├── cover6.png ├── The Summary of Linear Algebra and the Analytic Geometry.pdf ├── README.md ├── .gitignore ├── LICENSE └── The Summary of Linear Algebra and the Analytic Geometry.tex /cover6.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/ShevonKuan/The-Summary-of-Linear-Algebra-and-the-Analytic-Geometry/HEAD/cover6.png -------------------------------------------------------------------------------- /The Summary of Linear Algebra and the Analytic Geometry.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/ShevonKuan/The-Summary-of-Linear-Algebra-and-the-Analytic-Geometry/HEAD/The Summary of Linear Algebra and the Analytic Geometry.pdf -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # 线性代数与解析几何笔记 2 | #### Shevon Kuan 3 | ## 简介 4 | 5 | 这是我基于 Latex2$\varepsilon$ 的数学笔记,它是一个系列并将使用一致的模板(目前仅部分开源),将来我可能会对该模板作独立开源处理.这个笔记基于华南理工大学的线性代数课本进行总结.目前由于期末考前只完成了笔者最需要复习的部分,剩余部分仍未完工,并且目前暂不考虑完工,有意加入并希望帮忙完善的的请发邮件[联系我][contact] 6 | 7 | [contact]: mailto:331749486@qq.com 8 | 9 | **Demo:** 10 | ![element](https://github.com/ShevonKuan/Probability-and-Mathematical-Statistics/raw/master/readme_image/main.png) 11 | 12 | 13 | ---- 14 | 15 | ## 编译介绍 16 | ```Latex``` 编译链```XeLaTex -> bibtex -> makeindex -> texindy -> xeLaTex重复编译```对应```VS Code```设置如下: 17 | ```json 18 | "latex-workshop.latex.recipes": [ 19 | { 20 | "name": "完整编译链", 21 | "tools": [ 22 | "xelatex", 23 | "bibtex", 24 | "makeindex", 25 | "texindy", 26 | "xelatex", 27 | "xelatex" 28 | ] 29 | }, 30 | ] 31 | ``` 32 | 各命令具体配置如下： 33 | ```json 34 | "latex-workshop.latex.tools": [ 35 | { 36 | // 编译工具和命令 37 | "name": "xelatex", 38 | "command": "xelatex", 39 | "args": [ 40 | "-synctex=1", 41 | "-interaction=nonstopmode", 42 | "%DOCFILE%" 43 | ] 44 | }, 45 | { 46 | "name": "bibtex", 47 | "command": "bibtex", 48 | "args": [ 49 | "%DOCFILE%" 50 | ] 51 | }, 52 | { 53 | "name": "texindy", 54 | "command": "texindy", 55 | "args": [ 56 | "%DOCFILE%.idx" 57 | ] 58 | }, 59 | { 60 | "name": "makeindex", 61 | "command": "makeindex", 62 | "args": [ 63 | "%DOCFILE%.nlo", 64 | "-s", 65 | "nomencl.ist", 66 | "-o", 67 | "%DOCFILE%.nls" 68 | ] 69 | } 70 | ], 71 | ``` 72 | ---- 73 | ## TODO: 74 | 目前不打算继续完善. 75 | 76 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | ## Core latex/pdflatex auxiliary files: 2 | *.aux 3 | *.lof 4 | *.log 5 | *.lot 6 | *.fls 7 | *.out 8 | *.toc 9 | *.fmt 10 | *.fot 11 | *.cb 12 | *.cb2 13 | .*.lb 14 | 15 | ## Intermediate documents: 16 | *.dvi 17 | *.xdv 18 | *-converted-to.* 19 | # these rules might exclude image files for figures etc. 20 | # *.ps 21 | # *.eps 22 | # *.pdf 23 | 24 | ## Generated if empty string is given at "Please type another file name for output:" 25 | .pdf 26 | 27 | ## Bibliography auxiliary files (bibtex/biblatex/biber): 28 | *.bbl 29 | *.bcf 30 | *.blg 31 | *-blx.aux 32 | *-blx.bib 33 | *.run.xml 34 | 35 | ## Build tool auxiliary files: 36 | *.fdb_latexmk 37 | *.synctex 38 | *.synctex(busy) 39 | *.synctex.gz 40 | *.synctex.gz(busy) 41 | *.pdfsync 42 | 43 | ## Build tool directories for auxiliary files 44 | # latexrun 45 | latex.out/ 46 | 47 | ## Auxiliary and intermediate files from other packages: 48 | # algorithms 49 | *.alg 50 | *.loa 51 | 52 | # achemso 53 | acs-*.bib 54 | 55 | # amsthm 56 | *.thm 57 | 58 | # beamer 59 | *.nav 60 | *.pre 61 | *.snm 62 | *.vrb 63 | 64 | # changes 65 | *.soc 66 | 67 | # comment 68 | *.cut 69 | 70 | # cprotect 71 | *.cpt 72 | 73 | # elsarticle (documentclass of Elsevier journals) 74 | *.spl 75 | 76 | # endnotes 77 | *.ent 78 | 79 | # fixme 80 | *.lox 81 | 82 | # feynmf/feynmp 83 | *.mf 84 | *.mp 85 | *.t[1-9] 86 | *.t[1-9][0-9] 87 | *.tfm 88 | 89 | #(r)(e)ledmac/(r)(e)ledpar 90 | *.end 91 | *.?end 92 | *.[1-9] 93 | *.[1-9][0-9] 94 | *.[1-9][0-9][0-9] 95 | *.[1-9]R 96 | *.[1-9][0-9]R 97 | *.[1-9][0-9][0-9]R 98 | *.eledsec[1-9] 99 | *.eledsec[1-9]R 100 | *.eledsec[1-9][0-9] 101 | *.eledsec[1-9][0-9]R 102 | *.eledsec[1-9][0-9][0-9] 103 | *.eledsec[1-9][0-9][0-9]R 104 | 105 | # glossaries 106 | *.acn 107 | *.acr 108 | *.glg 109 | *.glo 110 | *.gls 111 | *.glsdefs 112 | *.lzo 113 | *.lzs 114 | 115 | # uncomment this for glossaries-extra (will ignore makeindex's style files!) 