线性相关性$$ \begin{equation*} \begin{split} v=\left[\begin{array

\begin{equation*} \begin{split} v=\left[\begin{array}{c}1 \\ 2 \end{array} \right], w=\left[\begin{array}{c}3 \\ 4 \end{array} \right] \\ 3v=\left[\begin{array}{c}3 \\ 6\end{array} \right] \\ v+w=\left[\begin{array}{c}4 \\ 6\end{array} \right] \end{split} \end{equation*}

\begin{equation} \begin{split} |\overrightarrow{v} \cdot \overrightarrow{w}| &\le ||\overrightarrow{v}|| \cdot ||\overrightarrow{w}|| \\ 向量\overrightarrow{v} \cdot \overrightarrow{w}构成的一个平行四边形面积 &\le ||\overrightarrow{v}|| \cdot ||\overrightarrow{w}||所构成的矩形面积 \\ ||\overrightarrow{v} + \overrightarrow{w}|| &\le ||\overrightarrow{v}|| + ||\overrightarrow{w}|| \\ 三角形两边&之和大于第三边 \end{split} \end{equation}

向量空间(列空间)

线性无关

\begin{equation} \begin{split} v=\left[\begin{array}{c} 0 \\1\\2\end{array}\right], w=\left[\begin{array}{c} -1 \\3\\4\end{array}\right], z=\left[\begin{array}{c} 0 \\-3\\6\end{array}\right] \end{split} \end{equation}

三个平面方便观察各向量的线性无关性

线性相关

\begin{equation} \begin{split} v=\left[\begin{array}{c} 1 \\1\\6\end{array}\right], w=\left[\begin{array}{c} 2 \\4\\0\end{array}\right], z=\left[\begin{array}{c} 3 \\5\\6\end{array}\right] \end{split} \end{equation}

线性方程组

齐次线性方程组

齐次线性方程组个方程结果为零

\begin{equation} \begin{split} \left\{ \begin{array}{c} x_1+2x_2+3x_3=0 \\ 2x_1+x_2+4x_3=0 \\ x_1+3x_2+2x_3=0 \end{array} \right. \end{split} \end{equation}

\begin{equation*} \begin{split} 齐次线性方程组\left\{ \begin{array}{lc} 只有零解\\ \\\\ 存在非零解 \end{array} \right. \end{split} \end{equation*} \\

例1(只有零解)

\begin{equation} \begin{split} &\left\{ \begin{array}{r} x_1+2x_2+3x_3&=0\\ 4x_2+5x_3&=0 \\ x_3&=0 \end{array} \right. \Rightarrow \left[ \begin{array}{c} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 1 \end{array} \right] \end{split} \end{equation}

矩阵非零行数等于未知数个数，则该线性方程组只有零解

例2(存在非零解)

\begin{equation} \begin{split} \left\{ \begin{array}{r} x_1+2x_2+3x_3&=0\\ 4x_2+5x_3&=0 \\ 0&=0 \end{array} \right. &\Rightarrow \left[ \begin{array}{c} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 0 \end{array} \right] \\ &\Downarrow \\ x=\left[ \begin{array}{c} -\frac{1}{2}x_3 \\ -\frac{5}{4}x_3 \\ x_3(为何等于X_3?) \end{array} \right] &\Rightarrow x=x_3\left[ \begin{array}{c} -\frac{1}{2} \\ -\frac{5}{4} \\ 1 \end{array} \right] \end{split} \end{equation}

$X_3$ 可以取任意值

非齐次线性方程组(每个方程组不全为零解)

\begin{equation} \begin{split} \left\{ \begin{array}{r} x_1+2x_2+3x_3&=5 \\ x_1+3x_2+4x_3&=6 \\ x_1+4x_2+5x_3&=7 \\ \end{array} \right. \end{split} \end{equation}

