Section03_向量组等价，极大组，秩

1. 向量组等价 $\text{I: } \boldsymbol{\alpha}_{1}, \cdots, \boldsymbol{\alpha}_{m}\text{; II: } \boldsymbol{\beta}_{1},\boldsymbol{\beta}_{2},\cdots, \boldsymbol{\beta}_{n}$
   • 若 $\begin{cases} \boldsymbol{\alpha}_{1} = k_{11}\boldsymbol{\beta}_{1} + \cdots + k_{1n}\boldsymbol{\beta}_{n} \\ \cdots \\ \boldsymbol{\alpha}_{m} = k_{m1}\boldsymbol{\beta}_{1} + \cdots + k_{mn}\boldsymbol{\beta}_{n} \\ \end{cases} \quad (1)$，称 $\text{I}$ 可由 $\text{II}$ 线性表示
   • 若 $\begin{cases} \boldsymbol{\beta}_{1} = \ell_{11}\boldsymbol{\alpha}_{1} + \cdots + \ell_{1m}\boldsymbol{\alpha}_{m} \\ \cdots \\ \boldsymbol{\beta}_{n} = \ell_{n1}\boldsymbol{\alpha}_{1} + \cdots + \ell_{nm}\boldsymbol{\alpha}_{m} \\ \end{cases} \quad (2)$，称 $\text{II}$ 可由 $\text{I}$ 线性表示
   • 若 $(1),(2)$ 皆成立，则称 $\text{I, II}$ 等价
2. 极大线性无关组(无关组) 若
   1. $\exists\ r$ 个向量线性无关
   2. $\forall\ r+1$ 个向量线性相关
   3. 称 $r$ 个无关向量为最大线性无关组，$r$ 称为向量组的秩

      Notes

      1. 极大组不一定唯一
      2. 向量组与极大组等价
      3. 重点 对于 $\boldsymbol{\alpha}_{1},\cdots, \boldsymbol{\alpha}_{n}$

         • Case 1 $\boldsymbol{\alpha}_{1},\cdots,\boldsymbol{\alpha}_{n}$ 无关 $\Leftrightarrow$ $\boldsymbol{\alpha}_{1},\cdots,\boldsymbol{\alpha}_{n}$ 的秩 $=n$
         • Case 2 $\boldsymbol{\alpha}_{1},\cdots,\boldsymbol{\alpha}_{n}$ 相关 $\Leftrightarrow$ $\boldsymbol{\alpha}_{1},\cdots,\boldsymbol{\alpha}_{n}$ 的秩 $<n$

      4. $\text{I: } \boldsymbol{\alpha}_{1},\cdots, \boldsymbol{\alpha}_{n}\text{, II: } \boldsymbol{\alpha}_{1},\cdots,\boldsymbol{\alpha}_{n}, \boldsymbol{\beta}$

         • Case 1 $\text{I}$ 的秩 $=$ $\text{II}$ 的秩 $\Leftrightarrow$ $\boldsymbol{\beta}$ 可由 $\boldsymbol{\alpha}_{1},\cdots,\boldsymbol{\alpha}_{n}$ 线性表示 $\Leftrightarrow$ $(**)$ 有解
         • Case 2 $\text{I}$ 的秩 + 1 $=$ $\text{II}$ 的秩 $\Leftrightarrow$ $\boldsymbol{\beta}$ 不可由 $\boldsymbol{\alpha}_{1},\cdots,\boldsymbol{\alpha}_{n}$ 线性表示 $\Leftrightarrow$ $(**)$ 无解

      5. $\boldsymbol{A}_{m\times n} = \left( \begin{matrix} \boldsymbol{\alpha}_{1} \\ \vdots \\ \boldsymbol{\alpha}_{m} \end{matrix} \right) = \left( \begin{matrix} \boldsymbol{\beta}_{1} & \cdots & \boldsymbol{\beta}_{n} \end{matrix} \right)$

         • $\boldsymbol{\alpha}_{1},\cdots, \boldsymbol{\alpha}_{m}$ 称为 $\boldsymbol{A}$ 的行向量组，其的秩成为 $\boldsymbol{A}$ 的行秩
         • $\boldsymbol{\beta}_{1},\cdots, \boldsymbol{\beta}_{n}$ 称为 $\boldsymbol{A}$ 的列向量组，其的秩成为 $\boldsymbol{A}$ 的列秩