116 | # *.ist 117 | 118 | # gnuplottex 119 | *-gnuplottex-* 120 | 121 | # gregoriotex 122 | *.gaux 123 | *.gtex 124 | 125 | # htlatex 126 | *.4ct 127 | *.4tc 128 | *.idv 129 | *.lg 130 | *.trc 131 | *.xref 132 | 133 | # hyperref 134 | *.brf 135 | 136 | # knitr 137 | *-concordance.tex 138 | # TODO Comment the next line if you want to keep your tikz graphics files 139 | *.tikz 140 | *-tikzDictionary 141 | 142 | # listings 143 | *.lol 144 | 145 | # luatexja-ruby 146 | *.ltjruby 147 | 148 | # makeidx 149 | *.idx 150 | *.ilg 151 | *.ind 152 | 153 | # minitoc 154 | *.maf 155 | *.mlf 156 | *.mlt 157 | *.mtc[0-9]* 158 | *.slf[0-9]* 159 | *.slt[0-9]* 160 | *.stc[0-9]* 161 | 162 | # minted 163 | _minted* 164 | *.pyg 165 | 166 | # morewrites 167 | *.mw 168 | 169 | # nomencl 170 | *.nlg 171 | *.nlo 172 | *.nls 173 | 174 | # pax 175 | *.pax 176 | 177 | # pdfpcnotes 178 | *.pdfpc 179 | 180 | # sagetex 181 | *.sagetex.sage 182 | *.sagetex.py 183 | *.sagetex.scmd 184 | 185 | # scrwfile 186 | *.wrt 187 | 188 | # sympy 189 | *.sout 190 | *.sympy 191 | sympy-plots-for-*.tex/ 192 | 193 | # pdfcomment 194 | *.upa 195 | *.upb 196 | 197 | # pythontex 198 | *.pytxcode 199 | pythontex-files-*/ 200 | 201 | # tcolorbox 202 | *.listing 203 | 204 | # thmtools 205 | *.loe 206 | 207 | # TikZ & PGF 208 | *.dpth 209 | *.md5 210 | *.auxlock 211 | 212 | # todonotes 213 | *.tdo 214 | 215 | # vhistory 216 | *.hst 217 | *.ver 218 | 219 | # easy-todo 220 | *.lod 221 | 222 | # xcolor 223 | *.xcp 224 | 225 | # xmpincl 226 | *.xmpi 227 | 228 | # xindy 229 | *.xdy 230 | 231 | # xypic precompiled matrices and outlines 232 | *.xyc 233 | *.xyd 234 | 235 | # endfloat 236 | *.ttt 237 | *.fff 238 | 239 | # Latexian 240 | TSWLatexianTemp* 241 | 242 | ## 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But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /The Summary of Linear Algebra and the Analytic Geometry.tex: -------------------------------------------------------------------------------- 1 | \documentclass[9pt,a4paper]{book} 2 | \usepackage{shevonNotebook} 3 | %标题配置————————————— 4 | \title{线性代数与解析几何\\ \large{ 重要知识点总结}\\ \small{The Summary of Linear Algebra and the Analytic Geometry}\\ 5 | \ \\ 6 | \begin{center} 7 | \includegraphics[width=10cm]{cover6.png} 8 | 9 | \end{center} 10 | } 11 | \author{关舒文\\\small{\kaishu{华南理工大学}}} 12 | \date{\small{Latest Update\ :\ \today}} 13 | %正文部分————————————— 14 | \begin{document} 15 | %目录与公式编号生成—————————— 16 | \numberwithin{equation}{section} 17 | \allowdisplaybreaks%强制自动换行 18 | \newgeometry{left=2cm,right=2cm,marginparwidth=0cm,marginparsep=0cm}%封面设置 19 | \pagenumbering{roman} 20 | \maketitle 21 | \thispagestyle{empty} 22 | %页面重新配置---------------------------- 23 | \restoregeometry 24 | {\printnomenclature 25 | \setcounter{page}{0}\pagenumbering{roman} 26 | \addcontentsline{toc}{chapter}{符号说明}} 27 | \newpage 28 | \pagenumbering{Roman} 29 | \setcounter{page}{0} 30 | \tableofcontents 31 | 32 | 33 | 34 | %正文开始—————————————可以使用\boldmath输入粗斜体与\unboldmath合用 35 | \chapter{行列式\\Determinant} 36 | \pagenumbering{arabic} 37 | \setcounter{page}{2} 38 | \section{二阶与三阶行列式} 39 | \begin{defination}[二阶行列式的定义] 40 | 令\[ D= \begin{vmatrix} 41 | a_{11}&a_{12}\\a_{21}&a_{22} 42 | \end{vmatrix}=a_{11}a_{22}-a_{12}a_{21} \]其中$ \begin{vmatrix} 43 | a_{11}&a_{12}\\a_{21}&a_{22} 44 | \end{vmatrix} $称为\textbf{二阶行列式}\index{EJHLS@二阶行列式},这里$ a_{11},a_{22} $所在的斜线称为二阶行列式的主对角线,相应的$ a_{11},a_{22} $称为\textbf{主对角线元素},而$ a_{12},a_{21} $称为\textbf{副对角线元素}.\index{ZDJXYS@主对角线元素}\index{FDJXYS@副对角线元素} 45 | \end{defination} 46 | \begin{defination}[三阶行列式的定义] 47 | 令\[D={\begin{vmatrix} 48 | {{a_{11}}}&{{a_{12}}}&{{a_{13}}} \\ 49 | {{a_{21}}}&{{a_{22}}}&{{a_{23}}} \\ 50 | {{a_{31}}}&{{a_{32}}}&{{a_{33}}} 51 | \end{vmatrix}} = {a_{11}}{a_{22}}{a_{33}} + {a_{12}}{a_{23}}{a_{31}} + {a_{13}}{a_{21}}{a_{32}} - {a_{11}}{a_{23}}{a_{32}} - {a_{12}}{a_{21}}{a_{33}} - {a_{13}}{a_{22}}{a_{31}},\]我们称之为\textbf{三阶行列式}.\index{SJHLS@三阶行列式} 52 | \end{defination} 53 | \section{$ n $阶排列及其逆序数,对换} 54 | \begin{defination}[排列的定义] 55 | 由自然数$ 1,2,\cdots,n $组成(不能重复)的任意一个$ n $元有序数组$ i_1i_2\cdots i_n $称为一个\textbf{$ n $阶排列},其中排列$ 12\cdots n $称为\textbf{自然排列}.\index{PL@排列}\index{ZRPL@自然排列} 56 | \end{defination} 57 | \begin{theorem} 58 | $ n $阶排列一共有$ n!=(n-1)(n-2)\cdots 3\cdot 2\cdot 1 $个. 59 | \end{theorem} 60 | \begin{defination} 61 | 在一个排列中,如果一个较大的数字排在了一个较小的数字之前,则称这两个数字构成一个\textbf{逆序}.否则,称这两个数字构成一个\textbf{顺序}.两个数字之间的关系只能是顺序或逆序.有以下定义 62 | \begin{enumerate} 63 | \item 在一个排列$ i_1i_2\cdots i_n $中,逆序的总数称为这个排列的\textbf{逆序数},记为$ \tau(i_1i_2\cdots i_n) $. 64 | \item 逆序数为奇数的排列称为\textbf{奇排列},逆序数为偶数的排列称为\textbf{偶排列}. 65 | \index{LX@逆序}\index{SX@顺序}\index{LXS@逆序数}\index{JPL@奇排列}\index{OPL@偶排列} 66 | \end{enumerate} 67 | 特别地,根据定义我们可得:\begin{enumerate} 68 | \item 逆序数最小的排列为自然排列,$ \tau(12\cdots n)=0 $; 69 | \item 逆序数最大的排列为反向自然排列,$ \displaystyle \tau\Bigl( n(n-1)\cdots321\Bigr) =\sum_{i=1}^{n-1}i=\frac{n(n-1)}{2} $. 70 | \end{enumerate} 71 | \end{defination} 72 | \begin{defination}[对换的定义] 73 | 把一个排列中的两个数字$ i,j $的位置互换而保持其余数字的位置不动,则称对这个排列施行了一次\textbf{对换}\index{DH@对换},记作$ (i,j) $.两个相邻位置数字的对换称为\textbf{不相邻对换},否则称为\textbf{一般对换}.\index{BXLDH@不相邻对换}\index{YBDH@一般对换} 74 | \end{defination} 75 | \begin{theorem} 76 | 对换具有可逆性. 77 | \end{theorem} 78 | \begin{theorem} 79 | 对换改变排列的奇偶性. 80 | \end{theorem} 81 | \begin{inference} 82 | 排列经过奇数次对换其奇偶性发生变化,经过偶数次对换其奇偶性不变. 83 | \end{inference} 84 | \begin{inference} 85 | 当$ n\geqslant 2 $时,在$ n $阶排列中,奇偶排列数目相等各有$ \dfrac{n!