 | $\lda_2 = 5$ with $\vx[2] = \bmx{c}-1\\7\emx$ 10 | 11 | \item $\lda_1 = -1/4$ with $\vx[1] = \bmx{c}7\\1\emx$; 12 | 13 | $\lda_2 = 1/5$ with $\vx[2] = \bmx{c}-2\\1\emx$ 14 | 15 | \item 1 16 | \item -20 17 | \end{enumerate} 18 | } -------------------------------------------------------------------------------- /exercises/04_02_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cc}5&30\\-1&-6\emx$} 2 | {\begin{enumerate} 3 | \item $\lda_1 = -1$ with $\vx[1] = \bmx{c}-5\\1\emx$; 4 | 5 | $\lda_2 = 0$ with $\vx[2] = \bmx{c}-6\\1\emx$ 6 | 7 | \item $\lda_1 = -1$ with $\vx[1] = \bmx{c}1\\6\emx$; 8 | 9 | $\lda_2 = 0$ with $\vx[2] = \bmx{c}1\\5\emx$ 10 | 11 | \item \tta is not invertible. 12 | 13 | \item -1 14 | 15 | \item 0 16 | \end{enumerate} 17 | } -------------------------------------------------------------------------------- /exercises/04_02_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{cc}-4&72\\-1&13\emx$} 2 | {\begin{enumerate} 3 | \item $\lda_1 = 4$ with $\vx[1] = \bmx{c}9\\1\emx$; 4 | 5 | $\lda_2 = 5$ with $\vx[2] = \bmx{c}8\\1\emx$ 6 | 7 | \item $\lda_1 = 4$ with $\vx[1] = \bmx{c}-1\\8\emx$; 8 | 9 | $\lda_2 = 5$ with $\vx[2] = \bmx{c}-1\\9\emx$ 10 | 11 | \item $\lda_1 = 1/4$ with $\vx[1] = \bmx{c}9\\1\emx$; 12 | 13 | $\lda_2 = 1/5$ with $\vx[2] = \bmx{c}8\\1\emx$ 14 | 15 | \item 9 16 | 17 | \item 20 18 | \end{enumerate} 19 | } -------------------------------------------------------------------------------- /exercises/04_02_ex_05.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc}5&-9&0\\1&-5&0\\2&4&3\emx$} 2 | {\begin{enumerate} 3 | \item $\lda_1 = -4$ with $\vx[1] = \bmx{c}-7\\-7\\6\emx$; 4 | 5 | $\lda_2 = 3$ with $\vx[2] = \bmx{c}0\\0\\1\emx$ 6 | 7 | $\lda_3 = 4$ with $\vx[3] = \bmx{c}9\\1\\22\emx$ 8 | 9 | \item $\lda_1 = -4$ with $\vx[1] = \bmx{c}-1\\9\\0\emx$; 10 | 11 | $\lda_2 = 3$ with $\vx[2] = \bmx{c}-20\\26\\7\emx$ 12 | 13 | $\lda_3 = 4$ with $\vx[3] = \bmx{c}-1\\1\\0\emx$ 14 | 15 | \item $\lda_1 = -1/4$ with $\vx[1] = \bmx{c}-7\\-7\\6\emx$; 16 | 17 | $\lda_2 = 1/3$ with $\vx[2] = \bmx{c}0\\0\\1\emx$ 18 | 19 | $\lda_3 = 1/4$ with $\vx[3] = \bmx{c}9\\1\\22\emx$ 20 | 21 | \item $3$ 22 | 23 | \item $-48$ 24 | 25 | \end{enumerate} 26 | } -------------------------------------------------------------------------------- /exercises/04_02_ex_06.tex: -------------------------------------------------------------------------------- 1 | {$\bmx{ccc}0&25&0\\1&0&0\\1&1&-3\emx$} 2 | {\begin{enumerate} 3 | \item $\lda_1 = -5$ with $\vx[1] = \bmx{c}-5\\1\\2\emx$; 4 | 5 | $\lda_2 = -3$ with $\vx[2] = \bmx{c}0\\0\\1\emx$ 6 | 7 | $\lda_3 = 5$ with $\vx[3] = \bmx{c}20\\4\\3\emx$ 8 | 9 | \item $\lda_1 = -5$ with $\vx[1] = \bmx{c}-1\\5\\0\emx$; 10 | 11 | $\lda_2 = -3$ with $\vx[2] = \bmx{c}1\\-11\\8\emx$ 12 | 13 | $\lda_3 = 5$ with $\vx[3] = \bmx{c}1\\5\\0\emx$ 14 | 15 | \item $\lda_1 = -1/5$ with $\vx[1] = \bmx{c}-5\\1\\2\emx$; 16 | 17 | $\lda_2 = -1/3$ with $\vx[2] = \bmx{c}0\\0\\1\emx$ 18 | 19 | $\lda_3 = 1/5$ with $\vx[3] = \bmx{c}20\\4\\3\emx$ 20 | 21 | \item $-3$ 22 | 23 | \item $75$ 24 | 25 | \end{enumerate} 26 | } -------------------------------------------------------------------------------- /exercises/04_02_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/04_02_exset_01} 2 | -------------------------------------------------------------------------------- /exercises/04_02_exset_01.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises } 2 | {, a matrix \tta\ is given. For each, 3 | \begin{list}{(\alph{enumi})}{}\usecounter{enumi} 4 | \item Find the \el s of \tta, and for each \el, find an \ev. 5 | \item Do the same for \ttat. 6 | \item Do the same for \ttai. 7 | \item Find tr(\tta). 8 | \item Find $\det{\tta}$. 9 | \end{list} 10 | Use Theorem \ref{thm:eigen_properties} to verify your results.} 11 | \exinput{exercises/04_02_ex_01} 12 | \exinput{exercises/04_02_ex_02} 13 | \exinput{exercises/04_02_ex_03} 14 | \exinput{exercises/04_02_ex_04} 15 | \exinput{exercises/04_02_ex_05} 16 | \exinput{exercises/04_02_ex_06} 17 | -------------------------------------------------------------------------------- /exercises/05_01_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$\vx = \bmx{c}3\\1\emx$, $\vy = \bmx{c} 1\\-2\emx$} 2 | {$\vx+\vy = \bmx{c}4\\-1\emx$, $\vx-\vy = \bmx{c} 2\\3\emx$ 3 | 4 | Sketches will vary depending on choice of origin of each vector. 5 | 6 | \begin{tikzpicture}[>=latex,scale=.5] 7 | % Draw grid 8 | \draw (-1,0)--(5,0); 9 | \draw (0,-3)--(0,2); 10 | \foreach \x in {1,...,4} 11 | {\draw (\x,-.1)--(\x,.1); 12 | } 13 | \foreach \x in {-2,...,1} 14 | {\draw (-.1,\x)--(.1,\x); 15 | }; 16 | \node[below] at (2,-0.1) {2}; 17 | \node[left] at (0,1) {1}; 18 | %Draw arrows 19 | \draw [->,thick] (0,0)--(3,1) node [above] {\vx}; 20 | \draw [->,thick] (0,0) -- (1,-2) node [below ] {\vy}; 21 | \draw [->,thick] (0,0) -- (4,-1) node [below] {$\vx+\vy$}; 22 | \draw [->,thick] (1,-2) -- (3,1) node [below right] {$\vx-\vy$}; 23 | \end{tikzpicture} 24 | } -------------------------------------------------------------------------------- /exercises/05_01_ex_09.tex: -------------------------------------------------------------------------------- 1 | {$\vx = \bmx{c} 2\\1\emx$, $a = 3$.