├── 01-方程组的几何解释
└── 01-方程组的几何解释.md
├── 02-矩阵消元
└── 02-矩阵消元.md
├── 03-乘法和逆矩阵
└── 03-乘法和逆矩阵.md
├── 04-A的LU分解
└── 04-A的LU分解.md
├── 05-转置-置换-向量空间R
└── 05-转置-置换-向量空间R.md
├── 06-列空间和零空间
└── 06-列空间和零空间.md
├── 07-求解Ax=0-主变量-特解
└── 07-求解Ax=0-主变量-特解.md
├── 08-求解Ax=b-可解性和解的结构
└── 08-求解Ax=b-可解性和解的结构.md
├── 09-线性相关性-基-维数
└── 09-线性相关性-基-维数.md
├── 10-四个基本子空间
└── 10-四个基本子空间.md
├── 11-矩阵空间-秩1矩阵和小世界图
└── 11-矩阵空间-秩1矩阵和小世界图.md
├── 12-图和网络
└── 12-图和网络.md
├── 13-复习一
└── 13-复习一.md
├── 14-正交向量与子空间
└── 14-正交向量与子空间.md
├── 15-子空间投影
└── 15-子空间投影.md
├── 16-投影矩阵和最小二乘
└── 16-投影矩阵和最小二乘.md
├── 17-正交矩阵和Gram-Schmidt正交化
└── 17-正交矩阵和Gram-Schmidt正交化.md
├── 18-行列式及其性质
└── 18-行列式及其性质.md
├── 19-行列式公式和代数余子式
└── 19-行列式公式和代数余子式.md
├── 20-克拉默法则-逆矩阵-体积
└── 20-克拉默法则-逆矩阵-体积.md
├── 21-特征值和特征向量
└── 21-特征值和特征向量.md
├── 22-对角化和A的幂
└── 22-对角化和A的幂.md
├── 23-微分方程和exp(At)
└── 23-微分方程和exp(At).md
├── 24-马尔可夫矩阵-傅立叶级数
└── 24-马尔可夫矩阵-傅立叶级数.md
├── 25-复习二
└── 25-复习二.md
├── 26-对称矩阵及正定性
└── 26-对称矩阵及正定性.md
├── 27-复数矩阵和快速傅里叶变换
└── 27-复数矩阵和快速傅里叶变换.md
├── 28-正定矩阵和最小值
└── 28-正定矩阵和最小值.md
├── 29-相似矩阵和若尔当形
└── 29-相似矩阵和若尔当形.md
├── 30-奇异值分解
└── 30-奇异值分解.md
├── 31-线性变换及对应矩阵
└── 31-线性变换及对应矩阵.md
├── 32-基变换和图像压缩
└── 32-基变换和图像压缩.md
├── 33-复习三
└── 33-复习三.md
├── 34-左右逆和伪逆
└── 34-左右逆和伪逆.md
├── 35-期末复习
└── 35-期末复习.md
├── LICENSE
├── README.md
├── SUMMARY.md
├── addition
└── 换个角度看线性代数1.md
├── images
├── 10
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├── 01
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├── A
│ └── 01
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├── LA_whole_1.png
└── LA_whole_2.png
└── index.html
/01-方程组的几何解释/01-方程组的几何解释.md:
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1 | # 一、方程组的几何解释
2 |
3 | ## 1、Overview (概述)
4 |
5 | 本节主要介绍 线性代数 的基础。
6 |
7 | 学习线性代数的应用之一就是求解复杂的方程问题，因此本节内容就从 解方程 开始。
8 |
9 | 本节核心内容是从 row picture (行图像) 和 column picture (列图像) 的角度求解方程。
10 |
11 | * row picture 的角度：将方程的行向量抽出来，画出其 row picture, 通过画图的的方式求出方程的解，即行向量的交点。
12 |
13 | * column picture 的角度：则将方程表示为xA + yB = b，将方程转化为列向量的线性组合，此时，方程的求解转换为找到 (x, y)，使得列向量 (A, B) 正确组合得到向量b。
14 |
15 | ## 2、方程组的几何解释基础
16 |
17 | ### 2.1、二维行图像（矩阵的角度）
18 |
19 | 我们首先通过一个例子了解二维方程组（2个未知数，2个方程），如下：
20 |
21 | 
22 |
23 | 我们首先按 row (行) 将方程组写成矩阵形式：
24 |
25 | 
26 |
27 | ```
28 | 系数矩阵(A): 将方程组系数按行提取出来，构造完成的一个矩阵。
29 | 未知向量(x): 将方程组的未知数提取出来，按列构成一个向量。
30 | 向量(b): 将等号右侧结果按列提取，构成一个向量。
31 | 因此，原方程可以表示为：
32 | Ax = b
33 | ```
34 |
35 | 构造完成相应的矩阵形式了，我们将对应的 行图像 画出来。
36 |
37 | 所谓 行图像，就是在系数矩阵上，一次取一行 构成方程，在坐标系上做出如下图。和我们在初等数学中学习的作图求解方程的过程无异。
38 |
39 | 
40 |
41 | ### 2.2、二维列图像（向量的角度）
42 |
43 | 从列图像的角度，我们再次求解上面的方程：
44 |
45 | 
46 |
47 | 在这一次的求解过程中，我们将方程按列提取，使用的矩阵为：
48 |
49 | 
50 |
51 | 如上所示，我们使用 列向量 构成系数矩阵，将问题转化为: 将向量  与向量  正确组合，使得其结果构成  , 这个过程称为列向量的线性组合（Linear Combination of Columns）
52 |
53 | 接下来我们使用 列图像 将方程组展现出来，并求解：
54 |
55 | 
56 |
57 | 即寻找合适的 x，y 使得 x 倍的 (2,-1) + y 倍的 (-1,2)得到最终的向量 (0,3)。很明显能看出来，1 倍 (2,-1) + 2 倍 (-1,2) 即满足条件。
58 |
59 | 反映在图像上，明显结果正确。
60 |
61 | 我们再想一下，仅仅对  这个方程，如果我们任意取 x 和 y ，那么我们得到的是什么呢？
62 |
63 | 很明显，能得到任意方向的向量，这些向量能够布满整个平面。
64 |
65 | 注：事实上，任取 x 和 y, 可以得到矩阵 A 的列向量的所有线性组合，这个线性组合构成了矩阵 A 的列空间，这个列空间构成了整个二维实空间 R2。从方程的角度看，对于任意的向量 b, Ax = b 始终有解。（**后续课程会介绍到**）
66 |
67 | ## 3、方程组的几何解释推广
68 |
69 | ### 3.1、高维行图像
70 |
71 | 我们将方程组的维数进行推广，从三维开始，给定三维矩阵如下：
72 |
73 | 
74 |
75 | 如果我们继续使用上面介绍的 做出行图像 来求解问题，那么会得到一个很复杂的图像。
76 |
77 | 矩阵如下：
78 |
79 | 
80 |
81 | 对应方程： Ax = b
82 |
83 | 
84 |
85 | 如果绘制行图像，很明显这是一个三个平面相交得到一点，我们想直接看出这个点的性质可谓是难上加难。
86 |
87 | 比较靠谱的思路是先联立其中两个平面，使其相交于一条直线，再研究这条直线与平面相交于哪个点，最后得到点坐标即为方程的解。
88 |
89 | 这个求解过程对于三维来说或许还算合理，那四维呢？五维甚至更高维数呢？
90 |
91 | 直观上很难直接绘制更高维数的图像，这种行图像受到的限制也越来越多。
92 |
93 | ### 3.2、高维列图像
94 |
95 | 我们使用上面同样的例子：
96 |
97 | 
98 |
99 | 如果我们使用列图像的思路进行计算，那矩阵形式就变为：
100 |
101 | 
102 |
103 | 左侧是（矩阵列向量的）线性组合，右侧是线性组合得到的结果，这样一来思路就清晰多了，“寻找正确的线性组合”成为了解题关键。
104 |
105 | 
106 |
107 | 很明显这个问题是一个特例，我们只需要取 x = 0, y = 0, z = 1 就得到了结果，这在行图像之中并不明显。
108 |
109 | 使用 列图像 求解方程， 其优势在于这种求解方法更加系统，只需 寻找正确的线性组合，而不用绘制每个行方程的图像之后寻找那个很难看出来的点。
110 |
111 | 另外一个优势在于这种方法更加灵活，如果我们改变向量 b，如：
112 |
113 | 
114 |
115 | 只需要重新寻找一个线性组合就够了，但是如果我们使用的是行图像呢？那意味着我 们要完全重画三个平面图像，就简便性来讲，两种方法高下立判。
116 |
117 | * 那么，对任意的 b 是不是始终能够求解 Ax = b 这个矩阵方程呢？
118 |
119 | 从线性组合的角度看，问题可以转化为：对 3*3 的系数矩阵 A，其列的线性组合是不是始终可以覆盖整个三维空间呢？
120 |
121 | 对于我们举的这个例子来说，一定可以，还有我们上面 2*2 的那个例子，也可以覆盖整个平面，但是有一些矩阵就是不行的。
122 |
123 | 例如，若矩阵A的所有的列向量都集中在同一个平面内的话，那么其线性组合也将集中在同一个平面内，因此对于大部分不在这个平面内的 b, 均是无法构造出来的，这种情形称为 **奇异** 。
124 |
125 | 例如，若三个列向量本身就构成了一个平面，那么这三个向量组合成的向量只能活动在这个平面上，肯定无法覆盖整个三维空间。
126 |
127 | 
128 |
129 | 这三个向量就构成了一个平面。
130 |
131 | 
132 |
133 | ### 3.3、矩阵乘法
134 |
135 | 以下从列向量与行向量的角度介绍一下矩阵乘法。
136 |
137 | 
138 |
139 | ## 4、小结
140 |
141 | 这部分内容主要是对线性代数概念的初步了解。
142 |
143 | 从解方程谈起，从行空间逐步过渡到列空间，可以看到从列空间角度求解方程可以将解方程问题转化为求列向量的线性组合的问题。
144 |
145 | 最后，从列向量与行向量的角度介绍一下矩阵乘法。
146 |
147 |
148 | 【[下一章：02-矩阵消元](../02-矩阵消元/02-矩阵消元.md)】
149 |
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/02-矩阵消元/02-矩阵消元.md:
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1 | # 02-矩阵消元
2 |
3 | ## 1、Overview (概述)
4 |
5 | 本节首先介绍了消元法的初等变换实现，然后进一步介绍了向量与矩阵的乘法，在此基础上，研究了消元法的矩阵实现（消元矩阵）。最后简单地引入了置换矩阵与逆矩阵。
6 |
7 | 具体内容如下：
8 |
9 | * Elimination（消元法）
10 |
11 | * Back-Substitution （回代）
12 |
13 | * Elimination Matrics（消元矩阵）
14 |
15 | * Matrix Multiplication（矩阵乘法）
16 |
17 | > 事实上，计算机语言实现方程的求解，就是通过 Elimination 实现的。
18 |
19 | ## 2、使用 消元法 求解方程：消元与回代
20 |
21 | ### 2.1、消元法介绍
22 |
23 | 消元法的使用场景主要分为以下 2 种情况:
24 | * 成功: 矩阵的主元不包含 0， 也就是说，是可逆矩阵。
25 | * 失败: 矩阵的主元有至少一个 0，也就是说，是不可逆矩阵。
26 |
27 | 注：(1)这里的矩阵，一般指方程的系数矩阵;(2)消元法其实不止应用于系数矩阵为可逆矩阵的情形，也可以应用于不可逆的情形。（后续会讲到）
28 |
29 | 对于一些 “好” 的系数矩阵（可逆矩阵） A 来说，我们可以使用消元法来求解方程 Ax=b ，我们还是从一个例子谈起。
30 |
31 | 
32 |
33 | 我们仍然使用矩阵计算，将方程写为矩阵形式 Ax=b 。如下:
34 |
35 | 
36 |
37 | 所谓矩阵的消元法，其实与我们在初等数学中学习的解二元一次方程组的消元法同理，都是通过将不同行的方程进行消元运算来简化方程，最后能得到简化的方程组。只不过这里我们把 系数 单独抽出来进行运算，寻找一种 矩阵 情况下的普遍规律而已。
38 |
39 | 
40 |
41 | 
42 |
43 | 
44 |
45 | 注：消元法针对的其实是列向量，进行的操作却是初等行变换。
46 |
47 | > 事实上，计算机使用消元法求解方程的时候，关键步骤就是求解从 A 到 U 的过程，然后通过算法不断优化这个过程，以提高方程求解的速度。
48 |
49 | 那么消元法何时失效呢？如果失效该如何处理呢？
50 |
51 | 事实上，并不是所有的 A 矩阵都可消元处理，需要注意在我们消元过程中，如果主元位置（左上角）为 0，
52 | 那么意味着这个主元不可取，需要进行 “换行” 处理：
53 |
54 | 首先看它的下一行对应位置是不是 0，如果不是，就将这两行位置互换，将非零数视为主元。
55 | 如果是，就再看下下行，以此类推。若其下面每一行都看到了，仍然没有非零数的话，那就意味着这个矩阵不可逆，
56 | 消元法求出的解不唯一。（后续我们会接着讨论这种解不唯一的情形，即系数矩阵不可逆的情形，这其实也是消元法的应用范围）
57 |
58 |
59 | 下面是 3 个例子:
60 |
61 | 
62 |
63 | ### 2.2、回带求解
64 |
65 | 其实回带求解应该和消元法同时进行，只不过在我们讲解的时候以及在一些软件工作原理中它们是先后进行的，所以我们这里分开讨论，下面我们首先介绍—— **增广矩阵**。
66 |
67 | 例如，
68 |
69 | 
70 |
71 | 此方程的 **增广矩阵** 形式为:
72 |
73 | 
74 |
75 | 可以一下就看出来，**增广矩阵** 就是把 **系数矩阵 A** 和 **向量 b** 拼接成一个矩阵就行了。
76 |
77 | 
78 |
79 | 从下向上开始求解，很容易求出 **x, y, z** 的值了。
80 |
81 |
82 | ## 3、消元法的矩阵视角：消元矩阵
83 |
84 | ### 3.1、向量相关的矩阵乘法： **矩阵行与列的线性组合** （虽然就简单，但这是个很重要的视角）
85 |
86 | 上面的消元法是从**初等行变换的角度**介绍了消元法的具体操作，接下来我们需要用矩阵来表示变换的步骤，这也十分有必要，因为这是一种 “系统地” 变换矩阵的方法。
87 |
88 | 
89 |
90 | 导致错误。其实学过矩阵之间的乘法之后这些东西都极为简单，但这里还是建议大家尽量从向量的角度去考虑问题。
91 |
92 | 以下举例说明。
93 |
94 | * 矩阵列向量的线性组合
95 |
96 | 
97 |
98 | * 矩阵行向量的线性组合
99 |
100 | 例1：
101 |
102 | 
103 |
104 | 例2：
105 |
106 | 
107 |
108 | ### 3.2、消元矩阵
109 |
110 | 学会了行向量与矩阵之间的乘法，我们就可以使用行向量对矩阵的行做操作了。
111 |
112 | 所谓 **消元矩阵，就是将消元过程中的初等行变换用矩阵乘法实现**。这其实就是从矩阵视角实现消元法，而不再是初等数学的方式了。
113 |
114 | 
115 |
116 | 我们消元过程是将第一行 乘以 -3 加到 第二行，这是对第二行的操作，那么就从单位阵的第二行着手:
117 |