}{2} $个. 86 | \end{inference} 87 | \begin{theorem} 88 | 自然排列$ 12\cdots n $可以与任意$ n $阶排列$ i_1i_2\cdots i_n $经过一系列对换相互转换,且所作对换次数与排列$ i_1i_2\cdots i_n $具有相同的奇偶性. 89 | \end{theorem} 90 | \section{$ n $阶行列式的定义} 91 | \begin{defination}[$ n $阶行列式的定义] 92 | \[ D = {\begin{vmatrix} 93 | {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ 94 | {{a_{21}}}&{{a_{22}}}& \cdots &{{a_{2n}}}\\ 95 | \vdots & \vdots &\;& \vdots \\ 96 | {{a_{n1}}}&{{a_{n2}}}& \cdots &{{a_{nn}}} 97 | \end{vmatrix}} \]表示一个\textbf{$ n $阶行列式}.其中元素$ a_{ij}\in \mathbb{C}\quad(i,j=1,2,\cdots,n) $.\index{NJHLS@$ n $阶行列式} 98 | \end{defination} 99 | \begin{defination} 100 | $ n $阶行列式等于所有来自不同行不同列的$ n $个元素乘积的代数和.由于代数和的项数为$ n! $个,为了表达方便,我们可以将每项中的$ n $个元素按行指标有小及大的顺序排列,即写作$ a_{1j_1}a_{2j_2}\cdots a_{nj_n} $的形式,并规定当列指标$ j_1j_2\cdots j_n $是偶排列时,此项前面带正号,若是奇排列则前面带负号.这样,$ n $阶行列式被定义为\[ D = {\begin{vmatrix} 101 | {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ 102 | {{a_{21}}}&{{a_{22}}}& \cdots &{{a_{2n}}}\\ 103 | \vdots & \vdots &\;& \vdots \\ 104 | {{a_{n1}}}&{{a_{n2}}}& \cdots &{{a_{nn}}} 105 | \end{vmatrix}}=\sum_{j_1j_2\cdots j_n}{{(-1)}^{\tau (j_1j_2\cdots j_n)} a_{1j_1}a_{2j_2}\cdots a_{nj_n}}. \]其中$ \displaystyle \sum_{j_1j_2\cdots j_n} $表示对所有可能的$ n $阶排列求和.上式称为\textbf{行列式的展开式}.\index{HLSDZJKS@行列式的展开式} 106 | 107 | 上述$ n $阶行列式通常记为$ D=\mathrm{det}(a_{ij}) $或着$ |a_{ij}| $.当$ n=1 $时我们规定$ |a_{11}|=a_{11} $. 108 | 109 | 若行或列指标均不为自然排列,则我们有以下展开式:\[ D = {\begin{vmatrix} 110 | {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ 111 | {{a_{21}}}&{{a_{22}}}& \cdots &{{a_{2n}}}\\ 112 | \vdots & \vdots &\;& \vdots \\ 113 | {{a_{n1}}}&{{a_{n2}}}& \cdots &{{a_{nn}}} 114 | \end{vmatrix}}=\sum_{j_1j_2\cdots j_n}{{(-1)}^{\tau(i_1i_2\cdots i_n)+\tau (j_1j_2\cdots j_n)} a_{i_1j_1}a_{i_2j_2}\cdots a_{i_nj_n}}. \] 115 | \end{defination} 116 | \begin{defination} 117 | 在行列式中,由左上角到右下角所形成的斜线称为\textbf{主对角线},由右上角到左上角所形成的斜线称为\textbf{副对角线}.在主对角线下方的元素全为零,称为\textbf{上三角行列式},如果主对角线上方的元素全为零,则称为\textbf{下三角行列式}.上三角行列式和下三角行列式统称为\textbf{三角行列式}.如果除了主对角线之外的元素全为零,则称为\textbf{对角行列式}. 118 | \index{ZDJX@主对角线} 119 | \index{FDJX@副对角线} 120 | \index{SSJHLS@上三角行列式} 121 | \index{XSJHLS@下三角行列式} 122 | \index{SJHLS@三角行列式} 123 | \index{DJHLS@对角行列式} 124 | \end{defination} 125 | \section{$ n $阶行列式的性质及运算} 126 | \begin{feature} 127 | 行与列互换,行列式的值不变,即 \[ {\begin{vmatrix} 128 | {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ 129 | {{a_{21}}}&{{a_{22}}}& \cdots &{{a_{2n}}}\\ 130 | \vdots & \vdots &{\rm{ }}& \vdots \\ 131 | {{a_{n1}}}&{{a_{n2}}}& \cdots &{{a_{nn}}} 132 | \end{vmatrix}} = {\begin{vmatrix} 133 | {{a_{11}}}&{{a_{21}}}& \cdots &{{a_{n1}}}\\ 134 | {{a_{12}}}&{{a_{22}}}& \cdots &{{a_{n2}}}\\ 135 | \vdots & \vdots &{\rm{ }}& \vdots \\ 136 | {{a_{1n}}}&{{a_{2n}}}& \cdots &{{a_{nn}}} 137 | \end{vmatrix}} .\]假设\[ D={\begin{vmatrix} 138 | {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ 139 | {{a_{21}}}&{{a_{22}}}& \cdots &{{a_{2n}}}\\ 140 | \vdots & \vdots &{\rm{ }}& \vdots \\ 141 | {{a_{n1}}}&{{a_{n2}}}& \cdots &{{a_{nn}}} 142 | \end{vmatrix}} \qquad D^{\mathrm{T}}{\begin{vmatrix} 143 | {{a_{11}}}&{{a_{21}}}& \cdots &{{a_{n1}}}\\ 144 | {{a_{12}}}&{{a_{22}}}& \cdots &{{a_{n2}}}\\ 145 | \vdots & \vdots &{\rm{ }}& \vdots \\ 146 | {{a_{1n}}}&{{a_{2n}}}& \cdots &{{a_{nn}}} 147 | \end{vmatrix}} \]那么称$ D^{\mathrm{T}} $为$ D $的\textbf{转置行列式}\index{ZZHLS@转置行列式},由此性质可知行列式中行与列的地位是对称的,具有相同的性质.上述性质可改写成$ D^{\mathrm{T}}=D $. 148 | \end{feature} 149 | \begin{feature} 150 | 在行列式中,如果某一行(列)元素全为零,则该行列式的值为零. 151 | \end{feature} 152 | \begin{feature} 153 | 交换任意两行(列)的位置,行列式的值变号,即\[ {\begin{vmatrix} 154 | {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ 155 | \vdots & \vdots &{\rm{ }}& \vdots \\ 156 | {{a_{i1}}}&{{a_{i2}}}& \cdots &{{a_{in}}}\\ 157 | \vdots & \vdots &{\rm{ }}& \vdots \\ 158 | {{a_{j1}}}&{{a_{j2}}}& \cdots &{{a_{jn}}}\\ 159 | \vdots & \vdots &{\rm{ }}& \vdots \\ 160 | {{a_{n1}}}&{{a_{n2}}}& \cdots &{{a_{nn}}} 161 | \end{vmatrix}} + {\begin{vmatrix} 162 | {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ 163 | \vdots & \vdots &{\rm{ }}& \vdots \\ 164 | {{a_{j1}}}&{{a_{j2}}}& \cdots &{{a_{jn}}}\\ 165 | \vdots & \vdots &{\rm{ }}& \vdots \\ 166 | {{a_{i1}}}&{{a_{i2}}}& \cdots &{{a_{in}}}\\ 167 | \vdots & \vdots &{\rm{ }}& \vdots \\ 168 | {{a_{n1}}}&{{a_{n2}}}& \cdots &{{a_{nn}}} 169 | \end{vmatrix}} .\] 170 | \end{feature} 171 | \begin{feature} 172 | 如果行列式有两行(列)完全相同,则行列式为零. 