} 2 | {$||\vx|| = \sqrt{5}$; $||a\vx|| = \sqrt{45} = 3\sqrt{5}$. The vector $a\vx$ is 3 times as long as \vx.} -------------------------------------------------------------------------------- /exercises/05_01_ex_10.tex: -------------------------------------------------------------------------------- 1 | {$\vx = \bmx{c} 4\\7\emx$, $a = -2$.} 2 | {$||\vx|| = \sqrt{65}$; $||a\vx|| = \sqrt{260} = 2\sqrt{65}$. The vector $a\vx$ is 2 times as long as \vx.} -------------------------------------------------------------------------------- /exercises/05_01_ex_11.tex: -------------------------------------------------------------------------------- 1 | {$\vx = \bmx{c} -3\\5\emx$, $a = -1$.} 2 | {$||\vx|| = \sqrt{34}$; $||a\vx|| = \sqrt{34}$. The vectors $a\vx$ and \vx\ are the same length (they just point in opposite directions).} -------------------------------------------------------------------------------- /exercises/05_01_ex_12.tex: -------------------------------------------------------------------------------- 1 | {$\vx = \bmx{c} 3\\-9\emx$, $a = \frac 13$.} 2 | {$||\vx|| = \sqrt{90} = 3\sqrt{10}$; $||a\vx|| = \sqrt{10}$. The vector $a\vx$ is one-third the length of \vx; equivalently, \vx\ is 3 times as long as $a\vx$.} -------------------------------------------------------------------------------- /exercises/05_01_ex_14.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{ccc} -16 & -28 & -19 \\ 42 & 69 & 46 \\ -42 & -72 & -49\emx,\quad \lambda = 5} 2 | {$\vx = \bmx{c} 3\\-7\\7\emx$} -------------------------------------------------------------------------------- /exercises/05_01_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/05_01_exset_01} 2 | \exsetinput{exercises/05_01_exset_02} 3 | \exsetinput{exercises/05_01_exset_03} 4 | \exinput{exercises/05_01_ex_13} 5 | \exsetinput{exercises/05_03_exset_01} 6 | 7 | -------------------------------------------------------------------------------- /exercises/05_01_exset_01.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, vectors \vx\ and \vy\ are given. Sketch \vx, \vy, $\vx+\vy$, and $\vx-\vy$ on the same Cartesian axes.} 3 | \exinput{exercises/05_01_ex_01} 4 | \exinput{exercises/05_01_ex_02} 5 | \exinput{exercises/05_01_ex_03} 6 | \exinput{exercises/05_01_ex_04} -------------------------------------------------------------------------------- /exercises/05_01_exset_02.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, vectors \vx\ and \vy\ are drawn. Sketch $2\vx$, $-\vy$, $\vx+\vy$, and $\vx-\vy$ on the same Cartesian axes.} 3 | \exinput{exercises/05_01_ex_05} 4 | \exinput{exercises/05_01_ex_06} 5 | \exinput{exercises/05_01_ex_07} 6 | \exinput{exercises/05_01_ex_08} -------------------------------------------------------------------------------- /exercises/05_01_exset_03.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, a vector \vx\ and a scalar $a$ are given. Using Definition \ref{def:vector_length}, compute the lengths of \vx\ and $a\vx$, then compare these lengths.} 3 | \exinput{exercises/05_01_ex_09} 4 | \exinput{exercises/05_01_ex_10} 5 | \exinput{exercises/05_01_ex_11} 6 | \exinput{exercises/05_01_ex_12} 7 | -------------------------------------------------------------------------------- /exercises/05_01_exset_04.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, vectors \vx\ and \vy\ are given. Compute the lengths of \vx, \vy, and $\vx+\vy$. } 3 | \exinput{exercises/05_01_ex_09} 4 | \exinput{exercises/05_01_ex_10} 5 | \exinput{exercises/05_01_ex_11} 6 | \exinput{exercises/05_01_ex_12} 7 | -------------------------------------------------------------------------------- /exercises/05_02_ex_01.tex: -------------------------------------------------------------------------------- 1 | {$\vx = \bmx{c}1\\-1\\2\emx$, $\vy = \bmx{c} 2\\3\\2\emx$} 2 | {$\vx+\vy = \bmx{c}3\\2\\4\emx$, $\vx-\vy = \bmx{c} -1\\-4\\0\emx$ 3 | 4 | Sketches will vary slightly depending on orientation. 5 | 6 | \begin{tikzpicture}[>=latex,x={(-.53cm,-.53cm)},y={(.75cm,0)},z={(0,.75cm)}] 7 | \drawxaxis{-1}{3}{1}{2}; 8 | \drawyaxis{-2}{2}{-1}{1}; 9 | \drawzaxis{0}{2.5}{1}{2}; 10 | \drawvect{2}{3}{4}{dotted}{->,thick}{above right}{\vy}; 11 | \drawvect{1}{-1}{2}{dotted}{->,thick}{above left}{\vx}; 12 | \drawvect{3}{2}{4}{dotted}{->,thick}{left}{$\vx+\vy$}; 13 | \drawvect{-1}{-4}{0}{dotted}{->,thick}{below left}{$\vx-\vy$}; 14 | \end{tikzpicture} 15 | } -------------------------------------------------------------------------------- /exercises/05_02_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$\vx = \bmx{c}2\\4\\-1\emx$, $\vy = \bmx{c} -1\\-3\\-1\emx$} 2 | {$\vx+\vy = \bmx{c}1\\1\\-2\emx$, $\vx-\vy = \bmx{c} 3\\7\\0\emx$ 3 | 4 | Sketches will vary slightly depending on orientation. 