118 | 
119 |
120 | ### 3.3.1、置换矩阵初探：行变换和列变换
121 |
122 | 上面我们谈到了初等矩阵 E，事实上，还有一类初等矩阵：置换矩阵（Permutation matrix）,主要作用是交换2行或者2列。
123 |
124 | 
125 |
126 | ### 3.3.2、逆矩阵初探
127 |
128 | 通过消元法，我们可以实现矩阵 A -> U, 那么我们考虑一个反过程，将消元法得到的 **矩阵 U** 变回到未经消元的 **矩阵 A** 。 那么如何实现呢？
129 |
130 | 答案就是 **乘上一个逆矩阵** ！！！。
131 |
132 | 
133 |
134 | ## 4、小结
135 |
136 | 本节首先介绍了消元法的初等变换实现，然后进一步介绍了向量与矩阵的乘法，在此基础上，研究了消元法的矩阵实现（消元矩阵）。最后简单地引入了置换矩阵与逆矩阵。
137 |
138 | 具体内容如下：
139 |
140 | * 初等数学的视角看待消元法求解方程
141 | * 矩阵变换的视角看待消元法求解方程（将初等变换的过程通过矩阵（乘法）表示，EA = U）
142 | * 向量与矩阵的乘法：矩阵行与列的线性组合
143 | * 置换矩阵初探：矩阵的行变换与列变换
144 | * 逆矩阵初探：如何从 U -> A
145 |
146 | 本章节的消元法以后会常用到，要熟练掌握才可以。
147 |
148 | 【[上一章：01-方程组的几何解释](../01-方程组的几何解释/01-方程组的几何解释.md)】【[下一章：03-乘法和逆矩阵](../03-乘法和逆矩阵/03-乘法和逆矩阵.md)】
149 |
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1 | # 03-乘法和逆矩阵
2 |
3 | ## 1、Overview（概述）
4 |
5 | 前面介绍了向量和矩阵的乘法，这一节我们要介绍一下两个矩阵之间的乘法。并讨论逆矩阵存在的条件。最后再介绍求解逆矩阵的方法。
6 |
7 | ## 2、矩阵乘法
8 |
9 | ### 2.1 矩阵乘法定义视角求解：矩阵乘法最常见的求解方式
10 |
11 | 
12 |
13 | 
14 |
15 | 
16 |
17 | ### 2.2、矩阵行向量与列项量线性组合的视角求解
18 |
19 | #### 2.2.1、列组合
20 |
21 | 还记得在 【[02-矩阵消元](https://github.com/chenyyx/notes-linear-algebra/blob/master/02-%E7%9F%A9%E9%98%B5%E6%B6%88%E5%85%83/02-%E7%9F%A9%E9%98%B5%E6%B6%88%E5%85%83.md)】中学过的 矩阵与列向量的乘积，得到一个列向量，如下:
22 |
23 | 
24 |
25 | #### 2.2.2、行组合
26 |
27 | 
28 |
29 | 同样,按照形式，这次将矩阵 A 看做行向量组合就行了:
30 |
31 | 
32 |
33 | ### 2.3、扩展视角：列乘以行
34 |
35 | 
36 |
37 | 
38 |
39 | ### 2.4、分块做乘法
40 |
41 | 分块乘法就是宏观上的矩阵乘法，比如现在有一个 50 * 50 的矩阵与 50 * 50 矩阵相乘，一个一个进行运算很麻烦，尤其是如果矩阵在某一区域上有一定的性质， 那么我们可以将其分块，如：
42 |
43 | 
44 |
45 | ## 3、矩阵的逆
46 |
47 | ### 3.1、矩阵的逆是否存在？不存在的条件是什么？
48 |
49 | 
50 |
51 | 
52 |
53 | ### 3.2、若存在，矩阵的逆如何求解？
54 |
55 | 
56 |
57 | #### 3.2.1、高斯-若尔当方法（Gauss-Jordan）
58 |
59 | 
60 |
61 | 接下来，我们论证一下它的合理性:
62 |
63 | 
64 |
65 | 
66 |
67 | ## 4、小结
68 |
69 | 本节具体内容如下：
70 |
71 | ### 矩阵乘法：AB = C
72 |
73 | 1.定义的视角
74 |
75 | 常规方法，行 * 列，即AB = sum of (rows of A) * (cols of B)
76 |
77 | 2.线性组合的视角:
78 |
79 | (1) 列的线性组合：C的每一个列向量是A的每一个列向量的线性组合，而B说明了如何进行线性组合，即说明了线性组合的系数。
80 |
81 | (2) 行的线性组合：C的每一个行向量是B的每一个行向量的线性组合，而A说明了线性组合的系数。
82 |
83 | 3.列 * 行的视角
84 |
85 | 扩展方法，列 * 行，即AB = sum of (cols of A) * (rows of B)
86 |
87 | 4.矩阵的分块乘法：满足以上所有的乘法方式
88 |
89 | ### 矩阵的逆（Inverses - For Square matrices）
90 |
91 | 1.矩阵的逆是否存在？如何判断不存在？
92 |
93 | > 对于矩阵A, 若存在向量x，使得Ax = 0 (x != 0)，则此矩阵为奇异矩阵（singular matrix），即矩阵A不可逆。即可以找到A的列向量的线性组合，使得其为0。
94 |
95 | 证明：记A的逆为B，若矩阵A可逆，则BA = I, 又Ax = 0, 因此 BAx= Ix = 0, 所以x = 0, 矛盾。#
96 |
97 | 注：若A为方阵，则AB = BA = I, 左逆 = 右逆。
98 |
99 | 2.若存在，矩阵的逆如何求解？（invertible, nonsingular（可逆，非奇异））
100 |
101 | > Gauss-Jordan (Solve 2 equs at once.)
102 |
103 | * AB = I (B为A的逆)
104 | * **[A | I] -> [I | B]**
105 |
106 | 证明： E [A | I] = [EA | E] = [I | ?], 因此EA = I, 因此E = B, 即E是A的逆。
107 |
108 | 在这个章节中，我们从不同的角度认识了矩阵的乘法，并介绍了逆矩阵的相关知识以及如何求解逆矩阵。
109 |
110 | 这个章节的内容很好地体现了线性代数这门课的优点之一: 少有繁琐的证明，更多的理解与类比。多多从向量、空间、线性组合的角度去认识矩阵之间的运算，这才是线性代数这门课的核心之一。
111 |
112 | 【[上一章：02-矩阵消元](../02-矩阵消元/02-矩阵消元.md)】【[下一章：04-A 的 LU 分解](../04-A的LU分解/04-A的LU分解.md)】
113 |
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/04-A的LU分解/04-A的LU分解.md:
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1 | # 04-A 的 LU 分解
2 |
3 | ## 1、Overview（概述）
4 |
5 | 在上一篇 「03-乘法和逆矩阵」 中，我们在末尾介绍了 **逆矩阵** 的一些内容，今天我们首先完善之前讲到的 **逆矩阵** 的内容，然后使用 **消元矩阵** 介绍 A 的 LU 分解，即: 将矩阵 A 分解为 下三角矩阵 L 与上三角矩阵 U ，介绍这种运算的普遍规律。
6 |
7 | 最后我们再一次提起之前介绍过的 “行交换矩阵”，引入 **置换矩阵** 概念。
8 |
9 | ## 2、逆矩阵性质
10 |
11 | 我们考虑一个问题: 若方阵 A, B 都是可逆矩阵的话， AB 的逆矩阵是什么呢？
12 |
13 | 矩阵A存在逆矩阵，根据矩阵逆的定义：
14 |
15 | 
16 |
17 | 要想求出方阵AB的逆，只需要找到满足矩阵满足以下式子：
18 |
19 | 
20 |
21 | 显然，下式满足：
22 |
23 | 
24 |
25 | 因此AB的逆为：
26 |
27 | 
28 |
29 | 由于我们的下一章要涉及到矩阵的转置问题，我们在这里一并讨论矩阵转置与矩阵的逆的关系。
30 |
31 | ### 2.1、转置矩阵基础
32 |
33 | 转置矩阵就是将原矩阵各行换成对应列，所得到的新矩阵，如:
34 |
35 | 
36 |
37 | 看起来就像是沿着左上角开始的一条对角线翻折了一样。
38 |
39 |
40 |
41 | 
42 |
43 | 注：单位阵的转置矩阵为单位阵。
44 |
45 | ### 2.2、转置矩阵和逆矩阵的关系
46 |
47 | 介绍完了转置矩阵，接下来我们看一看它和逆矩阵有什么联系？（我倒要看看你葫芦里卖的什么药）。
48 |
49 | 首先，逆矩阵满足：
50 |
51 | 
52 |
53 | 为了找到转置矩阵与逆矩阵之间的关系。我们对上式两边同时进行转置运算，得到
54 |
55 | 
56 |
57 | 上式说明了，**矩阵A的转置的逆矩阵是A的逆矩阵的转置矩阵**。即：
58 |
59 | 
60 |
61 | 上式说明对于方阵A，**转置运算和求逆运算的顺序可以颠倒。**
62 |
63 | > 为什么  会变换到  的前面来呢？我们想象一下，最后乘积所得的单位矩阵 I 中每个元素都是由 A 的行向量与  的列向量构成，当做转置运算时，I 沿对角线翻折，可以理解为整个乘法运算图形也要沿着 I 的对角线进行翻折，这样就解释了上式。
64 |
65 | 
66 |
67 | 
68 |
69 | 
70 |
71 |
72 | ## 3、A 的 LU 分解
73 |
74 | 在开始A的LU分解之前，先回顾一下消元法。对于方程 Ax = b，消元法的学习经过了以下几个过程：
75 |
76 | * 初等数学的视角看待消元法求解方程
77 | * 消元：通过初等变换将 A 转换为 U
78 | * 回代：然后使用同样的初等变换过程将向量 b 转换为 c
79 | * 或者也可以统一起来：**[A | b] -> [U | c]**
80 | * 矩阵变换的视角看待消元法求解方程
81 | * 矩阵视角看到消元法，本质上是将初等数学的方式用矩阵表示
82 | * A -> EA = U ，其中，E 为消元矩阵
83 |
84 | 无论哪一种视角，消元法本质都是对系数矩阵 A 进行初等行变换（EA = U） ，只是表示方式不同而已。
85 |
86 | * 下面我们介绍消元法的第三个视角，A = LU 分解的视角 （只是我这样看而已）
87 |
88 | 以二维矩阵 A 为例，若 A 可逆，则下式存在：
89 |
90 | 
91 |
92 | 其中，  这样的矩阵一定有逆矩阵，因为它本身就是单位阵变化过来的。所以原式可以改写成:
93 |
94 | 
95 |
96 | 这一形式即为 A = LU 形式，这个过程就是分解过程。
97 |
98 | 接下来，我们讨论一下3*3矩阵的情况（假设没有行互换），则有：
99 |
100 | 
101 |
102 | 因此：
103 |
104 | 
105 |
106 | **那么矩阵 L 是不是有什么特殊之处呢？**
107 |
108 | 我们通过下面一道例题来探讨一下。
109 |
110 | 
111 |
112 | 
113 |
114 | 从上面这个过程，我们可以看出，其实矩阵 L 是一个下三角矩阵（Lower），其下三角的系数刚好是初等行变换的主元位置的乘数。因此搞清楚如何进行初等行变换，就可以搞定 L 的求解，从而将 A 分解为 LU 的形式，其中，U为上三角矩阵(Upper)。
115 |
116 | **那么消元法的运算量有多大呢**？
117 |
118 | 比如现在我们有一个 100*100 的超级大的矩阵(无 0 元素）。
119 |
120 | 我们需要运算多少次（将一行乘一定倍数后加到另一行上消元，每一个这样的过程计为 一次运算）之后，才能将其化为上三角矩阵 U 呢？
121 |
122 | 
123 |
124 | 
125 |
126 |
127 | ## 4、置换矩阵
128 |
129 | **置换矩阵** 的定义：在数学上，特别是在矩阵理论中，置换矩阵是一个方形二进制矩阵，它在每行和每列中只有一个1，而在其他地方则为0。
130 |
131 | 我们之前接触过行变换所用到的矩阵，即是将单位阵 I 按照对应行变换方式进行操作之后得到的矩阵。它可以交换矩阵中的两行，代替矩阵行变换。什么时候 我们需要使用矩阵行变换呢？
132 |
133 | 一个经典的例子就是：在消元过程中，当矩阵主元位置上面不是 1 时，我们就需要用行变换将主元位置换回 1。
134 |
135 | 这样的由单位阵变换而来的矩阵，通过矩阵乘法可以使被乘矩阵行交换。
136 |
137 | 我们将这样的矩阵称为置换矩阵（Permutation, P）。我们通过一个例子来熟悉一下置换矩阵。
138 |
139 | 
140 |
141 | 推广到 n 阶矩阵，n 阶矩阵有 n！个置换矩阵，就是将单位矩阵 I 各行重新排列后所有可能的情况数量。我自己的理解是：单看第一行，有 n 种排列方式， 再看除去第一行，第一列的(n-1)阶矩阵，再看其第一行，有(n-1)种排列方式。 以此类推，直到最后的 1 阶，有 1 种排列方式，由乘法原理，就有了 n!个置换 矩阵。
142 |
143 |
144 |
145 | 那么问题来了：置换矩阵的逆该如何求解呢？
146 |
147 | 事实上，置换矩阵的逆就在其「**置换矩阵群**」中。
148 |
149 |
150 |
151 | ## 5、小结
152 |
153 | 线性代数的前面这部分基本是一些技巧的运算。本节我们对矩阵的转置，逆矩阵性质进行了部分介绍，学习了矩阵的 A = LU 分解，了解了这种分解方式的优点所在，并学会了直接构造 L 矩阵，简化消元过程。**这些技巧与知识都是我们接下来学习的重要基础**。
154 |
155 |
156 |
157 | 总结一下，本节其实主要是对消元法的一个扩展，即 A 的 LU 分解。到此，消元法就全部结束了。主要有：
158 |
159 | * 初等数学的视角看待消元法
160 | * 矩阵变换的视角看待消元法
161 | * 矩阵分解的视角看待消元法
162 |
163 | 其次，本节还在上一节求解单个矩阵 A 的逆的基础上，求解 AB 矩阵的逆。
164 |
165 | 然后，又引出了转置矩阵和置换矩阵两类矩阵。
166 |
167 |
168 |
169 | 【[上一章：03-乘法和逆矩阵](../03-乘法和逆矩阵/03-乘法和逆矩阵.md)】【[下一章：05-转置，置换，向量空间R](../05-转置-置换-向量空间R/05-转置-置换-向量空间R.md)】
170 |
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/05-转置-置换-向量空间R/05-转置-置换-向量空间R.md:
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1 | # 05-转置、置换、向量空间 R
2 |
3 | ## 1、Overview（概述）
4 |
5 | 在上一节 「04-A 的 LU 分解」中，我们谈到了 **置换矩阵** 和 **转置矩阵** ，这一节，我们再深入谈一下 **置换矩阵** 和 **转置矩阵** ，并简要介绍 **对称阵**。之后便进入学习 **linear-algebra （线性代数）** 的关键所在 —— **向量空间** 与 **子空间**，到此才算是开始进入线性代数的大门。
6 |
7 | 具体内容如下：
8 |
9 | * 置换矩阵（Permutations）
10 | * 转置矩阵（Transposes）
11 | * 对称矩阵（Symmetric）
12 | * 向量空间（Vector Spaces）及其子空间（Subspaces）
13 |
14 | ## 2、置换矩阵
15 |
16 | > Permutations P: Identity matrix with reordered rows, which are used to execute row exchange.