173 | \end{feature} 174 | \begin{feature} 175 | 行列式具有线性型,即 176 | \begin{enumerate} 177 | \item\[ {\begin{vmatrix} 178 | {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ 179 | \vdots & \vdots &{\rm{ }}& \vdots \\ 180 | {{b_1} + {c_1}}&{{b_2} + {c_2}}& \cdots &{{b_n} + {c_n}}\\ 181 | \vdots & \vdots &{\rm{ }}& \vdots \\ 182 | {{a_{n1}}}&{{a_{n2}}}& \cdots &{{a_{nn}}} 183 | \end{vmatrix}}{\rm{ = }} {\begin{vmatrix} 184 | {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ 185 | \vdots & \vdots &{\rm{ }}& \vdots \\ 186 | {{b_1}}&{{b_2}}& \cdots &{{b_n}}\\ 187 | \vdots & \vdots &{\rm{ }}& \vdots \\ 188 | {{a_{n1}}}&{{a_{n2}}}& \cdots &{{a_{nn}}} 189 | \end{vmatrix}} + {\begin{vmatrix} 190 | {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ 191 | \vdots & \vdots &{\rm{ }}& \vdots \\ 192 | {{c_1}}&{{c_2}}& \cdots &{{c_n}}\\ 193 | \vdots & \vdots &{\rm{ }}& \vdots \\ 194 | {{a_{n1}}}&{{a_{n2}}}& \cdots &{{a_{nn}}} 195 | \end{vmatrix}} ;\] 196 | \item\[ {\begin{vmatrix} 197 | {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ 198 | \vdots & \vdots &{\rm{ }}& \vdots \\ 199 | {k{a_{i1}}}&{k{a_{i2}}}& \cdots &{k{a_{in}}}\\ 200 | \vdots & \vdots &{\rm{ }}& \vdots \\ 201 | {{a_{n1}}}&{{a_{n2}}}& \cdots &{{a_{nn}}} 202 | \end{vmatrix}} = k {\begin{vmatrix} 203 | {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ 204 | \vdots & \vdots &{\rm{ }}& \vdots \\ 205 | {{a_{i1}}}&{{a_{i2}}}& \cdots &{{a_{in}}}\\ 206 | \vdots & \vdots &{\rm{ }}& \vdots \\ 207 | {{a_{n1}}}&{{a_{n2}}}& \cdots &{{a_{nn}}} 208 | \end{vmatrix}} .\] 209 | \end{enumerate} 210 | \end{feature} 211 | \begin{feature} 212 | 如果行列式有两行(列)成比例,则行列式为零,即\[ {\begin{vmatrix} 213 | {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ 214 | \vdots & \vdots &{\rm{ }}& \vdots \\ 215 | {{a_{i1}}}&{{a_{i2}}}& \cdots &{{a_{in}}}\\ 216 | \vdots & \vdots &{\rm{ }}& \vdots \\ 217 | {k{a_{i1}}}&{k{a_{i2}}}& \cdots &{k{a_{in}}}\\ 218 | \vdots & \vdots &{\rm{ }}& \vdots \\ 219 | {{a_{n1}}}&{{a_{n2}}}& \cdots &{{a_{nn}}} 220 | \end{vmatrix}} =0. \] 221 | \end{feature} 222 | \begin{feature} 223 | 行列式某一行(列)的$ k $倍加到令一行(列),行列式的值不变,即\[ {\begin{vmatrix} 224 | {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ 225 | \vdots & \vdots &{\rm{ }}& \vdots \\ 226 | {{a_{i1}}}&{{a_{i2}}}& \cdots &{{a_{in}}}\\ 227 | \vdots & \vdots &{\rm{ }}& \vdots \\ 228 | {{a_{j1}}}&{{a_{j2}}}& \cdots &{{a_{jn}}}\\ 229 | \vdots & \vdots &{\rm{ }}& \vdots \\ 230 | {{a_{n1}}}&{{a_{n2}}}& \cdots &{{a_{nn}}} 231 | \end{vmatrix}} {\rm{ = }} {\begin{vmatrix} 232 | {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ 233 | \vdots & \vdots &{\rm{ }}& \vdots \\ 234 | {{a_{i1}}}&{{a_{i2}}}& \cdots &{{a_{in}}}\\ 235 | \vdots & \vdots &{\rm{ }}& \vdots \\ 236 | {{a_{j1}} + k{a_{i1}}}&{{a_{j2}}{\rm{ + }}k{a_{i2}}}& \cdots &{{a_{jn}}{\rm{ + }}k{a_{in}}}\\ 237 | \vdots & \vdots &{\rm{ }}& \vdots \\ 238 | {{a_{n1}}}&{{a_{n2}}}& \cdots &{{a_{nn}}} 239 | \end{vmatrix}} . \] 240 | \end{feature} 241 | \begin{defination} 242 | 我们引入如下记号以描述行列式的变形: 243 | \begin{enumerate} 244 | \item $ r_i\div k $(或 $ c_i\div k $)表示从第$ i $行(列)提取公因子$ k $; 245 | \item $ r_i+kr_j $(或$ c_i+kc_j $)表示将第$ j $行(列)的$ k $倍加到第$ i $行(列); 246 | \item $ r_i\leftrightarrow r_j $(或$ c_i\leftrightarrow c_j $)表示交换第$ i $行(列)与第$ j $行(列)的位置. 247 | \end{enumerate} 248 | \end{defination} 249 | \section{行列式按一行展开及克拉默法则} 250 | \section{八大类型行列式及其解法} 251 | \subsection{} 252 | 253 | \chapter{矩阵\\Matrix} 254 | \section{矩阵及其运算} 255 | \section{矩阵的分块} 256 | \section{矩阵的秩} 257 | \section{矩阵的逆} 258 | \section{初等矩阵} 259 | \chapter{向量代数与几何应用\\Vector Algebra and Geometry} 260 | \section{向量的线性运算与空间直角坐标系} 261 | \section{向量的内积,外积与混合积} 262 | \section{空间平面及其方程} 263 | \section{空间直线及其方程} 264 | \chapter{线性方程组\\System of Linear Equations} 265 | \section{消元法} 266 | \begin{defination}[相容方程组与不相容方程组的定义] 267 | \index{XRFCZ@相容方程组} 268 | \index{BXRFCZ@不相容方程组} 269 | 有解的方程组称为\textbf{相容方程组};没有解的方程组称为\textbf{不相容方程组}. 270 | \end{defination} 271 | \begin{defination} 272 | 设给定两个线性方程组.如果第一个方程组的解都是第二个方程组的解,同时第二个方程组的解也都是第一个方程组的解,则称他们是\textbf{同解方程组}. 273 | \index{TJFCZ@同解方程组} 274 | 对于下述线性方程组: 275 | \begin{equation}\label{xxfcz} 276 | \left\{ 277 | \begin{array}{c} 278 | a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n=b_1\\ 279 | a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n=b_2\\ 280 | \vdots \\ 281 | a_{m1}x_1+a_{m2}x_2+\cdots+a_{mn}x_n=b_m 282 | \end{array} 283 | \right., 284 | \end{equation} 285 | \index{XSJZ@系数矩阵} 286 | \index{WZLJZ@未知量矩阵} 287 | \index{CSXJZ@常数项矩阵} 288 | 若令\[ 289 | \text{系数矩阵}\ \bm{A}=\begin{bmatrix} 290 | a_{11} & a_{12} & \cdots & a_{1n}\\ 291 | a_{21} & a_{22} & \cdots & a_{2n}\\ 292 | \vdots & \vdots & \ & \vdots\\ 293 | a_{m1} & a_{m2} & \cdots & a_{mn} 294 | \end{bmatrix}, 295 | \text{未知量矩阵}\ \bm{X}=\begin{bmatrix} 296 | x_1\\ 297 | x_2\\ 298 | \vdots\\ 299 | x_n 300 | \end{bmatrix}, 301 | \text{常数项矩阵}\ \bm{b}=\begin{bmatrix} 302 | b_1\\ 303 | b_2\\ 304 | \vdots\\ 305 | b_n 306 | \end{bmatrix}, 307 | \] 308 | 则方程组\ref{xxfcz}可以改写成矩阵的形式\[ \bm{AX}=\bm{b}, \] 309 | 称$ \bm{A} $为方程组\ref{xxfcz}的系数矩阵,而 310 | $$ 311 | \widetilde{\bm{A}}=[\bm{A}|\bm{b}]= 312 | \left[ 313 | \begin{array}{cccc|c} 314 | a_{11} & a_{12} & \cdots & a_{1n}&b_1\\ 315 | a_{21} & a_{22} & \cdots & a_{2n}&b_2\\ 316 | \vdots & \vdots & \ & \vdots&\vdots\\ 317 | a_{m1} & a_{m2} & \cdots & a_{mn}&b_m 318 | \end{array} 319 | \right] 320 | $$ 321 | 称为方程组\ref{xxfcz}的\textbf{增广矩阵}.