5 | 6 | \begin{tikzpicture}[>=latex,x={(-.53cm,-.53cm)},y={(.75cm,0)},z={(0,.75cm)}] 7 | \drawxaxis{-1}{3}{1}{2}; 8 | \drawyaxis{-3}{4}{-2}{3}; 9 | \drawzaxis{-2}{1}{-1}{-1}; 10 | \drawvect{2}{4}{-1}{dotted}{->,thick}{below right}{\vx}; 11 | \drawvect{-1}{-3}{-1}{dotted}{->,thick}{below left}{\vy}; 12 | \drawvect{1}{1}{-2}{dotted}{->,thick}{below}{$\vx+\vy$}; 13 | \drawvect{3}{7}{0}{dotted}{->,thick}{below }{$\vx-\vy$}; 14 | \end{tikzpicture} 15 | } -------------------------------------------------------------------------------- /exercises/05_02_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$\vx = \bmx{c}1\\1\\2\emx$, $\vy = \bmx{c} 3\\3\\6\emx$} 2 | {$\vx+\vy = \bmx{c}4\\4\\8\emx$, $\vx-\vy = \bmx{c} -2\\-2\\-4\emx$ 3 | 4 | Sketches will vary slightly depending on orientation. 5 | 6 | \begin{tikzpicture}[>=latex,x={(-.53cm,-.53cm)},y={(.75cm,0)},z={(0,.75cm)}] 7 | \drawxaxis{-2}{4}{-1}{3}; 8 | \drawyaxis{-2}{2.5}{-1}{2}; 9 | \drawzaxis{-2}{4}{-1}{3}; 10 | \drawvect{1}{1}{2}{dotted}{->,thick}{right}{\vx}; 11 | \drawvect{3}{3}{6}{dotted}{->,thick}{left}{\vy}; 12 | \drawvect{4}{4}{8}{dotted}{->,thick}{right}{$\vx+\vy$}; 13 | \drawvect{-2}{-2}{-4}{dotted}{->,thick}{left}{$\vx-\vy$}; 14 | \end{tikzpicture} 15 | } -------------------------------------------------------------------------------- /exercises/05_02_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$\vx = \bmx{c}0\\1\\1\emx$, $\vy = \bmx{c} 0\\-1\\1\emx$} 2 | {$\vx+\vy = \bmx{c}0\\0\\2\emx$, $\vx-\vy = \bmx{c} 0\\2\\0\emx$ 3 | 4 | Sketches may vary slightly. 5 | 6 | \begin{tikzpicture}[>=latex,x={(-.53cm,-.53cm)},y={(.75cm,0)},z={(0,.75cm)}] 7 | \drawxaxis{0}{2}{1}{1}; 8 | \drawyaxis{-1}{2}{1}{1}; 9 | \drawzaxis{0}{2}{1}{1}; 10 | \drawvect{0}{1}{1}{dotted}{->,thick}{above right}{\vx}; 11 | \drawvect{0}{-1}{1}{dotted}{->,thick}{above left}{\vy}; 12 | \drawvect{0}{0}{2}{dotted}{->,thick}{right}{$\vx+\vy$}; 13 | \drawvect{0}{2}{0}{dotted}{->,thick}{below}{$\vx-\vy$}; 14 | \end{tikzpicture} 15 | } -------------------------------------------------------------------------------- /exercises/05_02_ex_05.tex: -------------------------------------------------------------------------------- 1 | {\begin{tikzpicture}[>=latex,x={(-.53cm,-.53cm)},y={(.75cm,0)},z={(0,.75cm)},baseline=10pt] 2 | \drawxaxis{0}{2}{1}{1}; 3 | \drawyaxis{0}{2}{1}{1}; 4 | \drawzaxis{0}{2}{1}{1}; 5 | \drawjustvect{0}{2}{3}{}{->,thick}{above}{\vx}; 6 | \drawjustvect{2}{-1}{1}{}{->,thick}{above}{\vy}; 7 | \end{tikzpicture} 8 | } 9 | {Sketches may vary slightly. 10 | 11 | \begin{tikzpicture}[>=latex,x={(-.53cm,-.53cm)},y={(.75cm,0)},z={(0,.75cm)},scale=.75] 12 | \drawxaxis{0}{2}{1}{1}; 13 | \drawyaxis{0}{2}{1}{1}; 14 | \drawzaxis{0}{2}{1}{1}; 15 | \drawjustvect{0}{4}{6}{}{->,thick}{above}{2\vx}; 16 | \drawjustvect{-2}{1}{-1}{}{->,thick}{above}{-\vy}; 17 | \drawjustvect{2}{1}{4}{}{->,thick}{above}{$\vx+\vy$}; 18 | \drawjustvect{-2}{3}{2}{}{->,thick}{above}{$\vx-\vy$}; 19 | \end{tikzpicture} 20 | } -------------------------------------------------------------------------------- /exercises/05_02_ex_06.tex: -------------------------------------------------------------------------------- 1 | {\begin{tikzpicture}[>=latex,x={(-.53cm,-.53cm)},y={(.75cm,0)},z={(0,.75cm)},baseline=10pt] 2 | \drawxaxis{0}{2}{1}{1}; 3 | \drawyaxis{0}{2}{1}{1}; 4 | \drawzaxis{0}{2}{1}{1}; 5 | \drawjustvect{1}{2}{-1}{}{->,thick}{below}{\vx}; 6 | \drawjustvect{0}{3}{2}{}{->,thick}{above}{\vy}; 7 | \end{tikzpicture} 8 | } 9 | {Sketches may vary slightly. 10 | 11 | \begin{tikzpicture}[>=latex,x={(-.53cm,-.53cm)},y={(.75cm,0)},z={(0,.75cm)}] 12 | \drawxaxis{0}{2}{1}{1}; 13 | \drawyaxis{0}{2}{1}{1}; 14 | \drawzaxis{0}{2}{1}{1}; 15 | \drawjustvect{2}{4}{-2}{}{->,thick}{below}{2\vx}; 16 | \drawjustvect{0}{-3}{-2}{}{->,thick}{above}{-\vy}; 17 | \drawjustvect{1}{5}{1}{}{->,thick}{above}{$\vx+\vy$}; 18 | \drawjustvect{1}{-1}{-3}{}{->,thick}{below}{$\vx-\vy$}; 19 | \end{tikzpicture} 20 | } -------------------------------------------------------------------------------- /exercises/05_02_ex_07.tex: -------------------------------------------------------------------------------- 1 | {\begin{tikzpicture}[>=latex,x={(-.53cm,-.53cm)},y={(.75cm,0)},z={(0,.75cm)},baseline=10pt] 2 | \drawxaxis{0}{2}{1}{1}; 3 | \drawyaxis{0}{2}{1}{1}; 4 | \drawzaxis{0}{2}{1}{1}; 5 | \drawjustvect{2}{2}{0}{}{->,thick}{below}{\vx}; 6 | \drawjustvect{2}{0}{2}{}{->,thick}{above}{\vy}; 7 | \end{tikzpicture} 8 | } 9 | {Sketches may vary slightly. 