17 |
18 | **置换矩阵（Permutations, P） ** 是指对行进行重排的单位阵。主要用来进行行交换。
19 |
20 | * 置换矩阵可以通过对单位阵的行进行重排得到，以下举个例子说明：
21 |
22 | 
23 |
24 | * 那么对于 n * n阶矩阵来说，有多少个置换矩阵呢？
25 |
26 | 答案是： n! 种，也就是将单位阵 I 各行重新排列后所有可能的情况的数量。
27 |
28 | * 那么置换矩阵有什么性质呢？
29 |
30 | 所有的置换矩阵都满足：
31 |
32 | 因此，**所有的置换矩阵都是可逆矩阵**，其逆为其转置，即：
33 |
34 | 详细解释如下：
35 |
36 | 
37 |
38 |
39 |
40 | 
41 |
42 | * 置换矩阵有什么用呢？
43 |
44 | 前面在讲消元法的时候，主元位置为 0 是一件很让人头疼的事情，这时就需要考虑 **置换矩阵 P**来完成行交换，确保消元过程顺利进行。
45 |
46 | 上一节我们学习 **A = LU** 分解时，我们没有考虑要交换行的过程，如果我们想写出更加普适的 LU 分解式的话，必须把 行交换 情况考虑进去，即:
47 |
48 | **PA = LU**
49 |
50 | 先用 行交换 使得主元位置不为 0，行顺序正确。其后，再用 LU 分解。
51 |
52 | ## 3、转置矩阵与对称矩阵
53 |
54 | ### 3.1、转置矩阵
55 |
56 | **转置矩阵（Transposes）**：对于矩阵A，其转置矩阵满足：
57 |
58 | 示例如下：
59 |
60 | 
61 |
62 | ### 3.2、对称阵
63 |
64 | **对称矩阵（Symmetric Matrixs）**，顾名思义，就是对角线两侧元素对应相等的矩阵。
65 |
66 | * 对称矩阵满足以下性质：
67 |
68 | * 那么如何构造对称阵呢？
69 |
70 | 这里有个性质，对于任意矩阵 A, 矩阵 总是一个对称阵。
71 |
72 | 证明如下：
73 |
74 | 
75 |
76 | ## 4、向量空间与子空间
77 |
78 | ### 4.1、向量空间
79 |
80 | **向量空间（Vector Spaces） **表示一整个空间的向量。
81 |
82 | 但是要注意，不是任意向量的集合都能被称为 向量空间。
83 |
84 | 向量空间必须对以下运算满足封闭性，即两个向量进行向量的加法与数乘之后得到的向量仍然属于这个向量空间，即向量进行线性组合之后所得到的向量仍然属于这个向量空间。
85 |
86 | * 向量加法：V + W
87 | * 向量数乘：aV , 其中，a为标量。
88 |
89 | * 向量的线性组合：aV + bW。其中，a，b为标量。事实上，以上2种运算都是线性组合的特例。
90 |
91 | 任何向量空间及其子空间都需要满足封闭性。
92 |
93 | 以下举例说明：
94 |
95 | 
96 |
97 | 
98 |
99 | 
100 |
101 | 很明显，这部分空间无法满足 “线性组合仍在空间中” 的要求，比如 数乘运算 时，随便取个负数，向量就跑到第三象限去，脱离 D 空间范围内了。
102 |
103 | ### 4.2、子空间
104 | 上面的反例已经证明了。在向量空间里随便取其一部分，很可能得到的不是 向量空间。
105 |
106 | 那如果我们取向量空间的一部分，将其打乱，构成的有没有可能是 向量空间 呢？
107 |
108 | 
109 |
110 | 
111 |
112 | ---
113 |
114 | 总结一下，我们得到以下结论：
115 |
116 | * 对于二维实向量空间（R2），其子空间有：
117 | * R2本身
118 | * 零向量构成的空间，记为Z
119 | * 所有经过原点（零向量）的直线，记为L
120 | * 对于三维实向量空间（R3），其子空间有：
121 | * R3本身
122 | * 零向量构成的空间
123 | * 所有经过原点（零向量）的直线
124 | * 所有经过原点（零向量）的平面
125 |
126 | ### 4.3、列空间的简要介绍
127 |
128 | 那么如何构造子空间呢？
129 |
130 | 上面介绍的子空间都是基于已知的图像来寻找的，接下来我们来通过矩阵来构造出一个子空间。
131 |
132 | 比如: 列向量构造出的列空间。**对于矩阵A, 其所有列向量的线性组合构成了列空间，记为C(A)**。
133 |
134 | 
135 |
136 | 
137 |
138 | 
139 |
140 | 这里还需要注意列向量之间的性质，如果列向量之间就是共线的，那么其列空间就是一条过原点的直线。
141 |
142 |
143 |
144 | ## 5、小结
145 |
146 | 本节算是结束了之前部分对基本运算和基本概念的介绍。介绍了 **向量空间** 和 **子空间**，并由子空间引出了**通过具体的列向量构成的空间 —— 列空间**。如何理解 **空间** 十分重要，本节中对低维的空间做了图，目的主要是便于我们理解 “**空间**” 这一概念。
147 |
148 | 【[上一章：04-A 的 LU 分解](../04-A的LU分解/04-A的LU分解.md)】【[下一章：06-列空间和零空间](../06-列空间和零空间/06-列空间和零空间.md)】
149 |
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/06-列空间和零空间/06-列空间和零空间.md:
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1 | # 06-列空间和零空间
2 |
3 | ## 1、Overview（概述）
4 |
5 | 上一节 「05-转置-置换-向量空间R」介绍了 **向量空间与子空间** 。
6 |
7 | 本节在此基础上，总结了向量空间需要满足的条件以及子空间的性质。接下来介绍了两种**构建子空间**的方法，即列空间与子空间的构造方法。
8 |
9 | 具体内容如下：
10 |
11 | * 向量空间条件与子空间性质
12 | * 列空间：从矩阵的列向量出发，通过线性组合，构造列空间。
13 | * 零空间：对于零空间，刚开始并不知道其中有何向量，已知的信息只有这些向量需要满足的方程 Ax = 0， 通过让 x 满足特定条件而得到零空间。
14 |
15 | ## 2、向量空间与子空间
16 |
17 | ### 2.0、 向量空间
18 |
19 | 向量空间需要满足加法封闭性以及数乘封闭性，即给定向量空间的向量 v 和 w, 需要满足：
20 |
21 | * 加法封闭性：v + w 在该向量空间内
22 | * 数乘封闭性：a v 在该向量空间内
23 |
24 | 这个条件本质上其实是 v 和 w 的所有线性组合 av + bw 在该向量空间内。
25 |
26 | 注：任何向量空间必须包含零向量Z。
27 |
28 | ### 2.1、子空间回顾
29 |
30 | 
31 |
32 | 
33 |
34 | 很明显，**子空间直线 L** 或 **平面 P** 上，任取两个向量相加，得到的向量仍在该子空间中。而且将其上的向量做数乘伸长或缩短一定倍数，其结果也还在该子空间中。所以它们都对线性运算封闭。
35 |
36 | ### 2.2、子空间的 “交” 与 “并”
37 |
38 | 上面我们都是分别研究的两个子空间，那么接下来我们对两个空间之间联系展开讨论。
39 |
40 | #### 2.2.1、P∪L 空间
41 | 对于𝑅 3 的子空间 P 与 L，首先要研究的就是它们的并空间，即：现有一向量集合，包含了 P 与 L 中的所有向量，那么这个向量集合是否构成了子空间呢？
42 |
43 | **No!!!**
44 |
45 | 很明显，我们将直线 L 与平面 P 看做同一个集合 P∪L 之后，这个集合**对线性运算并不封闭**。
46 |
47 | 比如我们随便在直线 L 上取一个向量 a，在平面 P 上取一个向量 b。此时向量 a+b 方向就会夹在直线 L 与平面 P 之间，脱离了 P∪L 的范围。所以 P∪L 无法构成子空间。
48 |
49 | 
50 |
51 | 
52 |
53 | #### 2.2.2、P ∩ L 空间
54 |
55 | 
56 |
57 | 
58 |
59 | ## 3、列空间 C(A)
60 |
61 | ### 3.1、列空间回顾
62 |
63 | 
64 |
65 | 那么这个子空间有多大呢？这就需要用 Ax = b 方程来解释了。
66 |
67 | ### 3.2、 Ax = b 的空间解释（从 A 的角度）
68 |
69 | 
70 |
71 | 
72 |
73 | 
74 |
75 | 
76 |
77 | 总结：
78 |
79 | * 问题1: 对于所有的向量b，Ax = b 是否总是有解？
80 |
81 | 这个问题本质上其实相当于问：矩阵的 A 的列向量形成的列空间是否可以铺满整个四维实空间 R4？对于上面的方程 Ax = b 而言，四个方程，3个未知数，对于任意的向量 b 是不会总是有解的。事实上，3个列向量构成的构成的列空间 C(A) 只是 R4 的一个子空间，因此，对于任意的向量 b 方程是不会总是有解的。
82 |
83 | * 问题2：那么什么样的向量b，能够使得 Ax = b 有解呢？
84 |
85 | 这个问题相当于问：矩阵的 A 的列向量线性组合能够形成什么样的向量或者说什么样的向量在 C(A) 中？
86 |
87 | 因此，只要向量 b 在 C(A) 中，Ax = b 有解。
88 |
89 | * 问题3：去掉 A 的哪一列，剩下的列向量仍然可以得到相同的子空间 C(A) ?