其中增广矩阵唯一确定线性方程组. 322 | \index{ZGJZ@增广矩阵} 323 | \nomenclature{$ \widetilde{\bm{A}} $}{矩阵$ \bm{A} $的增广矩阵} 324 | \nomenclature{$\mathrm{det}(\bm{A}) $}{矩阵$ \bm{A} $的行列式} 325 | \end{defination} 326 | \begin{feature} 327 | 线性方程组的初等变换\CJKunderdot{一定}把方程组变成\textbf{同解的方程组}. 328 | \end{feature} 329 | \begin{theorem}[初等行变换对于增广矩阵的同解不变性定理]若线性方程组\ref{xxfcz}的增广矩阵$ \widetilde{\bm{A}} $经过一系列\CJKunderdot{有限次}的初等行变换变成$ \widetilde{\bm{B}} $,则以$ \widetilde{\bm{B}} $为增广矩阵的线性方程组与\ref{xxfcz}同解. 330 | \end{theorem} 331 | \begin{theorem}[线性方程组有解的充要条件] 332 | 线性方程组有解的\CJKunderdot{充分必要条件}是系数矩阵$ \bm{A} $的秩等于其对应的增广矩阵$ \widetilde{\bm{A}} $的秩,即\[ \mathrm{rank}(\bm{A})=\mathrm{rank}(\widetilde{\bm{A}}) .\] 333 | \end{theorem} 334 | \begin{theorem}[线性方程组解的结构判定充分必要条件] 335 | 线性方程组解的情况如下:\\ 336 | \begin{enumerate} 337 | \item 方程组有\textbf{唯一解}$\Leftrightarrow \mathrm{rank}(\widetilde{\bm{A}})=\mathrm{rank}(\bm{A})=n; $ 338 | \item 方程组有\textbf{无穷多个解}$\Leftrightarrow \mathrm{rank}(\widetilde{\bm{A}})=\mathrm{rank}(\bm{A})s $, 677 | \end{enumerate} 678 | 则向量组(I)线性相关. 679 | \end{theorem} 680 | \begin{inference} 681 | 若$ \bm{\alpha}_1,\bm{\alpha}_2,\cdots,\bm{\alpha}_r $可由$ \bm{\beta}_1,\bm{\beta}_2,\cdots,\bm{\beta}_s $线性表示,且$ \bm{\alpha}_1,\bm{\alpha}_2,\cdots,\bm{\alpha}_r $线性无关,则$ r\leqslant s $. 682 | \end{inference} 683 | \begin{inference} 684 | 任意$ n+1 $个$ n $维向量一定线性相关,从而当$ m>n $时任意$ m $个$ n $维向量一定线性相关. 685 | \end{inference} 686 | \begin{inference} 687 | 一个向量组的极大线性无关组所含向量个数相同. 688 | \end{inference} 689 | \begin{defination}[向量组的秩] 690 | 一个向量组$ \bm{A} $的极大线性无关向量组所含向量个数$ r $称为向量组$ \bm{A} $的秩,记为$ \mathrm{rank}(\bm{A}) $. 691 | \end{defination} 692 | \begin{inference} 693 | 等价的向量组有相同的秩. 694 | \end{inference} 695 | \subsection{矩阵秩的进一步讨论} 696 | \begin{defination} 697 | 设\[ \bm{A}_{m\times n}=\left[ {\begin{array}{*{20}{c}} 698 | {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ 699 | {{a_{21}}}&{{a_{22}}}& \cdots &{{a_{2n}}}\\ 700 | \vdots &{ \vdots} &{} & \vdots \\ 701 | {{a_{m1}}}&{{a_{m2}}}& \cdots &{{a_{mn}}} 702 | \end{array}} \right] \]是数域$ F $上的一个$ m\times n $矩阵. 703 | \begin{itemize} 704 | \item[\color{HotPink1} \textleaf ] {\color{HotPink1}\heiti{行向量}}\ 将 $ \bm{A} $的每一行看作$ F^n $的向量,称之为\textbf{行向量}.\index{HXL@行向量}设\[ \bm{\alpha}_i=[a_{i1},a_{i2},\cdots,a_{in}],\quad (i=1,2,\cdots,m), \]则\[ \bm{A}=\begin{bmatrix} 705 | \bm{\alpha}_1\\\bm{\alpha}_2\\\cdots\\\bm{\alpha}_m 706 | \end{bmatrix} ,\]这里$ \bm{\alpha}_1,\bm{\alpha}_2,\cdots,\bm{\alpha}_m $为$ \bm{A} $的\textbf{行向量组}\index{HXLZ@行向量组}.$ \bm{A} $可以看作由行向量组构成的分块矩阵. 707 | \item[\color{HotPink1} \textleaf ] {\color{HotPink1}\heiti{列向量}}\ 将 $ \bm{A} $的每一列看作$ F^m $的向量,称之为\textbf{列向量}.\index{LXL@列向量}设\[ \bm{\beta}_i=\begin{bmatrix} 708 | a_{1i}\\a_{2i}\\\vdots\\a_{mi} 709 | \end{bmatrix},\quad (i=1,2,\cdots,n), \]则\[ \bm{A}=[\bm{\beta}_1,\bm{\beta}_2,\cdots,\bm{\beta}_n] ,\]这里$\bm{\beta}_1,\bm{\beta}_2,\cdots,\bm{\beta}_n$为$ \bm{A} $的\textbf{列向量组}\index{LXLZ@列向量组}.$ \bm{A} $可以看作由列向量组构成的分块矩阵. 710 | \end{itemize} 711 | \end{defination} 712 | \begin{defination}[行秩与列秩的定义] 713 | 设矩阵\[ \bm{A}_{m\times n}=\left[ {\begin{array}{*{20}{c}} 714 | {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ 715 | {{a_{21}}}&{{a_{22}}}& \cdots &{{a_{2n}}}\\ 716 | \vdots &{ \vdots} &{} & \vdots \\ 717 | {{a_{m1}}}&{{a_{m2}}}& \cdots &{{a_{mn}}} 718 | \end{array}} \right] \]有$ m $个行向量与$ n $个列向量.矩阵$ \bm{A} $行向量组的秩称为矩阵$ \bm{A} $的\textbf{行秩},列向量组的秩称为矩阵$ \bm{A} $的\textbf{列秩}.\index{HZ@行秩}\index{LZ@列秩} 719 | \end{defination} 720 | \begin{theorem} 721 | 矩阵的行秩和列秩都等于矩阵的秩. 722 | \end{theorem} 723 | \begin{inference}[初等行变换后行向量组秩不变] 724 | 矩阵经数次初等行变换后,新矩阵的行向量组都与原矩阵的行向量组等价,故其秩相等. 725 | \end{inference} 726 | \begin{inference}[初等列变换后列向量组秩不变] 727 | 矩阵经数次初等列变换后,新矩阵的列向量组都与原矩阵的列向量组等价,故其秩相等. 728 | \end{inference} 729 | \begin{method}[求极大线性无关组与向量组的秩的方法] 730 | 我们以$ 4 $阶方阵 \[ \bm{A}_{4\times 4}=\begin{bNiceArrayRC}{CCCC}% 731 | [columns-width = auto, 732 | code-for-first-row = \color{HotPink1}, 733 | code-for-last-col = \color{HotPink1},] 734 | \bm{\beta}_1&\bm{\beta}_2&\bm{\beta}_3&\bm{\beta}_4\\ 735 | {{a_{11}}}&{{a_{12}}}& a_{13} &{{a_{14}}}&\bm{\alpha}_1\\ 736 | {{a_{21}}}&{{a_{22}}}& a_{23} &{{a_{24}}}&\bm{\alpha}_2\\ 737 | a_{31} &a_{32}&a_{33} & a_{34} &\bm{\alpha}_3\\ 738 | {{a_{41}}}&{{a_{42}}}& a_{43} &{{a_{44}}}&\bm{\alpha}_4\\ 739 | \end{bNiceArrayRC} \]来演示构造过程,其他矩阵同理.$ \bm{A} $中有行向量组$ \bm{\alpha}_1,\bm{\alpha}_2,\bm{\alpha}_3,\bm{\alpha}_4 $和列向量组$ \bm{\beta}_1,\bm{\beta}_2,\bm{\beta}_3,\bm{\beta}_4 $,现以列向量组为例,讨论其极大线性无关组和秩. 