10 | 11 | \begin{tikzpicture}[>=latex,x={(-.53cm,-.53cm)},y={(.75cm,0)},z={(0,.75cm)}] 12 | \drawxaxis{0}{2}{1}{1}; 13 | \drawyaxis{0}{2}{1}{1}; 14 | \drawzaxis{0}{2}{1}{1}; 15 | \drawjustvect{4}{4}{0}{}{->,thick}{below}{2\vx}; 16 | \drawjustvect{-2}{0}{-2}{}{->,thick}{above}{-\vy}; 17 | \drawjustvect{4}{2}{2}{}{->,thick}{above}{$\vx+\vy$}; 18 | \drawjustvect{0}{2}{-2}{}{->,thick}{right}{$\vx-\vy$}; 19 | \end{tikzpicture} 20 | } -------------------------------------------------------------------------------- /exercises/05_02_ex_08.tex: -------------------------------------------------------------------------------- 1 | {\begin{tikzpicture}[>=latex,x={(-.53cm,-.53cm)},y={(.75cm,0)},z={(0,.75cm)},baseline=10pt] 2 | \drawxaxis{0}{2}{1}{1}; 3 | \drawyaxis{0}{2}{1}{1}; 4 | \drawzaxis{0}{2}{1}{1}; 5 | \drawjustvect{0}{1.5}{0}{}{->,thick}{above}{\vx}; 6 | \drawjustvect{0}{0}{1.5}{}{->,thick}{right}{\vy}; 7 | \end{tikzpicture} 8 | } 9 | {Sketches may vary slightly. 10 | 11 | \begin{tikzpicture}[>=latex,x={(-.53cm,-.53cm)},y={(.75cm,0)},z={(0,.75cm)}] 12 | \drawxaxis{0}{2}{1}{1}; 13 | \drawyaxis{0}{2}{1}{1}; 14 | \drawzaxis{0}{2}{1}{1}; 15 | \drawjustvect{0}{3}{0}{}{->,thick}{above}{2\vx}; 16 | \drawjustvect{0}{0}{-1.5}{}{->,thick}{below}{-\vy}; 17 | \drawjustvect{0}{1.5}{1.5}{}{->,thick}{above}{$\vx+\vy$}; 18 | \drawjustvect{0}{1.5}{-1.5}{}{->,thick}{below}{$\vx-\vy$}; 19 | \end{tikzpicture} 20 | } -------------------------------------------------------------------------------- /exercises/05_02_ex_09.tex: -------------------------------------------------------------------------------- 1 | {$\vx = \bmx{c} 1\\-2\\5\emx$, $a = 2$ 2 | } 3 | {$||\vx|| = \sqrt{30}$, $||a\vx|| = \sqrt{120} = 2\sqrt{30}$ 4 | } -------------------------------------------------------------------------------- /exercises/05_02_ex_10.tex: -------------------------------------------------------------------------------- 1 | {$\vx = \bmx{c} -3\\4\\3\emx$, $a = -1$ 2 | } 3 | {$||\vx|| = \sqrt{34}$, $||a\vx|| = \sqrt{34}$ 4 | } -------------------------------------------------------------------------------- /exercises/05_02_ex_11.tex: -------------------------------------------------------------------------------- 1 | {$\vx = \bmx{c} 7\\2\\1\emx$, $a = 5$ 2 | } 3 | {$||\vx|| = \sqrt{54} = 3\sqrt{6}$, $||a\vx|| = \sqrt{270} = 15\sqrt{6}$ 4 | } -------------------------------------------------------------------------------- /exercises/05_02_ex_12.tex: -------------------------------------------------------------------------------- 1 | {$\vx = \bmx{c} 1\\2\\-2\emx$, $a = 3$ 2 | } 3 | {$||\vx|| = 3$, $||a\vx|| = 27$ 4 | } -------------------------------------------------------------------------------- /exercises/05_02_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/05_02_exset_01} 2 | \exsetinput{exercises/05_02_exset_02} 3 | \exsetinput{exercises/05_02_exset_03} 4 | 5 | -------------------------------------------------------------------------------- /exercises/05_02_exset_01.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, vectors \vx\ and \vy\ are given. Sketch \vx, \vy, $\vx+\vy$, and $\vx-\vy$ on the same Cartesian axes.} 3 | \exinput{exercises/05_02_ex_01} 4 | \exinput{exercises/05_02_ex_02} 5 | \exinput{exercises/05_02_ex_03} 6 | \exinput{exercises/05_02_ex_04} -------------------------------------------------------------------------------- /exercises/05_02_exset_02.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, vectors \vx\ and \vy\ are drawn. Sketch $2\vx$, $-\vy$, $\vx+\vy$, and $\vx-\vy$ on the same Cartesian axes.} 3 | \exinput{exercises/05_02_ex_05} 4 | \exinput{exercises/05_02_ex_06} 5 | \exinput{exercises/05_02_ex_07} 6 | \exinput{exercises/05_02_ex_08} -------------------------------------------------------------------------------- /exercises/05_02_exset_03.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, a vector \vx\ and a scalar $a$ are given. Using Definition \ref{def:3D_length}, compute the lengths of \vx\ and $a\vx$, then compare these lengths.} 3 | \exinput{exercises/05_02_ex_09} 4 | \exinput{exercises/05_02_ex_10} 5 | \exinput{exercises/05_02_ex_11} 6 | \exinput{exercises/05_02_ex_12} 7 | -------------------------------------------------------------------------------- /exercises/05_03_ex_01.