      6. $\boldsymbol{A}_{m\times n}, \boldsymbol{B}_{n\times s} = \left( \begin{matrix} \boldsymbol{\beta}_{1} & \cdots & \boldsymbol{\beta}_{s} \end{matrix} \right)$ $\Rightarrow$ $\boldsymbol{AB} = \boldsymbol{A} \left( \begin{matrix} \boldsymbol{\beta}_{1} & \cdots & \boldsymbol{\beta}_{s} \end{matrix} \right) = \left( \begin{matrix} \boldsymbol{A\beta}_{1} & \cdots & \boldsymbol{A\beta}_{s} \end{matrix} \right)$
      7. $\boldsymbol{A} \left( \begin{array}{c:c} \boldsymbol{B} & \boldsymbol{C} \end{array}\right) = \left( \begin{array}{c:c} \boldsymbol{AB} & \boldsymbol{AC} \end{array}\right)$
      8. 如 $\boldsymbol{\alpha}_{1},\boldsymbol{\alpha}_{2},\boldsymbol{\alpha}_{3}$，$\boldsymbol{A} = \left( \begin{matrix} \boldsymbol{\alpha}_{1} & \boldsymbol{\alpha}_{2} & \boldsymbol{\alpha}_{3} \end{matrix} \right)$, $$\begin{split} \boldsymbol{B} & = \left(\begin{smallmatrix} a_{1}\boldsymbol{\alpha}_{1} + a_{2}\boldsymbol{\alpha}_{2} + a_{3}\boldsymbol{\alpha}_{3} & b_{1}\boldsymbol{\alpha}_{1} + b_{2}\boldsymbol{\alpha}_{2} + b_{3}\boldsymbol{\alpha}_{3} & c_{1}\boldsymbol{\alpha}_{1} + c_{2}\boldsymbol{\alpha}_{2} + c_{3}\boldsymbol{\alpha}_{3} \end{smallmatrix}\right) \\ & = \boldsymbol{} \begin{pmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \\ \end{pmatrix} \end{split}$$

性质

1. 任何矩阵三秩相等
2. $\text{I: }\boldsymbol{\alpha}_{1},\cdots, \boldsymbol{\alpha}_{m}\text{, II: } \boldsymbol{\beta}_{1},\cdots, \boldsymbol{\beta}_{n}$，若 $\text{I}$ 组可由 $\text{II}$ 组线性表示，则 $\text{I}$ 的秩 $\le$ $\text{II}$ 的秩

   Proof $$\begin{array}{ll} & \boldsymbol{A} = \left( \begin{matrix} \boldsymbol{\alpha}_{1}& \cdots & \boldsymbol{\alpha}_{m} \end{matrix} \right) , \boldsymbol{B} = \left( \begin{matrix} \boldsymbol{\beta}_{1}& \cdots & \boldsymbol{\beta}_{n} \end{matrix} \right) \\ \because & \begin{cases} \boldsymbol{\alpha}_{1} = k_{11}\boldsymbol{\beta}_{1} + \cdots + k_{1n}\boldsymbol{\beta}_{n} \\ \cdots \\ \boldsymbol{\alpha}_{m} = k_{m1}\boldsymbol{\beta}_{1} + \cdots + k_{mn}\boldsymbol{\beta}_{n} \\ \end{cases} \\ \therefore & \boldsymbol{A} = \boldsymbol{B} \left( \begin{matrix} k_{11} & k_{21} & \cdots & k_{m1} \\ k_{12} & k_{22} & \cdots & k_{m2} \\ \vdots & \vdots & \ddots & \vdots \\ k_{1n} & k_{2n} & \cdots & k_{mn} \\ \end{matrix} \right) = \boldsymbol{BK}\\ \because & r(\boldsymbol{A}) = r(\boldsymbol{BK}) \le r(\boldsymbol{B}) \\ \therefore & \text{I的秩} \le \text{II的秩} \\ \end{array}$$

3. 向量组 $\text{I}$ 与向量组 $\text{II}$ 等价 $\nLeftarrow \Rightarrow$ $\text{I}$ 的秩 $=$ $\text{II}$ 的秩