740 | \begin{enumerate} 741 | \item 将上述矩阵$ \bm{A} $化为最简行阶梯矩阵(以下仅为其中一种可能): 742 | \[ \bm{A}_{4\times4}=\begin{bmatrix} 743 | {{a_{11}}}&{{a_{12}}}& a_{13} &{{a_{14}}}\\ 744 | {{a_{21}}}&{{a_{22}}}& a_{23} &{{a_{24}}}\\ 745 | a_{31} &a_{32}&a_{33} & a_{34} \\ 746 | {{a_{41}}}&{{a_{42}}}& a_{43} &{{a_{44}}}\\ 747 | \end{bmatrix} 748 | \overset{c}{\sim} 749 | \begin{bNiceArrayRC}{CCCC}% 750 | [name=bb,columns-width = auto, 751 | code-for-first-row = \color{HotPink1}] 752 | \bm{\beta}'_1&\bm{\beta}'_2&\bm{\beta}'_3&\bm{\beta}'_4\\ 753 | b_{11} & b_{12} & b_{13} & b_{14}\\ 754 | 0&0 & b_{23} & b_{24}\\ 755 | 0& 0 & 0 & b_{34}\\ 756 | 0&0 &0&0\\ 757 | \end{bNiceArrayRC} \] 758 | \tikzset{myoptions/.style={remember picture, 759 | overlay, 760 | name prefix =bb-, 761 | every node/.style = {fill =blue!15, 762 | blend mode = multiply}}} 763 | \begin{tikzpicture}[myoptions] 764 | \node [fit = (1-1) (1-4)] {} ; 765 | \node [fit = (2-3) (2-4)] {} ; 766 | \node [fit = (3-4) (3-4)] {} ; 767 | \end{tikzpicture}此时我们知道向量组$\bm{\beta}'_1,\bm{\beta}'_2,\bm{\beta}'_3,\bm{\beta}'_4$与向量组$ \bm{\beta}_1,\bm{\beta}_2,\bm{\beta}_3,\bm{\beta}_4 $等价,则我们可以得出原向量组的秩与向量组$ \bm{\beta}'_1,\bm{\beta}'_2,\bm{\beta}'_3,\bm{\beta}'_4 $的秩相等,故求出向量组的秩即为最简型矩阵的秩.同时$ \bm{\beta}'_1,\bm{\beta}'_2 $线性相关,考虑方程$ [\bm{\beta}_1,\bm{\beta}_2,\bm{\beta}_3,\bm{\beta}_4]\bm{X}=\bm{0} $与方程$ [\bm{\beta}'_1,\bm{\beta}'_2,\bm{\beta}'_3,\bm{\beta}'_4]\bm{X}=\bm{0} $同解,因此向量$ \bm{\beta}_1,\bm{\beta}_2,\bm{\beta}_3,\bm{\beta}_4 $之间的线性关系与向量$\bm{\beta}'_1,\bm{\beta}'_2,\bm{\beta}'_3,\bm{\beta}'_4$之间的线性关系相同.同时,同解保证了列与列之间的关系不变,那么$ \bm{\beta}'_1 $可=由向量$ \bm{\beta}'_2,\bm{\beta}'_3,\bm{\beta}'_4 $线性表出即\[ k_0\bm{\beta}'_1=k_1 \bm{\beta}'_2+k_2\bm{\beta}'_3+k_3\bm{\beta}'_4 \]以及\[ k_0\bm{\beta}_1=k_1 \bm{\beta}_2+k_2\bm{\beta}_3+k_3\bm{\beta}_4 ,\]若根据题意$ k_0\neq 0 $则可得$ \bm{\beta}'_1 $可由$ \bm{\beta}'_2,\bm{\beta}'_3,\bm{\beta}'_4 $线性表出同理可得$ \bm{\beta}'_2 $可由$ \bm{\beta}'_1,\bm{\beta}'_3,\bm{\beta}'_4 $线性表出,则我们构造极大线性无关组时,只需要选择$\bm{\beta}'_1,\bm{\beta}'_2$其中一个向量即可,对应原向量组向量$\bm{\beta}_1,\bm{\beta}_2$之一. 768 | \item 从不同阶梯中各选出一个向量,组成的向量组即为原向量组$ \bm{\beta}_1,\bm{\beta}_2,\bm{\beta}_3,\bm{\beta}_4 $的极大线性无关组.此处可选向量组$ \bm{\beta}_1,\bm{\beta}_3,\bm{\beta}_4 $或向量组$\bm{\beta}_2,\bm{\beta}_3,\bm{\beta}_4$即为极大线性无关组. 769 | \end{enumerate} 770 | \end{method} 771 | \begin{theorem} 772 | 设$ \bm{A},\bm{B} $分别为$ m\times n,n\times s $矩阵,则\[ \mathrm{rank}(\bm{AB})\leqslant \min{\left\lbrace \mathrm{rank}(\bm{A}),\mathrm{rank}(\bm{B})\right\rbrace }. \] 773 | \end{theorem} 774 | \begin{inference} 775 | 若$ \bm{A}=\displaystyle\prod_{i=1}^t{\bm{A}_i} $,则$ \mathrm{rank}(\bm{A})\leqslant\min\left\lbrace \mathrm{rank}(\bm{A}_1),\mathrm{rank}(\bm{A}_2),\cdots,\mathrm{rank}(\bm{A}_t)\right\rbrace $. 776 | \end{inference} 777 | \begin{inference} 778 | 设$ \bm{A} $是一个$ m\times n $矩阵,若$ \bm{P} $是$ m\times m $可逆矩阵,$ \bm{Q} $是$ n\times n$可逆矩阵,则\[ \mathrm{rank}(\bm{A})=\mathrm{rank}(\bm{PA})=\mathrm{rank}(\bm{AQ}). \] 779 | \end{inference} 780 | \subsection{向量组的秩与向量组特征的关系} 781 | \begin{feature} 782 | 线性相关性与秩的关系如下: 783 | \begin{enumerate} 784 | \item 向量组$ \bm{\alpha}_1,\bm{\alpha}_2,\cdots,\bm{\alpha}_m $线性无关$ \Leftrightarrow r( \bm{\alpha}_1,\bm{\alpha}_2,\cdots,\bm{\alpha}_m)=m $; 785 | \item 向量组$ \bm{\alpha}_1,\bm{\alpha}_2,\cdots,\bm{\alpha}_m $线性相关$ \Leftrightarrow r( \bm{\alpha}_1,\bm{\alpha}_2,\cdots,\bm{\alpha}_m),black!75](0,0,0)--(6,0,0) node[below left] {$\bm{\beta}_1,\bm{\alpha}_1$}; 1211 | \draw[->,black!75](0,0,0)--(0,6,0) node[right] {$\bm{\beta}_2$}; 1212 | \draw[->,black!75](0,0,0)--(0,0,6) node[right] {$\bm{\beta}_3$}; 1213 | \draw[-stealth,red](0,0,0)--(6,0,0); 1214 | \draw[-stealth,red](0,0,0)--(4,6,0) node[below right] {$\bm{\alpha}_2$}; 1215 | \draw[-stealth,red](0,0,0)--(2,-2,6) node[below right] {$\bm{\alpha}_3$}; 1216 | \draw[-stealth,blue!50](0,0,0)--(0,-2,0) node[left] {$\hat{\bm{\alpha}_3}\prod_{\bm{\beta}_2}{\bm{\alpha}_3}$}; 1217 | \draw[-stealth,purple!50](0,0,0)--(2,-2,0); 1218 | \draw[-stealth,orange](0,0,0)--(4,0,0) node[ left] {$\hat{\bm{\alpha}_2}\prod_{\bm{\beta}_1}{\bm{\alpha}_2}$}; 1219 | \draw[-stealth,blue!50](0,0,0)--(2,0,0) node[below right] {$\hat{\bm{\alpha}_3}\prod_{\bm{\beta}_1}{\bm{\alpha}_3}$}; 1220 | \draw[dashed,line width=0](4,6,0)--(4,0,0); 1221 | \draw[dashed,line width=0](4,6,0)--(0,6,0); 1222 | \draw[dashed,line width=0](2,0,0)--(2,-2,0); 1223 | \draw[dashed,line width=0](2,-2,0)--(0,-2,0); 1224 | \draw[dashed,line width=0](2,-2,6)--(2,-2,0); 1225 | \draw[dashed,line width=0](2,-2,6)--(2,0,0); 1226 | \draw[dashed,line width=0](2,-2,6)--(0,-2,0); 1227 | \draw[dashed,line width=0](2,-2,6)--(0,0,6); 1228 | \end{tikzpicture} 1229 | \end{minipage}} 1230 | \caption{施密特正交化的几何意义} 1231 | \end{figure} 1232 | \begin{multicols}{2} 1233 | \small 1234 | \kaishu 1235 | \begin{itemize} 1236 | \item 二维平面空间\ 如图\ref{fig-ew}\ 中,\\ 设$ \bm{\alpha}_1,\bm{\alpha}_2 $线性无关,尝试求出$ k $使得:$$ \bm{\alpha}_1\perp\bm{\alpha}_2+k\cdot\bm{\alpha}_1 $$ 1237 | \begin{align*} 1238 | \because&\bm{\alpha}_1\perp\bm{\alpha}_2+k\cdot\bm{\alpha}_1 \\ \therefore& (\bm{\alpha}_1,\bm{\alpha}_2+k\bm{\alpha}_1)=0\\ 1239 | \therefore& (\bm{\alpha}_1,\bm{\alpha}_2)+(\bm{\alpha}_1,\bm{\alpha}_1)k=0\\ 1240 | \therefore& k=-\frac{(\bm{\alpha}_2,\bm{\alpha}_1)}{(\bm{\alpha}_1,\bm{\alpha}_1)}\\ 1241 | \end{align*} 1242 | 此时我们令$ \bm{\gamma}=-k\bm{\alpha}_1 $则有$ \bm{\alpha}_1\perp\bm{\alpha}_2-\bm{\gamma} $,已求得与$ \bm{\alpha}_1,\bm{\alpha}_2 $等价的正交基$ \bm{\alpha}_1,\bm{\alpha}_2-\bm{\gamma} $.