增广矩阵无穷解的通解

\begin{split} &\left[ \begin{array}{c:c} \begin{matrix} 1&2&3 \\ 1&3&4 \\ 1&4&5 \\ \end{matrix} & \begin{matrix} 5 \\ 6 \\ 7 \end{matrix} \end{array} \right ] \overleftrightarrow{增广矩阵(A|b)} \left[ \begin{array}{c:c} \begin{matrix} 1&2&3 \\ 0&1&1 \\ 0&0&0 \\ \end{matrix} & \begin{matrix} 5 \\ 1 \\ 0 \end{matrix} \end{array} \right] \Rightarrow \left\{ \begin{array}{r} x_1+2x_2+3x_3&=5 \\ x_2+x_3&=1 \\ 0&=0 \end{array} \right. \Rightarrow \left\{ \begin{array}{r} x_1&=3-x_3 \\ x_2&=1-x_3 \\ x_3&=x_3 \end{array} \right. x_1,x_2,x_3均是x_3的因变量 \Rightarrow x=\left[ \begin{array}{c} 3-x_3 \\ 1-x_3 \\ x_3 \end{array} \right] = \left[ \begin{array}{c} 3 \\ 1 \\ 0 \end{array} \right] + x_3\left[ \begin{array}{c} -1 \\ -1 \\ 1 \end{array} \right] \end{split} \\ r(A|b)=r(A)=2 < 3(未知数个数),方程组有无穷解\\ \left\{ \begin{array}{r} x_1+2x_2+3x_3+x_4=3 \\ x_1-4x_2-x_3-x_4=1 \\ 2x_1+x_2+4x_3+x_4=5 \\ x_1-x_2+x_3=2 \end{array} \right . \Rightarrow \left[ \begin{array}{c:c} \begin{matrix} 1 &2 &3 &1 \\ 1 &-4 &-1 &-1 \\ 2 &1 &4 &1 \\ 1 &-1 &1 &0 \end{matrix} & \begin{matrix} 3 \\ 1 \\ 5 \\ 2 \end{matrix} \end{array} \right] \Rightarrow \left[ \begin{array}{c:c} \begin{matrix} 1 & 2 & 3 & 1 \\ 0 & 3 & 2 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} & \begin{matrix} 3 \\ 1 \\ 0 \\ 0 \end{matrix} \end{array} \right] \Rightarrow \left\{ \begin{array}{r} x_1+2x_2+3x_3+x_4=3 \\ 3x_2+2x_3+x_4=1 \end{array} \right. \\ general \quad equation(orthogonality) \\ x=k_1\left[ \begin{array}{c} 5 \\ 2 \\ -3 \\ 0 \end{array} \right] + k_2\left[ \begin{array}{c} 1 \\ 1 \\ 0 \\ 3 \end{array} \right] + \left[ \begin{array}{c} \frac{7}{3} \\ \frac{1}{3} \\ 0 \\ 0 \end{array} \right] \\(special \quad solution(the \quad variable \quad values \quad of \quad free \quad is\quad made\quad zero)) \\

r(A|b)=r(A)=3 = 3(未知数个数),方程组有唯一解

r(A|b)\ne r(A),方程组有无解

矩阵乘法

可视为向量的复合映射

$y=f(u),u=g(x),f和g均是线性映射$

%矩阵A A=\left[ \begin{array}{c} 1 & 2 \\ 3 & 4 \end{array} \right] \cdot \left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right] = \left[ \begin{array}{c} u_1 \\ u_2 \end{array} \right] \qquad (\quad g(x)\quad)

%矩阵B B=\left[ \begin{array}{c} 5 & 6 \\ 7 & 8 \end{array} \right] \cdot \left[ \begin{array}{c} u_1 \\ u_2 \end{array} \right] = \left[ \begin{array}{c} 9 \\ 10 \end{array} \right] \qquad (\quad f(u) \quad)

%矩阵AxB B\cdot A=\left[ \begin{array}{c} 5 & 6 \\ 7 & 8 \end{array} \right] \cdot \left[ \begin{array}{c} 1 & 2 \\ 3 & 4 \end{array} \right] \cdot \left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right]= \left[ \begin{array}{c} 9 \\ 10 \end{array} \right] \qquad (\quad f\circ g\quad)

逆矩阵

方阵才有逆矩阵
方阵的秩等于其行数(或列数),才可求逆矩阵
一个矩阵和其逆矩阵是可以交换的( $A\cdot A^{-1}=A^{-1}\cdot A=E$ )

假设有矩阵 $A=\begin{bmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \\ 1 & 4 & 9 \end{bmatrix}$ ,那么则有 $A\cdot A^{-1}=E$ $\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}$

求逆矩阵

同步进行矩阵 $A\xRightarrow{若干次初等变换} E,矩阵E\xRightarrow{若干次初等变换}A^{-1}$ ,即

A|E=\left[ \begin{array}{ccc:ccc} 1 & 3 & 5 & 1 & 0 & 0\\ 2 & 4 & 6 & 0 & 1 & 0\\ 1 & 4 & 9 & 0 & 0 & 1 \end{array} \right] \xRightarrow{若干次初等变换} E|A^{-1}\left[ \begin{array}{ccc:ccc} 1 & 0 & 0 & -3 & \frac{7}{4} & \frac{1}{2}\\ 0 & 1 & 0 & 3 & -1 & -1\\ 0 & 0 & 1 & -1 & \frac{1}{4} & \frac{1}{2} \end{array} \right]

类似的,例如数值 $a=3,a^{-1}=\frac{1}{3} \qquad 3|1 \xRightarrow{同乘\frac{1}{3}}1|\frac{1}{3} \quad a\cdot a^{-1}=1$