90 |
91 | 由上面的分析，我们可以看出来，第三列是前面两列的线性组合，因此第三列对子空间的形成没有任何贡献，因此第三列可以去掉。剩下的两列对于子空间的形成作出了贡献，我们成为 **主列**。
92 |
93 | 因此，**C(A) 是四维向量空间的一个二维子空间。**
94 |
95 | **注：对于矩阵 Am*n 而言，其列空间为 m 维向量空间 Rm 的一个子空间。**
96 |
97 | ## 4、零空间 N(A)
98 |
99 | ### 4.1、零空间介绍
100 |
101 | 
102 |
103 | 
104 |
105 | 
106 |
107 | ### 4.2、Ax = b 的空间解释(从 x 的角度)
108 |
109 | 那如果上面构造零空间的方程右侧变为任意向量的话，其解集 x 还能构成 向量空间 吗？
110 |
111 | 
112 |
113 | **注：对于矩阵 Am*n 而言，其零空间为 n 维向量空间 Rn 的一个子空间。**
114 |
115 | ## 5、小结
116 |
117 | 在本节中我们学习了**列空间**与**零空间**。从 Ax = b 入手，给出了**两种构建子空间**的方法:
118 | 1. 列空间：从矩阵的列向量出发，通过线性组合，构造列空间。
119 | 2. 零空间：对于零空间，刚开始并不知道其中有何向量，已知的信息只有这些向量需要满足的方程 Ax = 0， 通过让 x 满足特定条件而得到零空间。
120 |
121 | 这两种构造子空间的方法需要掌握。其实就是一个从 A 的列向量入手，一个从 x 的解集入手构建子空间的问题。
122 |
123 | 【[上一章：05-转置，置换，向量空间R](../05-转置-置换-向量空间R/05-转置-置换-向量空间R.md)】【[下一章：07-求解Ax=0、主变量、特解](../07-求解Ax=0-主变量-特解/07-求解Ax=0-主变量-特解.md)】
124 |
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/07-求解Ax=0-主变量-特解/07-求解Ax=0-主变量-特解.md:
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1 | # 07-求解Ax=0-主变量-特解
2 |
3 | ## 1、Overview（概述）
4 |
5 | 上一节我们讨论了矩阵的列空间和零空间的相关问题，那么这一节我们从它们的定义过渡到它们的计算，即如何求解出这些空间的一般形式。给出一种可以解出 Ax=0 中的 x 构成的零空间的算法。
6 |
7 | 具体内容如下：
8 |
9 | * 求解矩阵的零空间的算法（求解 Ax = 0 的算法）
10 | * 对于方程 Ax = 0，对 A 向下消元，碰到无法取得主元的列，不管它，接着消元，直到得到行阶梯矩阵 U（Row echelon form matrix）, 确定主列及自由列，确定主变量与自由变量，令自由变量为 0 和 1，求出特解，特解的线性组合即为零空间；接着向上消元，使得主元全部变为1，主元上方和下方的变量全部为 0，得到最简行阶梯矩阵 R(Reduced row echelon form matrix, rref)，由最简行阶梯矩阵即可得到零空间矩阵N（Null space Matrix）。由零空间矩阵 N 可以直接得到特解（因为零空间矩阵各列由特解组成）
11 | * Ax = 0 -> Ux = 0 -> Rx = 0 -> RN = 0
12 | * 这里的 R 用最简的形式包含了所有的信息。
13 | * 一些概念
14 | * 主行、主列 (Pivot cols, rows)
15 | * 主变量、自由变量（Pivot variables, Free variables)
16 | * 矩阵的秩（Rank）
17 |
18 | ## 2、求解方程Ax = 0：消元法求解零空间
19 |
20 | 在之前讲解「 使用消元法求解方程组 Ax=b」时，我们对一种情况是无法处理的，那就是矩阵 A 不可逆的情况。之前对这种情况的解释是：求出的解不唯一。这正好对应了我们现在学到的 “空间” 概念。
21 |
22 | 接下来我们使用消元法求解这种矩阵不可逆的情况。
23 |
24 | 我们从最简单的零空间（b = 0）的计算谈起，即求解 Ax = 0。
25 |
26 | ### 2.1、消元：消元法确定主变量与自由变量（A -> U）
27 |
28 | 
29 |
30 | 
31 |
32 | 
33 |
34 | ### 2.2、回代：对自由变量赋值覆盖零空间
35 |
36 | 
37 |
38 | 
39 |
40 | ### 2.3、算法总结
41 |
42 | 
43 |
44 | 
45 |
46 | 
47 |
48 | ## 3、简化行阶梯形式（U -> R）
49 |
50 | 
51 |
52 | 
53 |
54 | 
55 |
56 | 
57 |
58 | 
59 |
60 | 
61 |
62 | 
63 |
64 | ## 4、小结
65 |
66 | 这节学习的是计算 Ax = 0 中的 x 构成的零空间的方法，即：消元，找主变量与自由变量，为自由变量赋值，得到特解，特解线性组合得到零空间。后面又介绍了化简 U 变为 R，直接利用 R 的结构得到零空间矩阵 N 的方法。这节重在计算流程，需要加以练习才可以熟练掌握。
67 |
68 | 【[上一章：06-列空间和零空间](../06-列空间和零空间/06-列空间和零空间.md)】【[下一章/08-求解Ax=b、可解性和解的结构](../08-求解Ax=b-可解性和解的结构/08-求解Ax=b-可解性和解的结构.md)】
69 |
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/08-求解Ax=b-可解性和解的结构/08-求解Ax=b-可解性和解的结构.md:
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1 | # 08-求解 Ax=b 可解性和解的结构
2 |
3 | ## 1、Overview(概述)
4 |
5 | 上一节我们主要介绍方程 Ax = 0 的算法。这一节，我们主要介绍：
6 |
7 | * 方程 Ax = b 的完整算法。
8 |
9 | * 在此基础上，介绍矩阵的秩对方程组解的个数的影响
10 |
11 | ## 2、求解方程 Ax=b
12 |
13 | ### 2.1、可解性
14 |
15 | 这节我们要介绍如何来解 Ax=b ，但是这个方程并不一定有解。我们通过一个例子来说明下这个问题:
16 |
17 | 【例1】
18 |
19 | 
20 |
21 | 这里的 A 有一个特点，就是 1, 2 两行之和等于第三行，因此可以将第三行全部变为0。 根据「02-矩阵消元」提出的增广消元法，列增广矩阵后消元，得到矩阵 U:
22 |
23 | 
24 |
25 | 再看这个条件： ，它反映了一种线性组合特点，即 b 向量的第三个分量是前两个分量之和。反过来看 A 矩阵本身特点，发现 A 矩阵第三行也是前两行的和。记得之前我们说过， Ax=b 有解的条件是 b 在 A 的列空间中。这个例子再一次印证了这个条件。
26 |
27 | 总结一下：
28 |
29 | * **向量 b 满足什么条件时，方程 Ax = b 总有解？**
30 | * 列空间角度：当且仅当 b 属于 A 的列空间 C(A)。
31 | * 线性组合角度： b 是 A 的列向量的线性组合。
32 | * 初等变换的角度（消元法的角度）：如果 A 的各行线性组合得到零行，那么对 b 中元素线性组合，也必将得到自然数 0.
33 |
34 | 注：以上这3种说法等价。
35 |
36 | ### 2.2、求解 Ax = b 的算法
37 |
38 | * 方程 Ax = b 的通解 x = 特解 + 零空间向量
39 | * 特解xp：首先令所有的自由变量为0，然后求解主变量，主变量和自由变量组合得到了特解 xp。
40 | * 零空间向量 xn: 求解方程 Ax = 0 即可得到xn。
41 | * x = xp + xn
42 |
43 | > 证明：Axp = b Axn = 0 => A(xp + xn) = 0
44 |
45 | 对于 Ax=b 这个方程， **通解 = 矩阵零空间向量 + 矩阵特解** 。这很好理解，矩阵零空间向量代入方程最后结果等于 0 ，所以它不会影响等式，而是把方程的解向量扩展到一个类似子空间的空间上。
46 |
47 | 以下举例说明这个算法。
48 |
49 | 设  满足可解条件，我们来彻底求解方程。
50 |
51 | 上一节中，我们求解 Ax = 0 方程的特解时，分别将自由变量赋值为 0/1, 这是因为最特殊的赋值方式： **自由变量全部赋值为 0** 的方式在 Ax=0 中行不通，因为这样的赋值方式在 Ax=0 中得到的是零向量。我们最后求出的通解为:
52 |
53 | 
54 |
55 | 但是 Ax=b 这个方程不同，只要 b 不是 0，我们就可以将 **自由变量全部赋值为 0** 。本例中我们使用此方法得到特解。
56 |
57 | 以下是完整的过程：
58 |
59 | 
60 |
61 | 这个解集在几何角度的解释为：  上的一个二维平面，很显然，这个解集无法构成一个向量空间，因为解集中连零向量都没有。也就可以理解为：解集在空间中变现为  中的一个不过原点的平面。
62 |
63 |
64 | ## 3、m*n 的矩阵 A 的秩与解的关系
65 |
66 | 很明显在上面我们消元求 Ax=b 的过程中，矩阵 A 的秩对最后解的形式有至关重要的影响，下面我们就总结一下这方面的问题。
67 |
68 | ### 3.1、列满秩
69 |
70 | 列满秩是指：m*n 的矩阵 A 中，秩 R = n < m 。例如：
71 |
72 | 
73 |
74 | 主元。也就是说这样的矩阵零空间向量中只有一个向量：零向量。这样的矩阵 A 的构造的方程 Ax=b ，要么不满足可解条件，要么只有一种符合对应方程组的解。
75 |
76 | 解最后只有两种情况：
77 |
78 | * 有解且唯一
79 | * 无解，不满足可解条件
80 |
81 | ### 3.2、行满秩
82 |
83 | 行满秩是指： m*n 的矩阵中，秩 R = m < n 。例如：
84 |
85 | 
86 |
87 | 上一节我们介绍过，这样的矩阵消元之后会是 [I F] 形式（I 表示单位阵，F 表示其他部分），很明显由这样的矩阵构成的方程 Ax=b ，最后肯定是无穷多个解，因为这种矩阵中，永远有（n-R）个自由变量。
88 |
89 | ### 3.3、行列皆满秩
90 |
91 | 当 m*n 矩阵 A 是方阵时，若 A 满秩，则 R = m = n 。例如：
92 |
93 | 
94 |
95 | 这种矩阵经过消元，必可以化为单位阵 I ，自由变量个数为 0。只能得到一个全是主元的方程组。所以这种矩阵构成的 Ax=b 方程最后只能有唯一解。
96 |
97 | ### 3.4、不满秩
98 |
99 | 秩 R < n, 且 R < m 时，A 矩阵不满秩，此时 A 可化简为  形式，最后化简结果中有 0 行。如 【例1】 中的矩阵，b 的分量与零行牵扯出了可解条件的存在。所以这样的矩阵 A 所构成的 Ax=b 方程解有两种情况：
100 |
101 | * 不满足可解条件（零行导致的可解条件）
102 | * 解无穷多个（特解 + 零空间所有向量）
103 |
104 | ### 3.5、总结
105 |
106 | 观察以上情况，自由变量总为 (n-r) 个，所以先判断自由变量个数可以初步判断 Ax=b 的解的结构：
107 |
108 | 
109 |
110 | 而可解条件的产生是由于 A 消元之后的 0 行导致的，所以再判断 A 消元之后会不会有零行产生就可以确定解的结构：
111 |
112 | 
113 |
114 | ## 4、小结
115 |
116 | 本节基于上一节中零空间的求解，延伸介绍了 Ax=b 的一般解法。并从 A 矩阵秩的角度探讨了秩与方程解的结构之间的联系。至此我们已经学完了解方程 Ax=b 形式矩阵方程的所有问题，在这个过程中，我们需要注意的无非就是自由变元个数，以及通解和特解的问题，整体而言，这部分重在求解流程以及如何理解。正确理解向量空间之后，理解这种矩阵方程问题也就不是什么难事了！！
117 |
118 | 【[上一章：07-求解Ax=0、主变量、特解](../07-求解Ax=0-主变量-特解/07-求解Ax=0-主变量-特解.md)】【[下一章：09-先行先惯性、基、维数](../09-线性相关性-基-维数/09-线性相关性-基-维数.md)】
119 |
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/09-线性相关性-基-维数/09-线性相关性-基-维数.md:
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1 | # 09-线性相关性-基-维数
2 |
3 | ## 1、Overview（概述）
4 |
5 | 在之前的章节中，我们可能经常会发现矩阵中有时会有一行或几行本身就是前面几行的线性组合的情况，那么这一节，我们就从这种线性相关的特征入手，介绍空间中的几个重要的概念 。
6 |
7 | 具体内容如下：
8 |
9 | * 线性无关（Linear Independence）
10 | * 生成空间（Spanning a Space)
11 | * 空间的基（Basis）与维数（Dimension）
12 |
13 | 注：以上内容针对**向量组**
14 |
15 | ## 2、线性无关与线性相关
16 |
17 | ### 2.1、背景知识
18 |
19 | 我们从之前学习的 Ax=0 方程谈起。
20 |
21 | 给定 m*n 的矩阵 A （m < n）:
22 |
23 | 
24 |
25 | 方程 Ax = 0 中，，未知数一共 n 个，方程一共 m 个，未知数 x 的个数比方程的个数多。
26 |
27 | 所以矩阵 A 的零空间中除零向量以外还有其他非零向量，因为矩阵 A 最多 m 个主列，而 m < n，因此矩阵 A 至少有 n-m 个自由列，即矩阵 A 至少有 n-m 个自由向量，这就造成了矩阵 A 的零空间 N(A) 中有除了零向量之外的非零向量。
28 |
29 | ### 2.2、线性无关
30 |
31 | #### 1. 从向量的线性组合的角度看线性无关
32 |
33 | 设向量组为  。
34 |
35 | 对于任意的系数c（不全为0），各个向量的线性组合无法组合得到零向量，则称此向量组**线性无关**。
36 |
37 | 对于任意的系数c（不全为0），各个向量的线性组合可以等于零向量，则称此向量组**线性相关**。
38 |
39 | 因此，如果一个向量组中有零向量存在，那么这个向量组一定是线性相关的。例如，对于向量 v1, v2 (v2= 0), 其线性组合 0 v1 + 6 v2 = 0，因此此向量组线性相关，总能找到一个非零系数，使得向量的线性组合变成0，这都是零向量惹的祸。
40 |
41 | 举例说明：
42 |
43 | 
44 |
45 | 
46 |
47 | 显然，A 矩阵是 n > m 型的矩阵。根据背景知识 (2.1)， Ac = 0 这个方程对应的零空间中，除了零空间肯定还有其他非零向量，即存在一种 c 不全为 0 的情况，使 A 各列线性组合后得到 0 。因此矩阵 A 的各列：  线性相关。
48 |
49 | #### 2. 从矩阵的零空间看待线性无关
50 |
51 | 线性无关仅与向量组有关系，与矩阵毫无关系，但是我们可以将向量组放入矩阵之中，将向量组的线性无关与矩阵的零空间联系起来。
52 |
53 | 设向量组是矩阵 A 的列向量。
54 |
55 | 若向量组 **线性无关**，则矩阵 A 的零空间只包含零向量， N(A) = {0}
56 |
57 | 若向量组 **线性相关**，则矩阵 A 的零空间除了零向量之外还包含其他非零向量 c，N(A) = {0, c}，即 Ac = 0
58 |
59 | #### 3. 从矩阵的秩的角度看待线性无关
60 |
61 | 设向量组是矩阵 A （m*n）的列向量。
62 |
63 | 若向量组 **线性无关**，则 rank(A) = n，此时没有自由变量。
64 |
65 | 若向量组 **线性相关**，则 rank(A) < n，此时有 n-r 个自由变量。
66 |
67 | ## 3、生成空间与空间的基、维数
68 |
69 | ### 3.1 生成空间
70 |
71 | **生成空间** ：向量 `v1, v2, ..., vl `生成一个空间意味着这个空间是由这个向量组所有的线性组合构成的。
72 |
73 | 例如：给定矩阵 A, A 的所有列向量的线性组合生成了 A 的列空间 C(A)，即 C(A) 由列向量的所有线性组合构成。
74 |
75 | 那么这个列向量组是线性无关的吗? 也许是，也许不是。
76 |
77 | 我们更加关心的是这样的向量组：既能生成空间，又是线性无关的。这意味着向量组的个数必须是适当的个数。
78 |
79 | * 若个数不够，则向量组无法构成空间；
80 | * 若个数过多，则向量组可能不是线性无关的。
81 |
82 | 这就引出了空间的基的概念。
83 |
84 | ### 3.2 空间的基
85 |
86 | 空间的基是一组向量。那么这组向量具有什么性质呢？
87 |
88 | * 向量组线性无关
89 | * 向量组生成了整个空间
90 |
91 | 那么某个空间的基是不是唯一的呢？
92 |
93 | > 不是，因为我们始终可以找到不同的向量组，满足基的所有性质，因此某个空间的基有多个，并且基的个数是确定的，我们把某个空间中的基的个数称为空间的维数。
94 |
95 | 那么若将基的所有向量作为列向量组成一个矩阵，这个矩阵有何特征呢？
96 |
97 | > 将所有基的向量作为列向量组合成一个矩阵 A ，这个矩阵 A 必然是可逆的。因为所有的列向量都是线性无关的，这意味着给定系数矩阵 c, 矩阵 A 的所有列向量的线性组合要想得到零向量，必须使得 c = 0，即方程 Ax = 0 的零空间只有零向量，因此矩阵 A 必然是可逆的。此矩阵称为非奇异矩阵。 详细的证明见：[03-乘法和逆矩阵](../03-乘法和逆矩阵/03-乘法和逆矩阵.md)
98 |
99 |
100 | 以下详细解释。
101 |
102 | 
103 |
104 | 
105 |
106 | 
107 |
108 | ### 3.3 空间的维数
109 |
110 | **空间的维数** 是指空间的基的个数。
111 |
112 | 对于一个向量空间，可以有不同的基，但是基的总量是一定的，即不同的基包含的向量的个数是一定的。
113 |
114 | 理解维数也很简单，像我们的三维空间，其基一定是三个三维向量（三个向量，每个向量有三个分量），四维空间的基也一定是四个四维向量。
115 |
116 | 举例如下：
117 |
118 | 
119 |
120 | (2) 找出 A 列空间中的一个基
121 |
122 | 从 A 的结构看来：
123 |
124 | 
125 |
126 | 第3列 = 第1列 + 第2列
127 | 第4列 = 第1列
128 |
129 | 我们可以取前两列作为基。所以 A 的列空间的维数为：2。
130 |
131 | 再看 A 矩阵，显然 A 的秩为 2，因为 A 消元后只有两个主列。所以有：矩阵 A 的秩 = 矩阵 A 主列的个数 = A 列空间维数
132 |
133 | 这下我们就将矩阵的秩与列空间的维数联系了起来，而更重要的是，我们知道了列空间的维数，那么在这个列空间中随便找两个线性无关的向量，它们就可以构成一组基，这组基就可以生成这个列空间。
134 |
135 | (3) A 对应零空间的维数为多少？
136 |
137 | 所谓零空间维数，即是零空间基的个数，也是 Ax=0 的特解的个数，还可以理解为： Ax=0 的解中自由变量的个数。
138 |
139 | 最简单的方法是解 Ax=0 这个方程。经过消元，自由变量赋值，回代，最后得到两个特解：
140 |
141 | 
142 |
143 | 所以此零空间的维数为 2.