在二维空间,如图\ref{fig-ew}中$ \bm{\alpha}_2 $在$ \bm{\alpha}_1 $的投影即为$ \bm{\gamma} $. 1243 | \begin{align*} 1244 | \mbox{又}\because\bm{\gamma}&=\textstyle{\prod_{\bm{\alpha}_1}{\bm{\alpha}_2}}\displaystyle\cdot\hat{\bm{\alpha}_1}=\frac{(\bm{\alpha}_2,\bm{\alpha}_1)}{|\bm{\alpha}_1|}\cdot\hat{\bm{\alpha}_1}\\&=\frac{(\bm{\alpha}_2,\bm{\alpha}_1)}{|\bm{\alpha}_1|}\cdot\frac{\bm{\alpha}_1}{|\bm{\alpha}_1|}\\ 1245 | &=\frac{(\bm{\alpha}_2,\bm{\alpha}_1)}{|\bm{\alpha}_1|}\bm{\alpha}_1\\ 1246 | &=\frac{(\bm{\alpha}_2,\bm{\alpha}_1)}{(\bm{\alpha}_1,\bm{\alpha}_1)}\bm{\alpha}_1 1247 | \end{align*} 1248 | 则有平面内积空间$ \mathbb{R}^n $中的一组正交基$ \bm{\beta}_,\bm{\beta}_2 $其中 1249 | \begin{align*} 1250 | \bm{\beta}_1&=\bm{\alpha}_{1},\\ 1251 | \bm{\beta}_2&=\bm{\alpha}_2-\bm{\gamma}\\&=\bm{\alpha}_{2}-\dfrac{(\bm{\alpha}_2,\bm{\beta}_1)}{(\bm{\beta}_1,\bm{\beta}_1)}\bm{\beta}_1. \end{align*} 1252 | 1253 | \item 三维立体空间\ 如图\ref{fig-sw}\ 中,\\设$ \bm{\alpha}_1,\bm{\alpha}_2,\bm{\alpha}_3 $线性无关,以$ \bm{\alpha}_1 $作为正交基中的一个向量$ \bm{\beta}_1 $,并以该向量开始正交化.\\$ \bm{\alpha}_2 $所对应的正交基中的向量$ \bm{\beta}_2 $只需满足$ \bm{\beta}_2\perp \bm{\beta}_1$,按照二维空间的做法进行正交化:(1)作$ \bm{\alpha}_2 $在$ \bm{\alpha}_1 $($ \bm{\beta}_1 $)上的投影$\prod_{\bm{\beta}_1}{\bm{\alpha}_2}$(2)由此可得$ \bm{\beta}_2 =\bm{\alpha}_2-\prod_{\bm{\beta}_1}{\bm{\alpha}_2}\cdot\hat{\bm{\alpha}_2}$.\\$ \bm{\alpha}_3 $所对应的正交基中的向量$ \bm{\beta}_3 $需满足$ \bm{\beta}_3\perp \bm{\beta}_1$且$ \bm{\beta}_3\perp \bm{\beta}_2$因此作$ \bm{\alpha}_3 $在$ \bm{\beta}_1,\bm{\beta}_1 $上的投影$\prod_{\bm{\beta}_1}{\bm{\alpha}_3}$,$\prod_{\bm{\beta}_2}{\bm{\alpha}_3}$.由此可得$ \bm{\beta}_3=\bm{\alpha}_3-\hat{\bm{\alpha}_3}\left(\prod_{\bm{\beta}_1}{\bm{\alpha}_3}+\prod_{\bm{\beta}_2}{\bm{\alpha}_3} \right) $.综上所述有:\begin{align*} 1254 | \bm{\beta}_1&=\bm{\alpha}_{1},\\ 1255 | \bm{\beta}_2&=\bm{\alpha}_{2}-\dfrac{(\bm{\alpha}_2,\bm{\beta}_1)}{(\bm{\beta}_1,\bm{\beta}_1)}\bm{\beta}_1,\\ 1256 | \bm{\beta}_3&=\bm{\alpha}_{3}-\dfrac{(\bm{\alpha}_3,\bm{\beta}_1)}{(\bm{\beta}_1,\bm{\beta}_1)}\bm{\beta}_1-\dfrac{(\bm{\alpha}_3,\bm{\beta}_2)}{(\bm{\beta}_2,\bm{\beta}_2)}\bm{\beta}_2-\dfrac{(\bm{\alpha}_3,\bm{\beta}_{2})}{(\bm{\beta}_{2},\bm{\beta}_{2})}\bm{\beta}_{2}.\\ 1257 | \end{align*} 1258 | \end{itemize} 1259 | \end{multicols} 1260 | \end{theorem} 1261 | \begin{method}[标准正交化的方法] 1262 | 设$\bm{\alpha}_{1},\bm{\alpha}_{2},\bm{\alpha}_{3},\cdots,\bm{\alpha}_{n} $为内积空间$ \mathbb{R}^n $的一组基,则有以下\textbf{标准正交化}方法: 1263 | \begin{enumerate} 1264 | \item 由格拉姆-施密特正交化定理(定理\ref{gspt})对$\bm{\alpha}_{1},\bm{\alpha}_{2},\bm{\alpha}_{3},\cdots,\bm{\alpha}_{n} $实施正交化得到正交向量组$ \bm{\beta}_{1},\bm{\beta}_{2},\bm{\beta}_{3},\cdots,\bm{\beta}_{n} $; 1265 | \item 由单位化定义(定义\ref{zjhdy})依次对$ \bm{\beta}_{1},\bm{\beta}_{2},\bm{\beta}_{3},\cdots,\bm{\beta}_{n} $单位化,得标准正交基$ \bm{\eta}_1,\bm{\eta}_2,\cdots,\bm{\eta}_n $. 1266 | \end{enumerate} 1267 | \end{method} 1268 | 1269 | 1270 | \subsection{正交矩阵} 1271 | \begin{defination}[正交矩阵的定义] 1272 | 若$ n $阶\CJKunderdot{实矩阵}$ \bm{A} $满足:\[ \bm{A}^{\mathrm{T}}\bm{A}=\bm{E} ,\]则称$ \bm{A} $为\textbf{正交矩阵}. 1273 | \index{ZJJZ@正交矩阵} 1274 | \end{defination} 1275 | \begin{feature}[正交矩阵的性质] 1276 | $ n $阶正交矩阵$ \bm{A} $有以下性质: 1277 | \begin{enumerate} 1278 | \item $ \bm{A} $可逆,且$ \bm{A}^{-1}=\bm{A}^{\mathrm{T}} $; 1279 | \item$ \bm{A}^{-1},\bm{A}^{\mathrm{T}} $也是\textbf{正交矩阵}; 1280 | \item $ |\bm{A}|=\pm1; $ 1281 | \item 正交矩阵的乘积仍为正交矩阵. 1282 | \end{enumerate} 1283 | \end{feature} 1284 | \begin{theorem}[正交矩阵构造定理] 1285 | 设$ \bm{A} $为$ n $阶实矩阵,且$ \bm{A}=[\bm{\alpha}_{1},\bm{\alpha}_{2},\bm{\alpha}_{3},\cdots,\bm{\alpha}_{n}] $则$ \bm{A} $是正交矩阵的\CJKunderdot{充分必要条件}是列向量$ \bm{\alpha}_{1},\bm{\alpha}_{2},\bm{\alpha}_{3},\cdots,\bm{\alpha}_{n} $构成$ \mathbb{R}^n $的一组标准正交基. 1286 | \end{theorem} 1287 | \subsection{实对称矩阵的对角化} 1288 | \begin{theorem} 1289 | 实对称矩阵的特征值必都为常数,且可以取到实的特征向量. 1290 | \end{theorem} 1291 | \begin{theorem}\label{3.5} 1292 | 实对称矩阵$ \bm{A} $属于\CJKunderdot{不同特征值}的特征向量必正交.(由该定理可知若某个特征值只对应一个特征向量时,该特征向量可不进行正交化.) 1293 | \end{theorem} 1294 | \begin{theorem}\label{SDCJZBKDJH} 1295 | 设$ \bm{A} $是一个$ n $阶实对称矩阵,则必定存在\CJKunderdot{正交矩阵}$ \bm{T} $使得\[ \bm{T}^{-1} \bm{A} \bm{T}=\bm{\varLambda} \quad( \bm{\varLambda}\mbox{为对角矩阵}).\] 1296 | \end{theorem} 1297 | \begin{inference} 1298 | 由\ 定理\ref{SDCJZBKDJH}\ 可知:\textbf{实对称矩阵必可对角化}. 