   Proof

   $\Rightarrow$ $$\begin{array}{ll} \because & \text{I可由II线性表示} \Rightarrow \text{I的秩}\le \text{II的秩} \\ & \text{II可由I线性表示} \Rightarrow \text{II的秩}\le \text{I的秩} \\ \therefore & \text{I的秩} = \text{II的秩} \\ \end{array}$$ $\nLeftarrow$ 反例 $$\begin{array}{ll} & \text{I: } \boldsymbol{\alpha}_{1} = \left( \begin{smallmatrix} 1 \\ -1 \\ 0 \\ 0 \end{smallmatrix} \right), \boldsymbol{\alpha}_{2} = \left( \begin{smallmatrix} 2 \\ 1 \\ 0 \\ 0 \end{smallmatrix} \right) \\ & \text{II: } \boldsymbol{\alpha}_{1} = \left( \begin{smallmatrix} 0 \\ 0 \\ 1 \\ 2 \end{smallmatrix} \right), \boldsymbol{\alpha}_{2} = \left( \begin{smallmatrix} 0 \\ 0 \\ -1 \\ 3 \end{smallmatrix} \right) \end{array}$$

例题

1. $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}$ 线性无关，$\boldsymbol{\alpha}_{3}$ 不可由 $\boldsymbol{\alpha}_{1},\boldsymbol{\alpha}_{2}$ 线性表示，$\boldsymbol{\alpha}_{4}$ 可由 $\boldsymbol{\alpha}_{1},\boldsymbol{\alpha}_{2}$ 线性表示，证 $\boldsymbol{\alpha}_{1},\boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{3} + \boldsymbol{\alpha}_{4}$ 线性无关 $$\begin{array}{ll} \because & \boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2} \text{线性无关, 且} \boldsymbol{\alpha}_{3} \text{不可由} \boldsymbol{\alpha}_{1},\boldsymbol{\alpha}_{2}\text{线性表示} \\ \therefore & \boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2},\boldsymbol{\alpha}_{3}\text{线性无关} \\ & \boldsymbol{A} = \left( \begin{matrix} \boldsymbol{\alpha}_{1} &\boldsymbol{\alpha}_{2} &\boldsymbol{\alpha}_{3} \end{matrix} \right), r(\boldsymbol{A}) = 3 \\ \because & \boldsymbol{\alpha}_{4} \text{可由} \boldsymbol{\alpha}_{1},\boldsymbol{\alpha}_{2}\text{线性表示} \\ \therefore & \boldsymbol{\alpha}_{4} = k_{1}\boldsymbol{\alpha}_{1} + k_{2}\boldsymbol{\alpha}_{2} \\ \therefore & \boldsymbol{B} = \left( \begin{matrix} \boldsymbol{\alpha}_{1} & \boldsymbol{\alpha}_{2} & \boldsymbol{\alpha}_{3} + \boldsymbol{\alpha}_{4} \end{matrix} \right) = \boldsymbol{A} \left( \begin{matrix} 1 & 0 & k_{1} \\ 0 & 1 & k_{2} \\ 0 & 0 & 1 \\ \end{matrix} \right)\\ \because & \left\vert \begin{matrix} 1 & 0 & k_{1} \\ 0 & 1 & k_{2} \\ 0 & 0 & 1 \\ \end{matrix} \right\vert \ne 0\\ \therefore & r(\boldsymbol{B}) = r(\boldsymbol{A}) = 3\\ \therefore & \boldsymbol{\alpha}_{1},\boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{3} + \boldsymbol{\alpha}_{4} \text{线性无关} \\ \end{array}$$
2. $\text{I: } \boldsymbol{\alpha}_{1},\cdots, \boldsymbol{\alpha}_{r}\text{, II: } \boldsymbol{\beta}_{1},\cdots, \boldsymbol{\beta}_{s}$，$\text{I}$ 可由 $\text{II}$ 线性表示，且 $r>s$，证明 $\boldsymbol{\alpha}_{1},\cdots,\boldsymbol{\alpha}_{r}$ 线性相关 $$\begin{array}{ll} \because & \text{I可由II线性表示} \\ \therefore & \text{I的秩} \le \text{II的秩} \le s < r \\ \therefore & \boldsymbol{\alpha}_{1},\cdots,\boldsymbol{\alpha}_{r} \text{线性相关} \\ \end{array}$$