如何建立两个向量组之间的联系？

一、题目

若四维向量组 ${\mathit{\alpha }}_1,{\mathit{\alpha }}_2,{\mathit{\alpha }}_3,{\mathit{\alpha }}_4$ 线性无关, 且向量 ${\mathit{\beta }}_1={\mathit{\alpha }}_1+{\mathit{\alpha }}_3+{\mathit{\alpha }}_4$, ${\mathit{\beta }}_2={\mathit{\alpha }}_2-{\mathit{\alpha }}_4$, ${\mathit{\beta }}_3={\mathit{\alpha }}_3+{\mathit{\alpha }}_4$, ${\mathit{\beta }}_4={\mathit{\alpha }}_2+{\mathit{\alpha }}_3$, ${\mathit{\beta }}_5=2{\mathit{\alpha }}_1+{\mathit{\alpha }}_2+{\mathit{\alpha }}_3$. 则 $r({\mathit{\beta }}_1,{\mathit{\beta }}_2,{\mathit{\beta }}_3,{\mathit{\beta }}_4,{\mathit{\beta }}_5)=?$

难度评级：

二、解析

已知：

$$({\beta }_1,{\beta }_2,{\beta }_3,{\beta }_4)=({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4)\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 1\\ 1& 0& 1& 1\\ 1& -1& 1& 0\end{array}\right]$$

于是：

$$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 1\\ 1& 0& 1& 1\\ 1& -1& 1& 0\end{array}\right]\Rightarrow \left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 1\\ 0& 0& 1& 1\\ 0& -1& 1& 0\end{array}\right]\Rightarrow \left[\begin{array}{llll}1& 0& 0& 0\\ 0& 1& 0& 1\\ 0& 0& 1& 1\\ 0& 0& 0& 0\end{array}\right]$$

于是：

$$r({\mathit{\beta }}_1,{\mathit{\beta }}_2,{\mathit{\beta }}_3,{\mathit{\beta }}_4,{\mathit{\beta }}_5)=3$$

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