144 |
145 | **总结一下**：
146 |
147 | 对于矩阵 A (m*n)，rank(A) = r，则有：
148 |
149 | * 列空间的维数：dim(C(A)) = r = 主列的个数 = 主列的个数
150 | * 零空间的维数：dim(N(A)) = n - r = 自由变量的个数
151 |
152 | 一旦知道了一个空间的维数 dim，那么我们可以随便找 dim 个线性无关的向量构成空间的基。
153 |
154 | ## 6、小结
155 |
156 | 这一章节的内容相对来说，是比较简单的，就是几个概念的介绍：线性相关/无关，基，维数。这一节这几个概念都是用来描述空间的，了解了这几个概念之后，我们便将矩阵的秩，矩阵的自由变量等概念与空间的维数，基，线性相关/无关 的判定联系起来。便于我们接下来对向量空间的研究。
157 |
158 | 【[上一章：08-求解Ax=b、可解性和解的结构](../08-求解Ax=b-可解性和解的结构/08-求解Ax=b-可解性和解的结构.md)】【[下一章：10-四个基本子空间](../10-四个基本子空间/10-四个基本子空间.md)】
159 |
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/10-四个基本子空间/10-四个基本子空间.md:
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1 | # 10-四个基本子空间
2 |
3 | ## 1、Overview（概述）
4 | 在前几节中，我们介绍过列空间，零空间。但是这还远远不够，对一个矩阵来说，我们能从它身上挖掘出来的空间不止这些，所以这一节我们介绍四个基本子空间，也是对空间概念的补充，更便于我们接下来的讨论。
5 |
6 | 具体内容如下：
7 |
8 | 给定 m*n 的矩阵 A，其转置矩阵为A'。研究矩阵 A 的以下几个子空间：
9 |
10 | * 列空间 C(A) Rm
11 | * 零空间 N(A) Rn
12 | * 行空间 C(A') Rn
13 | * 左零空间 N(A') Rm
14 |
15 | 接下来，将 Rn 空间扩展到 Rn * n 空间，研究：
16 |
17 | * 矩阵空间：所有的 3*3 的矩阵构成的空间
18 |
19 | ## 2、四个基本子空间
20 |
21 | ### 2.1、四个基本子空间的维数与基
22 |
23 | 给定 m*n 的矩阵 A，其四个子空间的基本性质如下：
24 | * **列空间**
25 |
26 | 何为列空间？
27 |
28 | > 列空间就是矩阵 A 的列向量的所有线性组合构成的空间。因为每个列向量都有 m 个分量，因此列空间是 Rm的子空间。
29 |
30 | 如何求解列空间的基呢？
31 |
32 | > 对矩阵 A 进行初等行变换 (rows reduction)，通过消元法化简 A 得到矩阵 U，即可确认矩阵 A 的主列，这些主列列就是 C(A) 的一组基。
33 |
34 | 那么列空间的维数是多少呢？
35 |
36 | > 设矩阵 A 的秩为 r ，则 A 有 r 个主列，这 r 个主列就是列空间 C(A) 一组基，一组基里有 r 个向量，所以列空间维数为 r 。
37 |
38 | * **零空间**
39 |
40 | 何为零空间？
41 |
42 | > 零空间就是方程 Ax = 0 的所有特解线性组合构成的空间。因为每个特解都有 n 个分量，因此零空间是 Rn的子空间。
43 |
44 | 如何求解零空间的基呢？
45 |
46 | > 对矩阵 A 进行初等行变换 (rows reduction)，通过消元法化简 A 得到矩阵 U，即可确认矩阵 A 的自由列，从而确认矩阵 A 的自由变量，通过对自由变量赋值（0 或者 1），即可得到零空间的基向量，这些基向量就构成了零空间 N(A) 的一组基。
47 |
48 | 那么零空间的维数是多少呢？
49 |
50 | > 设矩阵 A 的秩为 r ，则 A 的自由列为 n-r 列。这 n-r 列决定了 x 中的 n-r 个自由变量，赋值后就构成了零空间的 n-r 个基向量，这n-r个基向量构成了零空间的基，故零空间维数为 n-r。
51 |
52 | * **行空间**
53 |
54 | 何为行空间？
55 |
56 | > 行空间就是矩阵 A 的行向量的所有线性组合构成的空间。因为每个行向量都有 n 个分量，因此行空间是 Rn的子空间。
57 |
58 | 如何求解行空间的基呢？
59 |
60 | > 方法1：将矩阵 A 转置，然后按照列空间的基的求法求解 A 的转置矩阵的列空间。
61 | >
62 | > 方法2：对矩阵 A 进行初等行变换 (rows reduction)，通过消元法化简 A 得到矩阵 A 的行最简形（reduced row echelon form）R，若矩阵 A 的秩为 r, 则矩阵 A 的行空间的基就是矩阵 R 的前 r 个向量。
63 | >
64 | > 注意：初等行变换会改变矩阵的列空间，但是不会改变矩阵的行空间。
65 |
66 | 以下举例说明：
67 |
68 | 
69 |
70 | 那么行空间的维数是多少呢？
71 |
72 | > 设矩阵 A 的秩为 r ，则根据以上分析可以知道，行空间维数为 r 。
73 |
74 | * **左零空间**
75 |
76 | 何为左零空间？
77 |
78 | > 左零空间就是矩阵 A 的转置构成的方程组 A'y = 0 的特解的所有的线性组合构成的空间。因为每个特解都有 m 个分量，因此零空间是 Rm的子空间。
79 | >
80 | > 左零空间本质上是矩阵 A 行向量线性组合的系数构成的向量线性组合构成的空间。即 y' A = 0 的特解的所有线性组合构成的空间。
81 |
82 | 如何求解左零空间的基呢？
83 |
84 | > Gauss-Jordan 消元法
85 | >
86 | > [A | I] -> [I | A-1]
87 | >
88 | > [A | I] -> [R | E]
89 | >
90 | > 找到矩阵中，使得方程 EA = 0 那些行就是左零空间的基。
91 |
92 | 以下通过示例具体介绍。
93 |
94 | 首先介绍一下左零空间，写成方程形式为  ，我们不处理  ，所以将
95 |
96 | 
97 |
98 | 
99 |
100 | 
101 |
102 | 注：Gauss-Jordan 消元的具体过程如下：
103 |
104 | 
105 |
106 | 那么左零空间的维数是多少呢？
107 |
108 | > 设矩阵 A 的秩为 r ，则根据以上分析可以知道，左零空间维数为 m-r 。
109 |
110 | ### 2.2、四个基础空间图像
111 |
112 | 
113 |
114 | ## 3、矩阵空间初探：新的向量空间
115 |
116 | 事实上，向量空间的元素并不一定都是由实数组成的向量，也可以是实数组成的 3 * 3 的矩阵。
117 |
118 | 矩阵空间：矩阵构成的空间。例如所有的 3*3 矩阵构成的空间。
119 |
120 | 那么矩阵空间是否满足空间要求的封闭性呢？
121 |
122 | > 给定 3*3 矩阵 A, B, 常数 c
123 | >
124 | > * 加法封闭性：A + B 仍然是 3 * 3 矩阵
125 | > * 数乘封闭性：cA 仍然是 3 * 3 矩阵
126 | >
127 | > 因此，矩阵空间满足空间所要求的封闭性。
128 |
129 | 因此，所有的 3 * 3 矩阵构成了一个线性空间，那么它的子空间有什么呢？
130 |
131 | > 上三角矩阵，对称矩阵，对角矩阵，这些矩阵都是其子空间。
132 |
133 | 很明显，上三角矩阵与对称矩阵的交集为对角矩阵(diag)。深入研究对角矩阵，就要给出它的基，
134 |
135 | 
136 |
137 | 注：矩阵空间是空间从Rn 到 Rn*n的扩展。
138 |
139 | ## 4、小结
140 |
141 | 这一节课基本是概念的介绍，介绍了四个基本空间，其中比较新的内容是左零空间，即行向量的线性组合得到零，这部分要好好理解。前面重点在于 2.2 的图，在我们以后的应用中会经常用到。另外还稍微说到了一下向量空间的概念，为我们下一节的内容埋下了一个小伏笔。
142 |
143 |
144 |
145 | 【[上一章：09-线性相关性、基、维数](../09-线性相关性-基-维数/09-线性相关性-基-维数.md)】【[下一章：11-矩阵空间、秩1矩阵和小世界图](../11-矩阵空间-秩1矩阵和小世界图/11-矩阵空间-秩1矩阵和小世界图.md)】
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1 | # 11-矩阵空间-秩1矩阵和小世界图
2 |
3 | ## 1、Overview（概述）
4 |
5 | 上节末尾我们介绍了矩阵空间，这是一种延伸的向量空间。这节我们从矩阵空间谈起，介绍矩阵空间的维数，基等问题。渗透一些微分方程与线性代数之间的联系，并介绍秩为 1 的矩阵特点。
6 |
7 | 具体内容如下：
8 |
9 | * 矩阵空间深入（新的向量空间）
10 | * 秩1矩阵与空间
11 | * 小世界图初探（Small world graphs）
12 |
13 | ## 2、矩阵空间深入
14 |
15 | 还是上一节中的问题，将所有的 3 * 3 的矩阵都看做 “向量空间” 中的元素，很明显，由所有 3 * 3 矩阵构成的集合中，矩阵之间加法与数乘矩阵都是封闭的，所以所有 3 * 3 矩阵构成的集合 M 是矩阵空间（一种新的向量空间）。
16 |
17 | 上节中我们介绍过，M 有两个基本的子空间：
18 | * 子空间S：3*3 对称矩阵构成的子空间
19 | * 子空间U: 3*3 上三角矩阵构成的子空间
20 |
21 | S 与 U 空间相交，可以得到M的另一个子空间D：由3*3对角阵构成的子空间。
22 |
23 | ### 2.1、矩阵空间的基与维数
24 |
25 | 讨论一个向量空间，需要搞清楚空间的基与维数。
26 |
27 | 以下讨论矩阵空间 M 、子空间 S、U、D以及空间 S+U 的基与维数。
28 |
29 | * 矩阵空间 M 类似于实数向量空间 R9，只是 M 的元素为 `3*3` 矩阵，而 R9 的元素为 `9*1` 向量，故二者的基与维数也很类似。
30 | * M = S + U
31 |
32 | 
33 |
34 | 
35 |
36 |
37 |
38 | ### 2.2、微分方程与线性代数的关系
39 |
40 | 
41 |
42 | ## 3、秩一矩阵
43 |
44 | ### 3.1、秩一矩阵
45 |
46 | 秩一矩阵是指秩为1的矩阵。秩一矩阵具有以下优点：
47 |
48 | * 可以分解为`列向量 * 行向量`的形式。
49 | * 可以像搭积木一样构建出其他矩阵。
50 |
51 | 以下举例说明。
52 |
53 | 
54 |
55 | 秩一矩阵的另外一个优点是它可以 “构建” 其他矩阵，比如秩为 4 的矩阵， 通过四个秩一矩阵就能搭建出来。具体过程类似于矩阵乘法中的“列乘行”形式， 通过一列一行搭出一个矩阵。
56 |
57 | ### 3.2、几个问题
58 |
59 | #### 3.2.1 问题1：同秩矩阵是否可以构成空间？
60 |
61 | 
62 |
63 | #### 3.2.2 问题2：子空间的转化
64 |
65 | 下面我们通过这样一个例子再加深一下对子空间的印象：
66 |
67 | 
68 |
69 | 因此维度为 **n-r=3** ， S 的零空间是三维空间。其基为 Av=0 的三个特解。
70 |
71 | 
72 |
73 | 因此列空间的维度为1，左零空间的维度为0。
74 |
75 | 再看看A 的行空间C(A')。A 的行空间为 A 的行向量的所有线性组合，其维度为1。
76 |
77 | ## 4、小世界图初探：图论与线性代数的关系
78 |
79 | 这一小节内容是对下一节 “图与网络” 的引出，主要讨论图论与线性代数的关系。
80 |
81 | **图** 是节点与边的集合，即 Graph={nodes, edges}。如图所示：
82 |
83 | 
84 |
85 | 这个图包括五个节点和六条边，可以用一个 5*6 的矩阵来表示其中的所有信息。具体内容我们下节课再说。
86 |
87 | 所谓的**“六度分割理论”**就可以通过图来描述。
88 |
89 | ## 5、小结
90 |
91 | 这一节中主要介绍了线性空间，一并介绍了类似于矩阵空间，解空间这一类空间的存在。另外，秩一矩阵将我们之前学习的矩阵乘法列乘行方式联系了起来，便于分解，可以搭建矩阵。
92 |
93 | 【[上一章：10-四个基本子空间](../10-四个基本子空间/10-四个基本子空间.md)】【[下一章：12-图和网络](../12-图和网络/12-图和网络.md)】
94 |
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1 | # 12-图和网络
2 |
3 | ## 1、Overview（概述）
4 |
5 | 这一节中我们主要介绍图与矩阵之间的关联，利用矩阵来说明图的特点。这一节中与之前几节的区别主要在于，前面例子中的矩阵中的元素大都是为了说明性质编造出来的，而本节中矩阵中的元素都是来源于实际问题，更能体现出我们之前介绍的性质在实际问题中有什么用。
6 |
7 | 本节主要以电路定律为例，介绍线性代数的应用。具体内容如下：
8 |
9 | * 图的深入：图和关联矩阵
10 | * 电路定律：欧姆定律、基尔霍夫定律与平衡方程
11 |
12 | ## 2、图和关联矩阵
13 |
14 | 线性代数在互联网、电路、信号系统、微积分等方面有着广泛的应用。本小节以电路网络为例讨论线性代数的实际应用。
15 |
16 | 在研究实际问题的过程中，我们首先需要将实际问题抽象成数学模型，然后用数学手段处理数学模型。在处理与图有关的实际问题的时候，首先需要将图抽象为关联矩阵。**由于关联矩阵源于实际问题，因此关联矩阵有效地描述了实际问题的拓扑结构。**
17 |
18 | 接下来，我们主要研究利用线性代数处理与图相关的实际问题。
19 |
20 | ### 2.1 关联矩阵的建立
21 |
22 | 图是由节点(nodes)和边(edges)组成的。图1是一个有向图，有4个节点，5条边。
23 |
24 | 
25 |
26 | 接下来，我们需要将此有向图抽象为一个关联矩阵。关联矩阵的每列表示一个节点，每行表示一条边，边的起点记为-1，边的终点记为1。因此此图的关联矩阵为：
27 |
28 | 
29 |
30 | 如果我们仔细观察矩阵 A 的话，会发现1 2 3行满足：行1+行2=行3，因此3行线性相关。回到图中发现，行1,2,3对应的边构成了一个回路，因此若图中的边构成回路的话，那么其对应的矩阵的行线性相关。
31 |
32 | 接下来，我们详细讨论一下矩阵A的特点。
33 |
34 | ### 2.2 关联矩阵的研究
35 |
36 | * 零空间
37 |
38 | 研究矩阵A的特点，很重要的一个方面就是A的零空间N(A)。接下来我们在电路系统的实际背景下，研究A的零空间。
39 |
40 | 假设列向量 x = [x1, x2, x3, x4]' 为每个节点的电势。
41 |
42 | 零空间主要研究的问题是：如何对列向量进行线性组合可以得到零向量？ 要研究A的零空间，本质上是研究A的各列的线性无关性。若线性无关，则N(A) = {O}，即零空间只包含一个零向量，此时零空间的维度为0。
43 |
44 | 零空间是 Ax = 0 的解的线性组合，因此我们构造方程Ax = 0，求解矩阵 A 的零空间。
45 |
46 | 
47 |
48 | 我们试着猜一下，发现当x = O 或者是 x = [1 1 1 1]‘ 时，Ax = 0。
49 |