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{cc} 1 & -1\\ 2 & 3\emx$} 2 | { 3 | \begin{tikzpicture}[>=latex,x={(.5cm,0)},y={(0,.5cm)}] 4 | \drawxlines{-3.5}{1.5}{-3,-2,-1,1}; 5 | \drawylines{-.5}{5.5}{1,...,5}; 6 | \draw[thick,->] (0,0) -- (1,1) node [right] {$\vx$}; 7 | \draw[thick,->] (0,0) -- (-1,2) node [above] {$\vy$}; 8 | \draw[thick,->] (0,0) -- (0,5) node [right] {$\tta\vx$}; 9 | \draw[thick,->] (0,0) -- (-3,4) node [above] {$\tta\vy$}; 10 | \end{tikzpicture} 11 | } -------------------------------------------------------------------------------- /exercises/05_03_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{cc} 2&0\\ -1 & 3\emx$} 2 | { 3 | \begin{tikzpicture}[>=latex,x={(.5cm,0)},y={(0,.5cm)}] 4 | \drawxlines{-2.5}{2.5}{-2,-1,1,2}; 5 | \drawylines{-.5}{7.5}{1,...,7}; 6 | \draw[thick,->] (0,0) -- (1,1) node [right] {$\vx$}; 7 | \draw[thick,->] (0,0) -- (-1,2) node [above] {$\vy$}; 8 | \draw[thick,->] (0,0) -- (2,2) node [right] {$\tta\vx$}; 9 | \draw[thick,->] (0,0) -- (-2,7) node [above] {$\tta\vy$}; 10 | \end{tikzpicture} 11 | } -------------------------------------------------------------------------------- /exercises/05_03_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{cc} 1&1\\ 1&1\emx$} 2 | { 3 | \begin{tikzpicture}[>=latex,x={(.5cm,0)},y={(0,.5cm)}] 4 | \drawxlines{-1.5}{2.5}{-1,1,2}; 5 | \drawylines{-.5}{2.5}{1,...,2}; 6 | \draw[thick,->] (0,0) -- (1,1) node [right] {$\vx$}; 7 | \draw[thick,->] (0,0) -- (-1,2) node [above] {$\vy$}; 8 | \draw[thick,->] (0,0) -- (2,2) node [right] {$\tta\vx$}; 9 | \draw[thick,->] (0,0) -- (1,1) node [above] {$\tta\vy$}; 10 | \end{tikzpicture} 11 | } -------------------------------------------------------------------------------- /exercises/05_03_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$\tta = \bmx{cc} 1&2\\ -1&-2\emx$} 2 | { 3 | \begin{tikzpicture}[>=latex,x={(.5cm,0)},y={(0,.5cm)}] 4 | \drawxlines{-1.5}{3.5}{-1,1,2,3}; 5 | \drawylines{-3.5}{2.5}{-3,...,2}; 6 | \draw[thick,->] (0,0) -- (1,1) node [right] {$\vx$}; 7 | \draw[thick,->] (0,0) -- (-1,2) node [above] {$\vy$}; 8 | \draw[thick,->] (0,0) -- (3,-3) node [right] {$\tta\vx$}; 9 | \draw[thick,->] (0,0) -- (3,-3) node [below] {$\tta\vy$}; 10 | \end{tikzpicture} 11 | } -------------------------------------------------------------------------------- /exercises/05_03_ex_05.tex: -------------------------------------------------------------------------------- 1 | {\begin{tikzpicture}[>=latex,x={(.5cm,0)},y={(0,.5cm)},baseline=10pt] 2 | \drawxlines{-1}{3.5}{1,2,3}; 3 | \drawylines{-1}{7.5}{1,...,7}; 4 | \node at (1,-.1) [below] {$1$}; 5 | \node at (0,1) [left] {$1$}; 6 | \unitsquare[thick,cm={1,3,2,4,(0,0)}]; 7 | \end{tikzpicture} 8 | } 9 | { 10 | $\tta = \bmx{cc}1&2\\3&4\emx$ 11 | } -------------------------------------------------------------------------------- /exercises/05_03_ex_06.tex: -------------------------------------------------------------------------------- 1 | {\begin{tikzpicture}[>=latex,x={(.5cm,0)},y={(0,.5cm)},baseline=10pt] 2 | \drawxlines{-1.5}{2.5}{1,2}; 3 | \drawylines{-1}{3.5}{1,...,3}; 4 | \node at (1,-.1) [below] {$1$}; 5 | \node at (0,1) [left] {$1$}; 6 | \unitsquare[thick,cm={-1,1,2,2,(0,0)}]; 7 | \end{tikzpicture} 8 | } 9 | { 10 | $\tta = \bmx{cc}-1&2\\1&2\emx$ 11 | } -------------------------------------------------------------------------------- /exercises/05_03_ex_07.tex: -------------------------------------------------------------------------------- 1 | {\begin{tikzpicture}[>=latex,x={(.5cm,0)},y={(0,.5cm)},baseline=10pt] 2 | \drawxlines{-1.5}{2.5}{1,2}; 3 | \drawylines{-1}{3.5}{1,...,3}; 4 | \node at (1,-.1) [below] {$1$}; 5 | \node at (0,1) [left] {$1$}; 6 | \unitsquare[thick,cm={1,1,2,2,(0,0)}]; 7 | \end{tikzpicture} 8 | } 9 | { 10 | $\tta = \bmx{cc}1&2\\1&2\emx$ 11 | } -------------------------------------------------------------------------------- /exercises/05_03_ex_08.tex: -------------------------------------------------------------------------------- 1 | {\begin{tikzpicture}[>=latex,x={(.5cm,0)},y={(0,.5cm)},baseline=10pt] 2 | \drawxlines{-1.5}{2.5}{-1,1,2}; 3 | \drawylines{-1}{1.5}{1}; 4 | \node at (1,-.1) [below] {$1$}; 5 | \node at (0,1) [left] {$1$}; 6 | \unitsquare[thick,cm={2,0,-1,0,(0,0)}]; 7 | \end{tikzpicture} 8 | } 9 | { 10 | $\tta = \bmx{cc}2&-1\\0&0\emx$ 11 | } -------------------------------------------------------------------------------- /exercises/05_03_ex_09.tex: -------------------------------------------------------------------------------- 1 | {\begin{enumerate} 2 | \item vertical shear by a factor of 2 3 | \item horizontal shear by a factor of 2 4 | \end{enumerate} 5 | } 6 | { 7 | $\tta = \bmx{cc}5&2\\2&1\emx$ 8 | } -------------------------------------------------------------------------------- /exercises/05_03_ex_10.tex: -------------------------------------------------------------------------------- 1 | {\begin{enumerate} 2 | \item horizontal shear by a factor of 2 3 | \item vertical shear by a factor of 2 4 | 5 | \end{enumerate} 6 | } 7 | { 8 | $\tta = \bmx{cc}1&2\\2&5\emx$ 9 | } -------------------------------------------------------------------------------- /exercises/05_03_ex_11.tex: -------------------------------------------------------------------------------- 1 | {\begin{enumerate} 2 | \item horizontal stretch by a factor of 3 3 | \item reflection across the line $y=x$ 4 | 5 | \end{enumerate} 6 | } 7 | { 8 | $\tta = \bmx{cc}0&1\\3&0\emx$ 9 | } -------------------------------------------------------------------------------- /exercises/05_03_ex_12.tex: -------------------------------------------------------------------------------- 1 | {\begin{enumerate} 2 | \item counterclockwise rotation by an angle of $45^\circ$ 3 | \item vertical stretch by a factor of $1/2$ 4 | 5 | \end{enumerate} 6 | } 7 | { 8 | $\tta = \bmx{cc}0.707&-0.707\\0.354&0.354\emx$ 9 | } -------------------------------------------------------------------------------- /exercises/05_03_ex_13.tex: -------------------------------------------------------------------------------- 1 | {\begin{enumerate} 2 | \item clockwise rotation by an angle of $90^\circ$ 3 | \item horizontal reflection across the $y$ axis 4 | \item vertical shear by a factor of 1 5 | \end{enumerate} 6 | } 7 | { 8 | $\tta = \bmx{cc}0&-1\\ -1&-1\emx$ 9 | } -------------------------------------------------------------------------------- /exercises/05_03_ex_14.tex: -------------------------------------------------------------------------------- 1 | {\begin{enumerate} 2 | \item vertical reflection across the $x$ axis 3 | \item horizontal reflection across the $y$ axis 4 | \item diagonal reflection across the line $y=x$ 5 | \end{enumerate} 6 | } 7 | { 8 | $\tta =\bmx{cc} 0&-1 \\ -1&0 \emx$. 9 | } -------------------------------------------------------------------------------- /exercises/05_03_ex_15.tex: -------------------------------------------------------------------------------- 1 | {\begin{enumerate} 2 | \item a horizontal reflection across the $y$ axis, followed by a vertical reflection across the $x$ axis, compared to 3 | \item a counterclockise rotation of $180^\circ$ 4 | \end{enumerate} 5 | } 6 | { 7 | Yes, these are the same; the transformation matrix in each is $\bmx{cc} -1 & 0 \\0&-1\emx$. 8 | } -------------------------------------------------------------------------------- /exercises/05_03_ex_16.tex: -------------------------------------------------------------------------------- 1 | {\begin{enumerate} 2 | \item a horizontal stretch by a factor of 2 followed by a reflection across the line $y=x$, compared to 3 | \item a vertical stretch by a factor of 2 4 | \end{enumerate} 5 | } 6 | { 7 | No, these are different. The first produces a transformation matrix $\bmx{cc} 0 & 1\\ 2 &0\emx$, while the second produces $\bmx{cc}1 & 0 \\ 0 & 2\emx$. 8 | } -------------------------------------------------------------------------------- /exercises/05_03_ex_17.tex: -------------------------------------------------------------------------------- 1 | {\begin{enumerate} 2 | \item a horizontal stretch by a factor of 1/2 followed by a vertical stretch by a factor of 3, compared to 3 | \item the same operations but in opposite order 4 | \end{enumerate} 5 | } 6 | { 7 | Yes, these are the same. Each produces the transformation matrix $\bmx{cc} 1/2 & 0 \\ 0 & 3\emx$. 8 | } -------------------------------------------------------------------------------- /exercises/05_03_ex_18.tex: -------------------------------------------------------------------------------- 1 | {\begin{enumerate} 2 | \item a reflection across the line $y=x$ followed by a reflection across the $x$ axis, compared to 3 | \item a reflection across the the $y$ axis, followed by a reflection across the line $y=x$. 4 | \end{enumerate} 5 | } 6 | { 7 | Yes, these are the same. Each produces the transformation matrix $\bmx{cc} 0 & 1 \\ -1 & 0\emx$. 8 | } -------------------------------------------------------------------------------- /exercises/05_03_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/05_03_exset_02} 2 | \exsetinput{exercises/05_03_exset_03} 3 | \exsetinput{exercises/05_03_exset_04} 4 | -------------------------------------------------------------------------------- /exercises/05_03_exset_01.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, a matrix \tta\ is given. Sketch \vx, \vy, \tta\vx\ and \tta\vy\ on the same Cartesian axes, where 3 | $$\vx = \bmx{c}1\\1\emx \ \text{ and } \ \vy=\bmx{c}-1\\2\emx.$$} 4 | \exinput{exercises/05_03_ex_01} 5 | \exinput{exercises/05_03_ex_02} 6 | \exinput{exercises/05_03_ex_03} 7 | \exinput{exercises/05_03_ex_04} -------------------------------------------------------------------------------- /exercises/05_03_exset_02.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, a sketch of transformed unit square is given. Find the matrix \tta\ that performs this transformation.} 3 | \exinput{exercises/05_03_ex_05} 4 | \exinput{exercises/05_03_ex_06} 5 | \exinput{exercises/05_03_ex_07} 6 | \exinput{exercises/05_03_ex_08} -------------------------------------------------------------------------------- /exercises/05_03_exset_03.