Section02_本质联系

$(*)$

$$\begin{array}{cl} \begin{cases} a_{11}x_{1} + \cdots + a_{1n}x_{n} = 0 \\ \cdots \\ a_{m1}x_{1} + \cdots + a_{mn}x_{n} = 0 \\ \end{cases} & \begin{cases} \text{1. 仅有零解} \\ \text{2. 有非零解} \\ \end{cases} \\ \Updownarrow \\ x_{1} \boldsymbol{\alpha}_{1} + x_{2}\boldsymbol{\alpha}_{2} + \cdots + x_{n}\boldsymbol{\alpha}_{n} = \boldsymbol{0} & \begin{cases} \text{1. }\boldsymbol{\alpha}_{1},\cdots, \boldsymbol{\alpha}_{n} \text{线性无关} \\ \text{2. }\boldsymbol{\alpha}_{1},\cdots, \boldsymbol{\alpha}_{n} \text{线性相关} \\ \end{cases} \\ \Updownarrow \\ \boldsymbol{AX} = \boldsymbol{0} & \begin{cases} \text{1. }r(\boldsymbol{A}) = n \\ \text{2. }r(\boldsymbol{A}) < n \\ \end{cases} \end{array}$$

$(**)$

$$\begin{array}{cl} \begin{cases} a_{11}x_{1} + \cdots + a_{1n}x_{n} = b_{1} \\ \cdots \\ a_{m1}x_{1} + \cdots + a_{mn}x_{n} = b_{m} \\ \end{cases} & \begin{cases} \text{1. 有解} \\ \text{2. 无解} \\ \end{cases} \\ \Updownarrow \\ x_{1} \boldsymbol{\alpha}_{1} + x_{2}\boldsymbol{\alpha}_{2} + \cdots + x_{n}\boldsymbol{\alpha}_{n} = \boldsymbol{\beta} & \begin{cases} \text{1. }\boldsymbol{\beta}\text{可由} \boldsymbol{\alpha}_{1},\cdots, \boldsymbol{\alpha}_{n} \text{线性表示} \\ \text{2. }\boldsymbol{\beta}\text{不可由} \boldsymbol{\alpha}_{1},\cdots, \boldsymbol{\alpha}_{n} \text{线性表示} \\ \end{cases} \\ \Updownarrow \\ \boldsymbol{AX} = \boldsymbol{\beta} & \begin{cases} \text{1. }r(\boldsymbol{A}) = r(\bar{\boldsymbol{A}}) \\ \text{2. }r(\boldsymbol{A}) < r(\bar{\boldsymbol{A}})\quad r(\bar{\boldsymbol{A}}) = r(\boldsymbol{A})+ 1 \\ \end{cases} \end{array}$$