50 | 通过求解，得到矩阵 A 的零空间为：c[1 1 1 1]‘，其基 x = [1 1 1 1]‘，其维度 dim N(A) = 1，其秩 rank = 4 - 1 = 3。 A 的零空间的形式说明了节点的电势由一个常数决定，c 决定了节点电势的上升或者下降。rank = 3 说明了任意3个节点的电势线性无关。只有电路中各节点之间存在电势差时，才能产生电流，因此，当每个节点的电势相同时，电路是无法产生电流的。
51 |
52 | * 左零空间
53 |
54 | 接下来我们再研究一下矩阵 A 的左零空间。要找到A的左零空间，需要求解方程 A'y = 0。
55 |
56 | 
57 |
58 | 因此，A的左零空间为所求解，其基为[1 1 -1 0 0]’ 和[0 0 1 -1 1]‘，其维度为 dim = m - r = 5 - 3 = 2。
59 |
60 | 我们仔细看一下， A'y = 0 这一个式子代表什么含义。A'y = 0 展开为：
61 |
62 | 
63 |
64 | 这个式子表示的图如下图所示。我们会发现每个式子都表示给定任意一个节点，其流入的电流等于流出的电流，说明节点不会积累电荷，电荷在整个电路中循环移动，这个定律就是**基尔霍夫定律**（KCL）。另外，我们发现 A 的转置矩阵的主列为1，2，4列，这三列线性无关，从图中可以看出这3列对应的边（红色标记的边）没有构成回路。因此没有构成回路的边在矩阵中构成的向量线性无关。
65 |
66 | 
67 |
68 | * 边、节点、回路的关系
69 |
70 | 
71 |
72 | ### 2.3 线性代数与电路系统
73 |
74 | 将以上内容总结如下：
75 |
76 | 
77 |
78 |
79 | ## 3、小结
80 |
81 | 可以看到，这一节与我们之前讲的内容联系较多，与实际应用联系也比较大。从一个图出发，联系实际物理问题，解释了如何用矩阵阐述欧姆定律以及基尔霍夫定律的。学习了我们本节的内容之后，我们才算真正明白了之前学习的各种空间具体到实际问题有什么作用。
82 |
83 | 【[上一章：11-矩阵空间、秩1矩阵和小世界图](../11-矩阵空间-秩1矩阵和小世界图/11-矩阵空间-秩1矩阵和小世界图.md)】【[下一章：13-复习一](../13-复习一/13-复习一.md)】
84 |
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1 | # 13-复习——经典题型的解法
2 |
3 | ## 1、Overview（概述）
4 |
5 | 这一节是习题课，主要回顾了一下之前学习的内容，需要掌握的一些经典题型的解法。
6 |
7 | ## 2、例题
8 |
9 | 
10 |
11 | 答案：
12 |
13 | 三个张量张开的空间，很明显维数只能是 0, 1, 2, 3. 本题中维数不可能是 0, 因为题设为非零向量。所以最后答案为： 1, 2, 3.
14 |
15 | 
16 |
17 | 答案：只有零向量。
18 |
19 | 
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21 | 
22 |
23 | 分析本题：
24 |
25 | 由秩为 3 可知原矩阵的列向量线性无关，也就是没有线性组合能得到零向量，所以其零空间中只有零向量。
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54 |
55 | ## 3、小结
56 |
57 | 这一节习题课结束后，线性代数的这一部分基础也差不多结束了，接下来的课程会围绕一些别的东西展开，比如 正交，特征值之类的知识，这一部分是基础，所以，大家一定要打好基础。
58 |
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60 | 【[上一章：12-图和网络](../12-图和网络/12-图和网络.md)】【[下一章：14-正交向量与子空间](../14-正交向量与子空间/14-正交向量与子空间.md)】
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3 | 【[上一章：25-复习二](../25-复习二/25-复习二.md)】【[下一章：27-复数矩阵和快速傅里叶变换](../27-复数矩阵和快速傅里叶变换/27-复数矩阵和快速傅里叶变换.md)】
4 |
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1 |
2 |
3 | 【[上一章：26-对称矩阵及正定性](../26-对称矩阵及正定性/26-对称矩阵及正定性.md)】【[下一章：28-正定矩阵和最小值](../28-正定矩阵和最小值/28-正定矩阵和最小值.md)】
4 |
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1 |
2 |
3 | 【[上一章：27-复数矩阵和快速傅里叶变换](../27-复数矩阵和快速傅里叶变换/27-复数矩阵和快速傅里叶变换.md)】【[下一章：29-相似矩阵和若尔当形](../29-相似矩阵和若尔当形/29-相似矩阵和若尔当形.md)】
4 |
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1 |
2 |
3 | 【[上一章：28-正定矩阵和最小值](../28-正定矩阵和最小值/28-正定矩阵和最小值.md)】【[下一章：30-奇异值分解](../30-奇异值分解/30-奇异值分解.md)】
4 |
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1 |
2 |
3 | 【[上一章：29-相似矩阵和若尔当形](../29-相似矩阵和若尔当形/29-相似矩阵和若尔当形.md)】【[下一章：31-线性变换及对应矩阵](../31-线性变换及对应矩阵/31-线性变换及对应矩阵.md)】
4 |
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1 |
2 |
3 | 【[上一章：30-奇异值分解](../30-奇异值分解/30-奇异值分解.md)】【[下一章：32-基变换和图像压缩](../32-基变换和图像压缩/32-基变换和图像压缩.md)】
4 |
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1 |
2 |
3 | 【[上一章：31-线性变换及对应矩阵](../31-线性变换及对应矩阵/31-线性变换及对应矩阵.md)】【[下一章：33-复习三](../33-复习三/33-复习三.md)】
4 |
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1 |
2 |
3 | 【[上一章：32-基变换和图像压缩](../32-基变换和图像压缩/32-基变换和图像压缩.md)】【[下一章：34-左右逆和伪逆](../34-左右逆和伪逆/34-左右逆和伪逆.md)】
4 |
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1 |
2 |
3 | 【[上一章：33-复习三](../33-复习三/33-复习三.md)】【[下一章：35-期末复习](../35-期末复习/35-期末复习.md)】
4 |
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1 |
2 |
3 | 【[上一章：34-左右逆和伪逆](../34-左右逆和伪逆/34-左右逆和伪逆.md)】【[下一章：附加-换个角度看问题](../addition/换个角度看线性代数1.md)】
4 |
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--------------------------------------------------------------------------------
/README.md:
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1 | # 中文 Linear Algebra （线性代数） 笔记
2 |
3 | [](https://opensource.org/licenses/mit-license.php)
4 |
5 | 麻省理工公开课 [Linear Algebra](http://open.163.com/special/opencourse/daishu.html) 学习笔记。
6 |
7 | ## Contents
8 |
9 | - [01. **方程组的几何解释**](https://github.com/guokaide/math/blob/master/01-%E6%96%B9%E7%A8%8B%E7%BB%84%E7%9A%84%E5%87%A0%E4%BD%95%E8%A7%A3%E9%87%8A/01-%E6%96%B9%E7%A8%8B%E7%BB%84%E7%9A%84%E5%87%A0%E4%BD%95%E8%A7%A3%E9%87%8A.md)
10 | * Row picture 角度解释方程组
11 | * Column picture 角度解释方程组
12 | * 矩阵乘法初探
13 |
14 | - [02. **矩阵消元**](https://github.com/guokaide/math/blob/master/02-%E7%9F%A9%E9%98%B5%E6%B6%88%E5%85%83/02-%E7%9F%A9%E9%98%B5%E6%B6%88%E5%85%83.md)
15 | * 初等数学的视角看待消元法求解方程
16 | * 矩阵变换的视角看待消元法求解方程（将初等变换的过程通过矩阵（乘法）表示，EA = U）
17 | * 向量与矩阵的乘法：矩阵行与列的线性组合
18 | * 置换矩阵初探：矩阵的行变换与列变换
19 | * 逆矩阵初探：如何从 U -> A
20 |
21 | - [03. **矩阵乘法和矩阵的逆**](https://github.com/guokaide/math/blob/master/03-%E4%B9%98%E6%B3%95%E5%92%8C%E9%80%86%E7%9F%A9%E9%98%B5/03-%E4%B9%98%E6%B3%95%E5%92%8C%E9%80%86%E7%9F%A9%E9%98%B5.md)
22 |
23 | - [04. **A的LU分解**](https://github.com/guokaide/math/blob/master/04-A%E7%9A%84LU%E5%88%86%E8%A7%A3/04-A%E7%9A%84LU%E5%88%86%E8%A7%A3.md)
24 |
25 | - [05. **转置-置换-向量空间R**](https://github.com/guokaide/math/blob/master/05-%E8%BD%AC%E7%BD%AE-%E7%BD%AE%E6%8D%A2-%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4R/05-%E8%BD%AC%E7%BD%AE-%E7%BD%AE%E6%8D%A2-%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4R.md)
26 |
27 | - [06. **列空间和零空间**](https://github.com/guokaide/math/blob/master/06-%E5%88%97%E7%A9%BA%E9%97%B4%E5%92%8C%E9%9B%B6%E7%A9%BA%E9%97%B4/06-%E5%88%97%E7%A9%BA%E9%97%B4%E5%92%8C%E9%9B%B6%E7%A9%BA%E9%97%B4.md)
28 |
29 | - [07. **求解Ax=0:主变量,特解**]()
30 |
31 | - [08. **求解Ax=b:可解性和解的结构**](https://github.com/guokaide/linear-algebra/blob/master/08-%E6%B1%82%E8%A7%A3Ax%3Db-%E5%8F%AF%E8%A7%A3%E6%80%A7%E5%92%8C%E8%A7%A3%E7%9A%84%E7%BB%93%E6%9E%84/08-%E6%B1%82%E8%A7%A3Ax%3Db-%E5%8F%AF%E8%A7%A3%E6%80%A7%E5%92%8C%E8%A7%A3%E7%9A%84%E7%BB%93%E6%9E%84.md)
32 |
33 | - [09. **线性相关性、基、维数**](https://github.com/guokaide/linear-algebra/blob/master/09-%E7%BA%BF%E6%80%A7%E7%9B%B8%E5%85%B3%E6%80%A7-%E5%9F%BA-%E7%BB%B4%E6%95%B0/09-%E7%BA%BF%E6%80%A7%E7%9B%B8%E5%85%B3%E6%80%A7-%E5%9F%BA-%E7%BB%B4%E6%95%B0.md)
34 | - 线性无关（向量组）
35 | - 生成空间（向量组）
36 | - 空间的基
37 | - 空间的维数
38 |
39 | - [10. **四个基本子空间**](https://github.com/guokaide/linear-algebra/blob/master/10-%E5%9B%9B%E4%B8%AA%E5%9F%BA%E6%9C%AC%E5%AD%90%E7%A9%BA%E9%97%B4/10-%E5%9B%9B%E4%B8%AA%E5%9F%BA%E6%9C%AC%E5%AD%90%E7%A9%BA%E9%97%B4.md)
40 |
41 | * 向量空间
42 | * 矩阵的列空间
43 | * 矩阵的零空间
44 | * 矩阵的行空间
45 | * 矩阵的左零空间
46 |
47 | - 矩阵空间初探：所有的 3*3 的矩阵构成的空间
48 |
49 | - [11. **矩阵空间、秩1矩阵和小世界图**](https://github.com/guokaide/linear-algebra/blob/master/11-%E7%9F%A9%E9%98%B5%E7%A9%BA%E9%97%B4-%E7%A7%A91%E7%9F%A9%E9%98%B5%E5%92%8C%E5%B0%8F%E4%B8%96%E7%95%8C%E5%9B%BE/11-%E7%9F%A9%E9%98%B5%E7%A9%BA%E9%97%B4-%E7%A7%A91%E7%9F%A9%E9%98%B5%E5%92%8C%E5%B0%8F%E4%B8%96%E7%95%8C%E5%9B%BE.md)
50 |
51 | - 矩阵空间深入
52 | - 秩一矩阵
53 | - 小世界图初探：图论与线性代数
54 |
55 | - [12. **图和网络**](https://github.com/guokaide/linear-algebra/blob/master/12-%E5%9B%BE%E5%92%8C%E7%BD%91%E7%BB%9C/12-%E5%9B%BE%E5%92%8C%E7%BD%91%E7%BB%9C.md)
56 |
57 | - 图的深入：图和关联矩阵
58 | - 电路定律：欧姆定律、基尔霍夫定律与平衡方程
59 |
60 | - [*13. 复习一*](https://github.com/apachecn/math/blob/master/13-%E5%A4%8D%E4%B9%A0%E4%B8%80/13-%E5%A4%8D%E4%B9%A0%E4%B8%80.md)
61 |