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, a list of transformations is given. Find the matrix \tta\ that performs those transformations, in order, on the Cartesian plane.} 3 | \exinput{exercises/05_03_ex_09} 4 | \exinput{exercises/05_03_ex_10} 5 | \exinput{exercises/05_03_ex_11} 6 | \exinput{exercises/05_03_ex_12} 7 | \exinput{exercises/05_03_ex_13} 8 | \exinput{exercises/05_03_ex_14} -------------------------------------------------------------------------------- /exercises/05_03_exset_04.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, two sets of transformations are given. Sketch the transformed unit square under each set of transformations. Are the transformations the same? Explain why/why not.} 3 | \exinput{exercises/05_03_ex_15} 4 | \exinput{exercises/05_03_ex_16} 5 | \exinput{exercises/05_03_ex_17} 6 | \exinput{exercises/05_03_ex_18} -------------------------------------------------------------------------------- /exercises/05_04_ex_01.tex: -------------------------------------------------------------------------------- 1 | {$T\left(\bmx{c}x_1\\x_2\emx\right) = \bmx{c}x_1+x_2\\3x_1-x_2\emx$} 2 | {Yes} -------------------------------------------------------------------------------- /exercises/05_04_ex_02.tex: -------------------------------------------------------------------------------- 1 | {$T\left(\bmx{c}x_1\\x_2\emx\right) = \bmx{c}x_1+x_2^2\\x_1-x_2\emx$} 2 | {No; cannot have a squared term.} -------------------------------------------------------------------------------- /exercises/05_04_ex_03.tex: -------------------------------------------------------------------------------- 1 | {$T\left(\bmx{c}x_1\\x_2\emx\right) = \bmx{c}x_1+1\\x_2+1\emx$} 2 | {No; cannot add a constant.} -------------------------------------------------------------------------------- /exercises/05_04_ex_04.tex: -------------------------------------------------------------------------------- 1 | {$T\left(\bmx{c}x_1\\x_2\emx\right) = \bmx{c}1\\1\emx$} 2 | {No; cannot add a constant.} -------------------------------------------------------------------------------- /exercises/05_04_ex_05.tex: -------------------------------------------------------------------------------- 1 | {$T\left(\bmx{c}x_1\\x_2\emx\right) = \bmx{c}0\\0\emx$} 2 | {Yes.} -------------------------------------------------------------------------------- /exercises/05_04_ex_06.tex: -------------------------------------------------------------------------------- 1 | {$T\left(\bmx{c}x_1\\x_2\emx\right) = \bmx{c}x_1+x_2\\x_1-x_2\emx$} 2 | {$[\, T\, ] = \bmx{cc} 1&1\\1&-1\emx$} -------------------------------------------------------------------------------- /exercises/05_04_ex_07.tex: -------------------------------------------------------------------------------- 1 | {$T\left(\bmx{c}x_1\\x_2\emx\right) = \bmx{c}x_1+2x_2\\ 3x_1-5x_2\\2x_2\emx$} 2 | {$[\, T\, ] = \bmx{cc} 1&2\\3&-5\\0&2\emx$} -------------------------------------------------------------------------------- /exercises/05_04_ex_08.tex: -------------------------------------------------------------------------------- 1 | {$T\left(\bmx{c}x_1\\x_2\\x_3\emx\right) = \bmx{c}x_1+2x_2-3x_3\\ 0\\ x_1+4x_3\\5x_2+x_3\emx$} 2 | {$[\, T\, ] = \bmx{ccc} 1&2&-3\\0&0&0\\1&0&4\\0&5&1\emx$} -------------------------------------------------------------------------------- /exercises/05_04_ex_09.tex: -------------------------------------------------------------------------------- 1 | {$T\left(\bmx{c}x_1\\x_2\\x_3\emx\right) = \bmx{c}x_1+3x_3\\ x_1-x_3\\ x_1+x_3\emx$} 2 | {$[\, T\, ] = \bmx{ccc} 1&0&3\\1&0&-1\\1&0&1\emx$} -------------------------------------------------------------------------------- /exercises/05_04_ex_10.tex: -------------------------------------------------------------------------------- 1 | {$T\left(\bmx{c}x_1\\x_2\emx\right) = \bmx{c}0\\0\emx$} 2 | {$[\, T\, ] = \bmx{cc} 0&0\\0&0\emx$} -------------------------------------------------------------------------------- /exercises/05_04_ex_11.tex: -------------------------------------------------------------------------------- 1 | {$T\left(\bmx{c}x_1\\x_2\\x_3\\x_4\emx\right) = \bmx{c}x_1+2x_2+3x_3+4x_4\emx$} 2 | {$[\, T\, ] = \bmx{cccc} 1&2&3&4\emx$} -------------------------------------------------------------------------------- /exercises/05_04_exercises.tex: -------------------------------------------------------------------------------- 1 | \exsetinput{exercises/05_04_exset_01} 2 | \exsetinput{exercises/05_04_exset_02} 3 | -------------------------------------------------------------------------------- /exercises/05_04_exset_01.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, a transformation $T$ is given. Determine whether or not $T$ is linear; if not, state why not.