行列式

可以看成矩阵的一个属性,他是一个数值,表示的是矩阵在某维度的比例映射系数

\left| \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} \right | \left| \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} \right |

$该行列式值为1\times5\times9+2\times6\times7+3\times4\times8-3\times5\times7-1\times6\times8-2\times4\times9=-2$

上三角行列式和下三角行列式

$A\left| \begin{array}{ccc} 1 & 2 & 3 \\ 0 & 5 & 6 \\ 0 & 0 & 9 \end{array} \right | \Rightarrow 1\times 5\times 9+2\times6\times0+3\times0\times0-3\times5\times0-1\times6\times0+2\times0\times9=45$

$B\left| \begin{array}{ccc} 1 & 0 & 0 \\ 2 & 5 & 0 \\ 3 & 6 & 9 \end{array} \right | \Rightarrow 1\times 5\times 9+2\times6\times0+3\times0\times0-3\times5\times0-1\times6\times0+2\times0\times9=45$

$上三角和下三角行列式都等于\underline{主对角线之积},A=(B)^T$

初等行列变换会对行列式的值产生影响,矩阵只是一个数值阵列,而行列式是一个数值

$\left| \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right| =A \qquad A\xRightarrow{行变换} \left | \begin{array}{cc} 3 & 4 \\ 1 & 2 \end{array} \right| =2 \qquad A\xRightarrow{列变换} \left | \begin{array}{cc} 2 & 1 \\ 4 & 3 \end{array} \right| =2$ 初等行列变换(一次行变换或一次列变换会改变行列式的值, $-1^{n}\ ,n为变换次数$ ) $\left| \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right| =A \qquad A\xRightarrow{\times 5} \left| \begin{array}{cc} 1\times 5 & 2\times 5 \\ 3 & 4 \end{array} \right| =-10$ 数乘 $\left| \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right| =A \qquad A\xRightarrow{-2\times A_{i1}+A_{i2}} \left | \begin{array}{cc} 1 & 0 \\ 3 & -2 \end{array} \right| =-2 \qquad A\xRightarrow{-2\times A_{1j+A_{2j}}} \left | \begin{array}{cc} 1 & 2 \\ 1 & 0 \end{array} \right| =-2$ 倍加

行列式降价与展开

行列式的初等行变换或列变换只能与相邻的行或列进行交换

在上三角行列式中 $A=\left| \begin{array}{ccc} 1 & 2 & 3 \\ 0 & 5 & 6 \\ 0 & 0 & 9 \end{array} \right | \Rightarrow 1\times 5\times 9=45,其中5\times9是行列式 \left| \begin{array}{ccc} 5 & 6 \\ 0 & 9 \end{array} \right |的值$

$设a_{ij}是行列式中第i行,第j列$

行列式的降价

$设行列式 D=\left| \begin{array}{ccc} 1 & 3 & 4 \\ 5 & 8 & 7 \\ 1 & 2 & 4 \end{array} \right | \xRightarrow{D_{1j} - D_{3j}} \left| \begin{array}{ccc} 1 & 3 & 4 \\ 5 & 8 & 7 \\ 0 & 1 & 0 \end{array} \right | \xRightarrow{初等行列变换\times3} (-1)^{3}\left| \begin{array}{ccc} 1 & 0 & 0 \\ 3 & 1 & 4 \\ 8 & 5 & 7 \end{array} \right | = (-1)^{3}\times1\times \left | \begin{array}{cc} 1 & 4 \\ 5 & 7 \end{array} \right| =13$

行列式的降价定理

$设a_{ij}是行列式D第i行,第j列的元素,M_{ij}是其余子式,对应的代数余子式是A_{ij}=(-1)^{i+j}\cdot M_{ij},如果第i行或第j列除a_{ij}以外的元素都为0,那么原行列式值D=a_{ij}\cdot A_{ij}\Leftrightarrow a_{ij}\cdot (-1)^{i+j}\cdot M_{ij}$

$因为元素a_{ij}到行列式的左上角需垂直移动(i-1)次,水平移动(j-1)次,总共移动i+j-2次,则D=(-1)^{i+j-2}\cdot a_{ij}\cdot M_{ij},\; (-1)^{i+j-2}=(-1)^{i+j},所以D=(-1)^{i+j}\cdot a_{ij} \cdot M_{ij}$

$例:D=\left | \begin{array}{ccc} 1 & 3 & 4 \\ 5 & 8 & 7 \\ 0 & 1 & 0 \end{array} \right | =(-1)^{3+2}\cdot 1\cdot \left | \begin{array}{cc} 1 & 4 \\ 5 & 7 \end{array} \right | =13$

线性相关性