62 | - [14. **正交向量与子空间**](https://github.com/apachecn/math/blob/master/14-%E6%AD%A3%E4%BA%A4%E5%90%91%E9%87%8F%E4%B8%8E%E5%AD%90%E7%A9%BA%E9%97%B4/14-%E6%AD%A3%E4%BA%A4%E5%90%91%E9%87%8F%E4%B8%8E%E5%AD%90%E7%A9%BA%E9%97%B4.md)
63 |
64 | - [15. **子空间投影**](https://github.com/apachecn/math/blob/master/15-%E5%AD%90%E7%A9%BA%E9%97%B4%E6%8A%95%E5%BD%B1/15-%E5%AD%90%E7%A9%BA%E9%97%B4%E6%8A%95%E5%BD%B1.md)
65 |
66 | - [16. **投影矩阵和最小二乘**](https://github.com/apachecn/math/blob/master/16-%E6%8A%95%E5%BD%B1%E7%9F%A9%E9%98%B5%E5%92%8C%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98/16-%E6%8A%95%E5%BD%B1%E7%9F%A9%E9%98%B5%E5%92%8C%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98.md)
67 |
68 | - [17. **正交矩阵和Gram-Schmidt正交化**](https://github.com/apachecn/math/blob/master/17-%E6%AD%A3%E4%BA%A4%E7%9F%A9%E9%98%B5%E5%92%8CGram-Schmidt%E6%AD%A3%E4%BA%A4%E5%8C%96/17-%E6%AD%A3%E4%BA%A4%E7%9F%A9%E9%98%B5%E5%92%8CGram-Schmidt%E6%AD%A3%E4%BA%A4%E5%8C%96.md)
69 |
70 | - [18. **行列式及其性质**](https://github.com/apachecn/math/blob/master/18-%E8%A1%8C%E5%88%97%E5%BC%8F%E5%8F%8A%E5%85%B6%E6%80%A7%E8%B4%A8/18-%E8%A1%8C%E5%88%97%E5%BC%8F%E5%8F%8A%E5%85%B6%E6%80%A7%E8%B4%A8.md)
71 |
72 | - [19. **行列式公式和代数余子式**](https://github.com/apachecn/math/blob/master/19-%E8%A1%8C%E5%88%97%E5%BC%8F%E5%85%AC%E5%BC%8F%E5%92%8C%E4%BB%A3%E6%95%B0%E4%BD%99%E5%AD%90%E5%BC%8F/19-%E8%A1%8C%E5%88%97%E5%BC%8F%E5%85%AC%E5%BC%8F%E5%92%8C%E4%BB%A3%E6%95%B0%E4%BD%99%E5%AD%90%E5%BC%8F.md)
73 |
74 | - [20. **克拉默法则、逆矩阵、体积**](https://github.com/apachecn/math/blob/master/20-%E5%85%8B%E6%8B%89%E9%BB%98%E6%B3%95%E5%88%99-%E9%80%86%E7%9F%A9%E9%98%B5-%E4%BD%93%E7%A7%AF/20-%E5%85%8B%E6%8B%89%E9%BB%98%E6%B3%95%E5%88%99-%E9%80%86%E7%9F%A9%E9%98%B5-%E4%BD%93%E7%A7%AF.md)
75 |
76 | - [21. **特征值和特征向量**](https://github.com/apachecn/math/blob/master/21-%E7%89%B9%E5%BE%81%E5%80%BC%E5%92%8C%E7%89%B9%E5%BE%81%E5%90%91%E9%87%8F/21-%E7%89%B9%E5%BE%81%E5%80%BC%E5%92%8C%E7%89%B9%E5%BE%81%E5%90%91%E9%87%8F.md)
77 |
78 | - [22. **对角化和A的幂**](https://github.com/apachecn/math/blob/master/22-%E5%AF%B9%E8%A7%92%E5%8C%96%E5%92%8CA%E7%9A%84%E5%B9%82/22-%E5%AF%B9%E8%A7%92%E5%8C%96%E5%92%8CA%E7%9A%84%E5%B9%82.md)
79 |
80 | - [23. **微分方程和exp(At)**](https://github.com/apachecn/math/blob/master/23-%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E5%92%8Cexp(At)/23-%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E5%92%8Cexp(At).md)
81 |
82 | - [24. **马尔可夫矩阵;傅立叶级数**](https://github.com/apachecn/math/blob/master/24-%E9%A9%AC%E5%B0%94%E5%8F%AF%E5%A4%AB%E7%9F%A9%E9%98%B5-%E5%82%85%E7%AB%8B%E5%8F%B6%E7%BA%A7%E6%95%B0/24-%E9%A9%AC%E5%B0%94%E5%8F%AF%E5%A4%AB%E7%9F%A9%E9%98%B5-%E5%82%85%E7%AB%8B%E5%8F%B6%E7%BA%A7%E6%95%B0.md)
83 |
84 | - [*25. 复习二*](https://github.com/apachecn/math/blob/master/25-%E5%A4%8D%E4%B9%A0%E4%BA%8C/25-%E5%A4%8D%E4%B9%A0%E4%BA%8C.md)
85 |
86 | - [26. **对称矩阵及正定性**](https://github.com/apachecn/math/blob/master/26-%E5%AF%B9%E7%A7%B0%E7%9F%A9%E9%98%B5%E5%8F%8A%E6%AD%A3%E5%AE%9A%E6%80%A7/26-%E5%AF%B9%E7%A7%B0%E7%9F%A9%E9%98%B5%E5%8F%8A%E6%AD%A3%E5%AE%9A%E6%80%A7.md)
87 |
88 | - [27. **复数矩阵和快速傅里叶变换**](https://github.com/apachecn/math/blob/master/27-%E5%A4%8D%E6%95%B0%E7%9F%A9%E9%98%B5%E5%92%8C%E5%BF%AB%E9%80%9F%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2/27-%E5%A4%8D%E6%95%B0%E7%9F%A9%E9%98%B5%E5%92%8C%E5%BF%AB%E9%80%9F%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2.md)
89 |
90 | - [28. **正定矩阵和最小值**](https://github.com/apachecn/math/blob/master/28-%E6%AD%A3%E5%AE%9A%E7%9F%A9%E9%98%B5%E5%92%8C%E6%9C%80%E5%B0%8F%E5%80%BC/28-%E6%AD%A3%E5%AE%9A%E7%9F%A9%E9%98%B5%E5%92%8C%E6%9C%80%E5%B0%8F%E5%80%BC.md)
91 |
92 | - [29. **相似矩阵和若尔当形**](https://github.com/apachecn/math/blob/master/29-%E7%9B%B8%E4%BC%BC%E7%9F%A9%E9%98%B5%E5%92%8C%E8%8B%A5%E5%B0%94%E5%BD%93%E5%BD%A2/29-%E7%9B%B8%E4%BC%BC%E7%9F%A9%E9%98%B5%E5%92%8C%E8%8B%A5%E5%B0%94%E5%BD%93%E5%BD%A2.md)
93 |
94 | - [30. **奇异值分解**](https://github.com/apachecn/math/blob/master/30-%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3/30-%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3.md)
95 |
96 | - [31. **线性变换及对应矩阵**](https://github.com/apachecn/math/blob/master/31-%E7%BA%BF%E6%80%A7%E5%8F%98%E6%8D%A2%E5%8F%8A%E5%AF%B9%E5%BA%94%E7%9F%A9%E9%98%B5/31-%E7%BA%BF%E6%80%A7%E5%8F%98%E6%8D%A2%E5%8F%8A%E5%AF%B9%E5%BA%94%E7%9F%A9%E9%98%B5.md)
97 |
98 | - [32. **基变换和图像压缩**](https://github.com/apachecn/math/blob/master/32-%E5%9F%BA%E5%8F%98%E6%8D%A2%E5%92%8C%E5%9B%BE%E5%83%8F%E5%8E%8B%E7%BC%A9/32-%E5%9F%BA%E5%8F%98%E6%8D%A2%E5%92%8C%E5%9B%BE%E5%83%8F%E5%8E%8B%E7%BC%A9.md)
99 |
100 | - [*33. 复习三*](https://github.com/apachecn/math/blob/master/33-%E5%A4%8D%E4%B9%A0%E4%B8%89/33-%E5%A4%8D%E4%B9%A0%E4%B8%89.md)
101 |
102 | - [34. **左右逆和伪逆**](https://github.com/apachecn/math/blob/master/34-%E5%B7%A6%E5%8F%B3%E9%80%86%E5%92%8C%E4%BC%AA%E9%80%86/34-%E5%B7%A6%E5%8F%B3%E9%80%86%E5%92%8C%E4%BC%AA%E9%80%86.md)
103 |
104 | - [*35. 期末复习*](https://github.com/apachecn/math/blob/master/35-%E6%9C%9F%E6%9C%AB%E5%A4%8D%E4%B9%A0/35-%E6%9C%9F%E6%9C%AB%E5%A4%8D%E4%B9%A0.md)
105 |
106 |
107 |
108 |
109 |
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/SUMMARY.md:
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1 | - [MIT 线性代数中文笔记](README.md)
2 | - [01. 方程组的几何解释](01-%E6%96%B9%E7%A8%8B%E7%BB%84%E7%9A%84%E5%87%A0%E4%BD%95%E8%A7%A3%E9%87%8A/01-%E6%96%B9%E7%A8%8B%E7%BB%84%E7%9A%84%E5%87%A0%E4%BD%95%E8%A7%A3%E9%87%8A.md)
3 | - [02. 矩阵消元](02-%E7%9F%A9%E9%98%B5%E6%B6%88%E5%85%83/02-%E7%9F%A9%E9%98%B5%E6%B6%88%E5%85%83.md)
4 | - [03. 乘法和逆矩阵](03-%E4%B9%98%E6%B3%95%E5%92%8C%E9%80%86%E7%9F%A9%E9%98%B5/03-%E4%B9%98%E6%B3%95%E5%92%8C%E9%80%86%E7%9F%A9%E9%98%B5.md)
5 | - [04. A的LU分解](04-A%E7%9A%84LU%E5%88%86%E8%A7%A3/04-A%E7%9A%84LU%E5%88%86%E8%A7%A3.md)
6 | - [05. 转置-置换-向量空间R](05-%E8%BD%AC%E7%BD%AE-%E7%BD%AE%E6%8D%A2-%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4R/05-%E8%BD%AC%E7%BD%AE-%E7%BD%AE%E6%8D%A2-%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4R.md)
7 | - [06. 列空间和零空间](06-%E5%88%97%E7%A9%BA%E9%97%B4%E5%92%8C%E9%9B%B6%E7%A9%BA%E9%97%B4/06-%E5%88%97%E7%A9%BA%E9%97%B4%E5%92%8C%E9%9B%B6%E7%A9%BA%E9%97%B4.md)
8 | - [07. 求解Ax=0:主变量,特解](07-%E6%B1%82%E8%A7%A3Ax=0-%E4%B8%BB%E5%8F%98%E9%87%8F-%E7%89%B9%E8%A7%A3/07-%E6%B1%82%E8%A7%A3Ax=0-%E4%B8%BB%E5%8F%98%E9%87%8F-%E7%89%B9%E8%A7%A3.md)
9 | - [08. 求解Ax=b:可解性和解的结构](08-%E6%B1%82%E8%A7%A3Ax=b-%E5%8F%AF%E8%A7%A3%E6%80%A7%E5%92%8C%E8%A7%A3%E7%9A%84%E7%BB%93%E6%9E%84/08-%E6%B1%82%E8%A7%A3Ax=b-%E5%8F%AF%E8%A7%A3%E6%80%A7%E5%92%8C%E8%A7%A3%E7%9A%84%E7%BB%93%E6%9E%84.md)
10 | - [09. 线性相关性、基、维数](09-%E7%BA%BF%E6%80%A7%E7%9B%B8%E5%85%B3%E6%80%A7-%E5%9F%BA-%E7%BB%B4%E6%95%B0/09-%E7%BA%BF%E6%80%A7%E7%9B%B8%E5%85%B3%E6%80%A7-%E5%9F%BA-%E7%BB%B4%E6%95%B0.md)
11 | - [10. 四个基本子空间](10-%E5%9B%9B%E4%B8%AA%E5%9F%BA%E6%9C%AC%E5%AD%90%E7%A9%BA%E9%97%B4/10-%E5%9B%9B%E4%B8%AA%E5%9F%BA%E6%9C%AC%E5%AD%90%E7%A9%BA%E9%97%B4.md)
12 | - [11. 矩阵空间、秩1矩阵和小世界图](11-%E7%9F%A9%E9%98%B5%E7%A9%BA%E9%97%B4-%E7%A7%A91%E7%9F%A9%E9%98%B5%E5%92%8C%E5%B0%8F%E4%B8%96%E7%95%8C%E5%9B%BE/11-%E7%9F%A9%E9%98%B5%E7%A9%BA%E9%97%B4-%E7%A7%A91%E7%9F%A9%E9%98%B5%E5%92%8C%E5%B0%8F%E4%B8%96%E7%95%8C%E5%9B%BE.md)