} 3 | \exinput{exercises/05_04_ex_01} 4 | \exinput{exercises/05_04_ex_02} 5 | \exinput{exercises/05_04_ex_03} 6 | \exinput{exercises/05_04_ex_04} 7 | \exinput{exercises/05_04_ex_05} -------------------------------------------------------------------------------- /exercises/05_04_exset_02.tex: -------------------------------------------------------------------------------- 1 | {\noin In Exercises} 2 | {, a linear transformation $T$ is given. Find $[\, T\, ]$.} 3 | \exinput{exercises/05_04_ex_06} 4 | \exinput{exercises/05_04_ex_07} 5 | \exinput{exercises/05_04_ex_08} 6 | \exinput{exercises/05_04_ex_09} 7 | \exinput{exercises/05_04_ex_10} 8 | \exinput{exercises/05_04_ex_11} 9 | -------------------------------------------------------------------------------- /exercises/titles.txt: -------------------------------------------------------------------------------- 1 | Volume in drive C has no label. 2 | Volume Serial Number is 0041-BA85 3 | 4 | Directory of C:\ 5 | 6 | -------------------------------------------------------------------------------- /exercises/troy1.tex: -------------------------------------------------------------------------------- 1 | {Which of the following are valid and useable variable names? Explain your answers 2 | 3 | \begin{enumerate} 4 | \item[a.] Homework 1 5 | \item[b.] 1Homework 6 | \item[c.] Homework\#1 7 | \item[d.] Homework\_1 8 | \item[e.] HoMeWoRkNuMbEr1 9 | \end{enumerate} 10 | } 11 | {} -------------------------------------------------------------------------------- /fundamentals_format.tex: -------------------------------------------------------------------------------- 1 | \input{headers/Matrix_Algebra_Text_Header.tex} -------------------------------------------------------------------------------- /headers/APEX_Format_Examples.answers: -------------------------------------------------------------------------------- 1 | \small \raggedright \begin {multicols}{2} 2 | -------------------------------------------------------------------------------- /headers/APEX_Format_Examples.idx: -------------------------------------------------------------------------------- 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-------------------------------------------------------------------------------- 1 | \BOOKMARK [1][-]{section.1}{1 Introduction}{}% 1 2 | \BOOKMARK [1][-]{section.2}{2 Writing Exercises}{}% 2 3 | \BOOKMARK [1][-]{section.3}{3 An Environment For Creating Examples}{}% 3 4 | \BOOKMARK [2][-]{subsection.3.1}{3.1 Basic Principles}{section.3}% 4 5 | \BOOKMARK [2][-]{subsection.3.2}{3.2 Options}{section.3}% 5 6 | -------------------------------------------------------------------------------- /headers/APEX_Format_Manual.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/APEXCalculus/Fundamentals-of-Matrix-Algebra/6f2d536d1259395a5bff80ae95623b0719d8d455/headers/APEX_Format_Manual.pdf -------------------------------------------------------------------------------- /headers/APEX_Format_Manual.toc: -------------------------------------------------------------------------------- 1 | \contentsline {section}{\numberline {1}Introduction}{1}{section.1} 2 | \contentsline {section}{\numberline {2}Writing Exercises}{1}{section.2} 3 | \contentsline {section}{\numberline {3}An Environment For Creating Examples}{3}{section.3} 4 | \contentsline {subsection}{\numberline {3.1}Basic Principles}{3}{subsection.3.1} 5 | \contentsline {subsection}{\numberline {3.2}Options}{4}{subsection.3.2} 6 | -------------------------------------------------------------------------------- /headers/Header_APEX.idx: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/APEXCalculus/Fundamentals-of-Matrix-Algebra/6f2d536d1259395a5bff80ae95623b0719d8d455/headers/Header_APEX.idx -------------------------------------------------------------------------------- /text/by-nc.pdf: -------------------------------------------------------------------------------- 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\noindent\begin{minipage}{2in} 3 | \includegraphics{text/by-nc} 4 | \end{minipage} 5 | \begin{minipage}{3in} 6 | \noindent Copyright \copyright\ 2011 Gregory Hartman 7 | 8 | Licensed to the public under Creative Commons Attribution-Noncommercial 3.0 United States License 9 | \end{minipage} 10 | 11 | \vspace{\stretch{1}} -------------------------------------------------------------------------------- /text/title_page.tex: -------------------------------------------------------------------------------- 1 | \vspace*{\stretch{.3}} 2 | \begin{flushright} 3 | 4 | \textsc{\huge Fundamentals of Matrix Algebra} \\ 5 | 6 | \textsl{Third Edition}, 7 | {\small Version 3.1110}\\ 8 | 9 | \Large 10 | \vspace{1in} 11 | 12 | Gregory Hartman, Ph.D.\\ 13 | 14 | \emph{\small Department of Mathematics and Computer Science}\\ 15 | 16 | \emph{\small Virginia Military Institute} 17 | 18 | \normalsize 19 | \end{flushright} 20 | 21 | \vspace{\stretch{1}} 22 | 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