Section01_线性方程组的三种形式

1. 基本形式 $$\begin{array}{ll} \begin{cases} a_{11} x_{1} + \cdots + a_{1n}x_{n} = 0 \\ \cdots \\ a_{m1} x_{1} + \cdots + a_{mn}x_{n} = 0 \\ \end{cases} & (*) \\ \\ \begin{cases} a_{11} x_{1} + \cdots + a_{1n}x_{n} = b_{1} \\ \cdots \\ a_{m1} x_{1} + \cdots + a_{mn}x_{n} = b_{m} \\ \end{cases} & (*) \\ \end{array}$$
2. 矩阵形式 $$\begin{array}{c} \boldsymbol{A} = \left( \begin{matrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{matrix} \right) \quad \boldsymbol{X} = \left( \begin{matrix} x_{1} \\ x_{2} \\ \vdots \\ x_{n} \end{matrix} \right) \quad \boldsymbol{\beta} = \left( \begin{matrix} b_{1} \\ b_{2} \\ \vdots \\ b_{m} \end{matrix} \right) \\\\ \boldsymbol{AX} = \boldsymbol{0}\quad (*)\\ \boldsymbol{AX} = \boldsymbol{\beta}\quad (**) \end{array}$$
3. 向量形式 $$\begin{array}{c} \boldsymbol{\alpha}_{1} = \left( \begin{matrix} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \end{matrix} \right)\quad \boldsymbol{\alpha}_{2} = \left( \begin{matrix} a_{12} \\ a_{22} \\ \vdots \\ a_{m2} \end{matrix} \right)\quad \cdots \quad \boldsymbol{\alpha}_{n} = \left( \begin{matrix} a_{1n} \\ a_{2n} \\ \vdots \\ a_{mn} \end{matrix} \right) \\\\ x_{1}\boldsymbol{\alpha}_{1} + x_{2} \boldsymbol{\alpha}_{2} + \cdots + x_{n}\boldsymbol{\alpha}_{n} = \boldsymbol{0}\quad (*) \\ x_{1}\boldsymbol{\alpha}_{1} + x_{2} \boldsymbol{\alpha}_{2} + \cdots + x_{n}\boldsymbol{\alpha}_{n} = \boldsymbol{\beta}\quad (**) \end{array}$$