13 | - [12. 图和网络](12-%E5%9B%BE%E5%92%8C%E7%BD%91%E7%BB%9C/12-%E5%9B%BE%E5%92%8C%E7%BD%91%E7%BB%9C.md)
14 | - [13. 复习一](13-%E5%A4%8D%E4%B9%A0%E4%B8%80/13-%E5%A4%8D%E4%B9%A0%E4%B8%80.md)
15 | - [14. 正交向量与子空间](14-%E6%AD%A3%E4%BA%A4%E5%90%91%E9%87%8F%E4%B8%8E%E5%AD%90%E7%A9%BA%E9%97%B4/14-%E6%AD%A3%E4%BA%A4%E5%90%91%E9%87%8F%E4%B8%8E%E5%AD%90%E7%A9%BA%E9%97%B4.md)
16 | - [15. 子空间投影](15-%E5%AD%90%E7%A9%BA%E9%97%B4%E6%8A%95%E5%BD%B1/15-%E5%AD%90%E7%A9%BA%E9%97%B4%E6%8A%95%E5%BD%B1.md)
17 | - [16. 投影矩阵和最小二乘](16-%E6%8A%95%E5%BD%B1%E7%9F%A9%E9%98%B5%E5%92%8C%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98/16-%E6%8A%95%E5%BD%B1%E7%9F%A9%E9%98%B5%E5%92%8C%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98.md)
18 | - [17. 正交矩阵和Gram-Schmidt正交化](17-%E6%AD%A3%E4%BA%A4%E7%9F%A9%E9%98%B5%E5%92%8CGram-Schmidt%E6%AD%A3%E4%BA%A4%E5%8C%96/17-%E6%AD%A3%E4%BA%A4%E7%9F%A9%E9%98%B5%E5%92%8CGram-Schmidt%E6%AD%A3%E4%BA%A4%E5%8C%96.md)
19 | - [18. 行列式及其性质](18-%E8%A1%8C%E5%88%97%E5%BC%8F%E5%8F%8A%E5%85%B6%E6%80%A7%E8%B4%A8/18-%E8%A1%8C%E5%88%97%E5%BC%8F%E5%8F%8A%E5%85%B6%E6%80%A7%E8%B4%A8.md)
20 | - [19. 行列式公式和代数余子式](19-%E8%A1%8C%E5%88%97%E5%BC%8F%E5%85%AC%E5%BC%8F%E5%92%8C%E4%BB%A3%E6%95%B0%E4%BD%99%E5%AD%90%E5%BC%8F/19-%E8%A1%8C%E5%88%97%E5%BC%8F%E5%85%AC%E5%BC%8F%E5%92%8C%E4%BB%A3%E6%95%B0%E4%BD%99%E5%AD%90%E5%BC%8F.md)
21 | - [20. 克拉默法则、逆矩阵、体积](20-%E5%85%8B%E6%8B%89%E9%BB%98%E6%B3%95%E5%88%99-%E9%80%86%E7%9F%A9%E9%98%B5-%E4%BD%93%E7%A7%AF/20-%E5%85%8B%E6%8B%89%E9%BB%98%E6%B3%95%E5%88%99-%E9%80%86%E7%9F%A9%E9%98%B5-%E4%BD%93%E7%A7%AF.md)
22 | - [21. 特征值和特征向量](21-%E7%89%B9%E5%BE%81%E5%80%BC%E5%92%8C%E7%89%B9%E5%BE%81%E5%90%91%E9%87%8F/21-%E7%89%B9%E5%BE%81%E5%80%BC%E5%92%8C%E7%89%B9%E5%BE%81%E5%90%91%E9%87%8F.md)
23 | - [22. 对角化和A的幂](22-%E5%AF%B9%E8%A7%92%E5%8C%96%E5%92%8CA%E7%9A%84%E5%B9%82/22-%E5%AF%B9%E8%A7%92%E5%8C%96%E5%92%8CA%E7%9A%84%E5%B9%82.md)
24 | - [23. 微分方程和exp(At)](23-%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E5%92%8Cexp(At)/23-%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E5%92%8Cexp(At).md)
25 | - [24. 马尔可夫矩阵;傅立叶级数](24-%E9%A9%AC%E5%B0%94%E5%8F%AF%E5%A4%AB%E7%9F%A9%E9%98%B5-%E5%82%85%E7%AB%8B%E5%8F%B6%E7%BA%A7%E6%95%B0/24-%E9%A9%AC%E5%B0%94%E5%8F%AF%E5%A4%AB%E7%9F%A9%E9%98%B5-%E5%82%85%E7%AB%8B%E5%8F%B6%E7%BA%A7%E6%95%B0.md)
26 | - [25. 复习二](25-%E5%A4%8D%E4%B9%A0%E4%BA%8C/25-%E5%A4%8D%E4%B9%A0%E4%BA%8C.md)
27 | - [26. 对称矩阵及正定性](26-%E5%AF%B9%E7%A7%B0%E7%9F%A9%E9%98%B5%E5%8F%8A%E6%AD%A3%E5%AE%9A%E6%80%A7/26-%E5%AF%B9%E7%A7%B0%E7%9F%A9%E9%98%B5%E5%8F%8A%E6%AD%A3%E5%AE%9A%E6%80%A7.md)
28 | - [27. 复数矩阵和快速傅里叶变换](27-%E5%A4%8D%E6%95%B0%E7%9F%A9%E9%98%B5%E5%92%8C%E5%BF%AB%E9%80%9F%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2/27-%E5%A4%8D%E6%95%B0%E7%9F%A9%E9%98%B5%E5%92%8C%E5%BF%AB%E9%80%9F%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2.md)
29 | - [28. 正定矩阵和最小值](28-%E6%AD%A3%E5%AE%9A%E7%9F%A9%E9%98%B5%E5%92%8C%E6%9C%80%E5%B0%8F%E5%80%BC/28-%E6%AD%A3%E5%AE%9A%E7%9F%A9%E9%98%B5%E5%92%8C%E6%9C%80%E5%B0%8F%E5%80%BC.md)
30 | - [29. 相似矩阵和若尔当形](29-%E7%9B%B8%E4%BC%BC%E7%9F%A9%E9%98%B5%E5%92%8C%E8%8B%A5%E5%B0%94%E5%BD%93%E5%BD%A2/29-%E7%9B%B8%E4%BC%BC%E7%9F%A9%E9%98%B5%E5%92%8C%E8%8B%A5%E5%B0%94%E5%BD%93%E5%BD%A2.md)
31 | - [30. 奇异值分解](30-%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3/30-%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3.md)
32 | - [31. 线性变换及对应矩阵](31-%E7%BA%BF%E6%80%A7%E5%8F%98%E6%8D%A2%E5%8F%8A%E5%AF%B9%E5%BA%94%E7%9F%A9%E9%98%B5/31-%E7%BA%BF%E6%80%A7%E5%8F%98%E6%8D%A2%E5%8F%8A%E5%AF%B9%E5%BA%94%E7%9F%A9%E9%98%B5.md)
33 | - [32. 基变换和图像压缩](32-%E5%9F%BA%E5%8F%98%E6%8D%A2%E5%92%8C%E5%9B%BE%E5%83%8F%E5%8E%8B%E7%BC%A9/32-%E5%9F%BA%E5%8F%98%E6%8D%A2%E5%92%8C%E5%9B%BE%E5%83%8F%E5%8E%8B%E7%BC%A9.md)
34 | - [33. 复习三](33-%E5%A4%8D%E4%B9%A0%E4%B8%89/33-%E5%A4%8D%E4%B9%A0%E4%B8%89.md)
35 | - [34. 左右逆和伪逆](34-%E5%B7%A6%E5%8F%B3%E9%80%86%E5%92%8C%E4%BC%AA%E9%80%86/34-%E5%B7%A6%E5%8F%B3%E9%80%86%E5%92%8C%E4%BC%AA%E9%80%86.md)
36 | - [35. 期末复习](35-%E6%9C%9F%E6%9C%AB%E5%A4%8D%E4%B9%A0/35-%E6%9C%9F%E6%9C%AB%E5%A4%8D%E4%B9%A0.md)
37 |
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/addition/换个角度看线性代数1.md:
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1 | # 换个角度看线性代数（一）
2 |
3 | ## 1、过往
4 |
5 | 第一次接触线性代数是在大一第一学期，学完后分数不低，但是结束考试过后就感觉啥都没学到，对线性代数还是那样一个模糊的概念，记忆中只有那零零碎碎的计算公式。真正慢慢理解是毕业之后，参加 [ApacheCN](http://www.apachecn.org) 之后，从头开始学习 **MachineLearning(机器学习)** 的时候，遇到矩阵知识真的是什么都不会。所以私下里偷偷从头学习了一下，也在网上找了好多资料，知道了好多大佬，稍微对线性代数的实质有了一点自己的理解。
6 |
7 | ## 2、开干
8 |
9 | ### 2.1、向量
10 |
11 | 线性代数最基础，最根源的组成部分就是向量，向量的线性组合构成了线性代数的基本运算。下面我们首先介绍向量。
12 |
13 | 直观上，向量通常被标示为一个带箭头的线段。线段的长度可以表示向量的大小，而向量的方向也就是箭头所指的方向。—— 维基百科
14 |
15 | 从上面的维基百科的定义中，我们可以得到决定一个向量的是它的长度和它所指的方向共同决定，只要保持着两个量（方向和长度）不变，那么你可以随意移动向量而使它不变。
16 |
17 | 实际上，向量在科学中的表示，通用的有三种表示，第一种用带有箭头的线段表示，第二种用符号表示，第三种用有序数组表示，它们看似不同，实际上只是向量的不同描述~如下图:
18 |
19 | 
20 |
21 | 那么问题来了，先上图:
22 |
23 | 
24 |
25 | 上面我们说到，向量是可以随意移动的，只要方向和长度不变即可，那么如何确定  是如何与图上向量一一对应的呢。
26 |
27 | 答: 在线性代数中，向量的起点往往是以原点开始，那么每一个有序数组都能在坐标系中唯一表示一个向量。
28 |
29 | **总而言之，向量就是一个从原点出发的箭头，有着方向和大小。可以用有序数组表示，并且由于默认起点是原点，一个有序数组唯一代表了一个向量。**
30 |
31 | ### 2.2、空间
32 |
33 | 首先我们来看一下的线性组合表示为:
34 |
35 | 
36 |
37 | 那么下面我们给出空间的定义:
38 |
39 | 所有表示为给定向量线性组合的向量的集合被称为**给定向量张成的空间**。我们来考虑两种情况，当 共线与不共线两种情况，它们所张成空间的情况。
40 | * 当不共线的时候，大部分情况还是这样，它们线性组合构成的向量集合是什么？
41 |
42 | 它们随着俩个向量前面系数的改变，它们张成的空间是所有二维向量的集合（也就是整个二维空间都会被充满）.下面我们来解释一下.
43 |
44 | 当 向量前面系数不变， 前面系数在实数范围内改变的时候，我们可以得到平行于向量的所有二维向量.如下图：
45 |
46 | 
47 |
48 | 它们组合的向量终点是在白线上（后续会说明为什么向量还可以用一个点表示，其实也很容易理解，起点默认了是原点了）
49 |
50 | 那么我们可以这样理解，对于每一个 前面系数在实数范围内变化，我们可以看成这条白线往 方向来回平移，这样二维平面内的向量都能被充满.这样也就解释了为什么俩个不共线的向量能够线性组合为所有的二维向量.也就是说，当俩个向量不共线的时候，它们张成的空间是整个二维空间
51 | * 当 与  共线的时候，它们线性组合构成的向量集合（也就是张成的空间是什么）是什么？
52 |
53 | 当两个初始向量共线的时候，所产生的向量的终点被限制在一条过原点的直线上。
54 |
55 | 
56 |
57 | 这非常好理解，可以把这两个向量旋转到 x 轴，平行的两个向量，无论你怎样线性组合，组合出的新的向量都是在 x 轴上。**当两个向量共线的时候，它们所张成的空间是平行于  向量的一维直线**。
58 |
59 | 下面我们来考虑三维的情况，先上图
60 |
61 | 
62 |
63 | **很显然，我们类比也能得出，当三个向量不共面的时候，它们张成的空间是整个三维空间，当三个向量不共线但是共面的时候，张成的空间是它们形成的二维平面，当他们共线的时候，那么张成的空间就是那条直线.下面我解释一下为什么这样，以便能够更好的推广到高维空间**.
64 |
65 | * 三个向量不共面的情况
66 |
67 | 我们首先可以看在三维空间中两个向量不共线的情况如下:
68 |
69 | 
70 |
71 | 根据我们前面的分析，我们很容易得到，**这两个张成的空间是三维空间中的一个平面**，如下图:
72 |
73 | 
74 |
75 | **那么当第三个向量如果在前俩个向量张成的空间内，那么这三个向量张成的空间就是这个平面**
76 |
77 | **当第三个向量不在前俩个向量张成的空间内，那么它们三个向量张成的空间是整个三维空间.这又如何思考呢？**
78 |
79 | 当前俩个向量张成一个空间的时候，第三个向量（下图是红色向量表示）控制了最后张成的空间，可以这里理解：当你缩放第三个向量时，它将前俩个向量张成的平面沿它的方向来回移动，从而扫过整个空间.over！这就解释了它们三个向量张成的空间是整个三维空间，下图显示：
80 |
81 | 
82 |
83 | 
84 |
85 | **当第三个向量（红色向量).上下移动的时候，前俩个向量构成的空间沿着这个向量方向移动，从而扫过整个空间.最后解释一下三个向量共线的情况，张成的空间就是那条直线上的一维空间**.由于我们生活在三维空间中，四维以上的空间我们无法想象到，可以通过类比上去，这也是线性代数的一个很重要的思想.
86 |
87 | * 为什么有时候向量可以用点来表示
88 |
89 | 想象落在一条直线上的一些向量的时候，你会觉得比较拥挤
90 |
91 | 
92 |
93 | 同理想象所有二维向量填满平面时，你会觉得非常拥挤
94 |
95 | 
96 |
97 | **所以为了对付这种情况，通常我们就用向量的终点来代表该向量**，如下图：
98 |
99 | 
100 |
101 | 这也就解释了为什么向量有时可以用点来表示~
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