1299 | \end{inference} 1300 | \begin{method}[求正交矩阵的方法] 1301 | \begin{enumerate} 1302 | \item 求出$ \bm{A} $的全部特征值$\lambda_1,\lambda_2,\lambda_3,\cdots,\lambda_n $和属于每一个$ \lambda_i\,(i=1,2,3,\cdots,t) $的特征向量$ \bm{\alpha}_{i1},\bm{\alpha}_{i2},\bm{\alpha}_{i3},\cdots,\bm{\alpha}_{is} $; 1303 | \item 将属于$ \lambda_i\,(i=1,2,3,\cdots,t) $的特征向量$ \bm{\alpha}_{i1},\bm{\alpha}_{i2},\bm{\alpha}_{i3},\cdots,\bm{\alpha}_{is} $正交化和单位化,记为$\bm{\eta}_{i1},\bm{\eta}_{i2},\bm{\eta}_{i3},\cdots,\bm{\eta}_{is} $,他们仍然是属于$ \lambda_i $的线性无关的特征向量; 1304 | \item 由\ 定理\ref{3.5}\ 知,上一步处理后所有特征向量合并之后的向量组仍然为正交的单位向量组,且所含的向量总个数仍为$ n $,以这$ n $个向量为列向量构成的矩阵$ \bm{T} $即为所求. 1305 | \end{enumerate} 1306 | \end{method} 1307 | 1308 | 1309 | \chapter{二次型与二次曲面\\Homogeneous Quadratic Polynomials and Quadrics} 1310 | \section{二次型及其标准型} 1311 | \subsection{二次型及其矩阵} 1312 | \begin{defination}[二次型及二次型矩阵的定义] 1313 | 含有$ n $个变量$ x_1,x_2,\cdots,x_n $且系数属于数域$ F $的二次齐次多项式 1314 | \begin{align*} 1315 | f( x_1,x_2,\cdots,x_n )=a_{11}x_1^2+2a_{12}x_1x_2&+\cdots+2a_{1n}x_1x_n\\ 1316 | +a_{22}x_2^2&+\cdots+2a_{2n}x_2x_n\\ 1317 | &+\cdots\\ 1318 | &+a_{nn}x_n^2 1319 | \end{align*} 1320 | 称为\textbf{关于数域$ F $的一个$ n $元二次型},简称\textbf{二次型}. 1321 | $ F=\mathbb{R} $时的二次型称为\textbf{实二次型},$ F=\mathbb{C} $时的二次型称为\textbf{复二次型}. 1322 | \index{ECX@二次型} 1323 | \index{SECX@实二次型} 1324 | \index{FECX@复二次型} 1325 | 1326 | 下面我们给出二次型的另外一表述方式.令$ a_{ij}=a_{ji}\quad (i0$,则称$f(x_{1},x_{2},\cdots,x_{n})$为\textbf{正定二次型};\index{ZDECX@正定二次型} 1549 | \item $f(c_{1},c_{2},\cdots,c_{n})<0$,则称$f(x_{1},x_{2},\cdots,x_{n})$为\textbf{负定二次型};\index{FDECX@负定二次型} 1550 | \end{itemize} 1551 | \item 1552 | 若对任意$n$个\CJKunderdot{实数}$c_{1},c_{2},\cdots,c_{n}$,总有: 1553 | \begin{itemize} 1554 | \item $f(c_{1},c_{2},\cdots,c_{n})\geqslant0$,则称$f(x_{1},x_{2},\cdots,x_{n})$为\textbf{半正定二次型};\index{BZDECX@半正定二次型} 1555 | \item $f(c_{1},c_{2},\cdots,c_{n})\leqslant0$,则称$f(x_{1},x_{2},\cdots,x_{n})$为\textbf{半负定二次型}.\index{BFDECX@半负定二次型} 1556 | \end{itemize} 1557 | \end{enumerate} 1558 | 1559 | 因此,正定二次型一定是半正定二次型,半正定二次型不一定是正定二次型,即\[ \mbox{正定二次型} \subsetneqq \mbox{半正定二次型}.\] 1560 | 显然$f(x_{1},x_{2},\cdots,x_{n})$为负定的当且仅当$-f(x_{1},x_{2},\cdots,x_{n})$为正定的.所以,负定二次型的问题总可以借助正定二次型的理论解决. 1561 | \end{defination} 1562 | \begin{example} 1563 | 二次型$f(x_{1},x_{2},\cdots,x_{n})=x_1^2 + x_2^2 + x_3^2 + \cdots + x_{n - 1}^2$是半正定的,但不是正定二次型.实际上,令$ ({x_1},{x_2}, \cdots ,{x_{n - 1}},{x_n}) = (0,0, \cdots ,0,1) \ne 0 $则$ f(x_{1},x_{2},\cdots,x_{n}) $的值为$ 0 $. 1564 | \end{example} 1565 | \begin{feature} 1566 | 非退化线性替换将正定实二次型(负定实二次型)仍变为正定实二次型(负定实二次型). 1567 | \end{feature} 1568 | \begin{theorem}[正定二次型的判定定理] 1569 | 关于正定二次型的判定,有以下\CJKunderdot{等价条件},他们可以互相等价,因此常被用于判定正定二次型. 1570 | 1571 | 设有$ n $元实二次型$f(x_{1},x_{2},\cdots,x_{n})=\bm{X}^{\mathrm{T}}\bm{AX}$,则下列命题相互等价: 1572 | \begin{enumerate} 1573 | \item $f(x_{1},x_{2},\cdots,x_{n})$为正定二次型; 1574 | \item $ \bm{A} $的所有特征值都是\CJKunderdot{正实数}; 1575 | \item $ \bm{A} $的秩和正惯性指数都是$ n $; 1576 | \item $ \bm{A} $与单位矩阵$ \bm{E} $合同; 1577 | \item 存在可逆矩阵$ \bm{P} $,使得$ \bm{A}=\bm{P}^{\mathrm{T}}\bm{P} $. 1578 | \end{enumerate} 1579 | \end{theorem} 1580 | \begin{defination} 1581 | 如果一个$ n\times n $实对称矩阵$ \bm{A} $所对应的二次型为正定二次型,那么$ \bm{A} $称为\textbf{正定矩阵}.\index{ZDJZ@正定矩阵} 1582 | \end{defination} 1583 | \begin{feature} 1584 | 实正对称矩阵$ \bm{A} $的行列式大于$ 0 $,即$ |\bm{A}|>0 $. 1585 | \end{feature} 1586 | \begin{defination}[$ \bm{A} $的$ k $阶顺序主子式的定义] 1587 | $ n $阶顺序$ \bm{A}=(a_{ij})_{nn} $的$ k\ (k=1,2,\cdots,n) $阶子式: 1588 | \[{p_k} = \left| {\begin{array}{cccc} 1589 | {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1k}}}\\ 1590 | {{a_{21}}}&{{a_{22}}}& \cdots &{{a_{2k}}}\\ 1591 | \vdots & \vdots &{}& \vdots \\ 1592 | {{a_{k1}}}&{{a_{k2}}}& \cdots &{{a_{kk}}} 1593 | \end{array}} \right|\] 1594 | 称为$ \bm{A} $的$ k $阶\textbf{顺序主子式}.\index{KJSXZZS@$ k $阶顺序主子式} 1595 | \end{defination} 1596 | \begin{theorem}[实对称矩阵与正定矩阵的关系] 1597 | \,\\ 1598 | \begin{itemize} 1599 | \item 实对称矩阵是\CJKunderdot{正定矩阵}$ \Leftrightarrow P_k>0 \ (k=1,2,\cdots,n) $恒成立; 1600 | \item 实对称矩阵是\CJKunderdot{负定矩阵}$ \Leftrightarrow (-1)^kP_k>0 \ (k=1,2,\cdots,n) $恒成立. 1601 | \end{itemize} 1602 | \end{theorem} 1603 | 1604 | \section{曲面及其方程} 1605 | 我们认为:\textbf{曲面}是由\CJKunderdot{一个动点}或一条\CJKunderdot{动曲线}按照一定的条件或规律运动所形成的轨迹.由此可以导出曲面上动点$ P $的坐标$ (x,y,z) $所满足的方程为\index{QM@曲面} 1606 | \[ 1607 | F(x,y,z)=0. 1608 | \] 1609 | 1610 | 当且仅当$ P $为曲面上的点时其坐标才满足上述方程,那么曲面的几何性质必然可以由该方程反映.因此可以用方程来表述曲面.则把上述方程称为\textbf{曲面的方程}\index{QMDFC@曲面的方程},把上述曲面称为\textbf{方程的图形}\index{FCDTX@方程的图形}. 1611 | \subsection{球面及其方程} 1612 | \begin{defination}[球面定义] 1613 | 空间中到定点的距离等于定长的点的集合称为\textbf{球面}.定点称为\textbf{球心},定长称为\textbf{半径}. 1614 | \index{QM@球面} 1615 | \index{QX@球心} 1616 | \index{BJ@半径} 1617 | \end{defination} 1618 | 1619 | 1620 | 1621 | 1622 | 1623 | 1624 | %打印索引————————————— 1625 | \newpage 1626 | \addcontentsline{toc}{chapter}{附录} 1627 | \addcontentsline{toc}{section}{索引} 1628 | \appendix 1629 | \kaishu 1630 | \printindex 1631 | %——————————————— 1632 | \end{document} --------------------------------------------------------------------------------