如何建立两个向量组之间的联系？

一、题目

若四维向量组 $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{4}$ 线性无关, 且向量 $\boldsymbol{\beta}_{1}=\boldsymbol{\alpha}_{1}+\boldsymbol{\alpha}_{3}+\boldsymbol{\alpha}_{4}$, $\boldsymbol{\beta}_{2}=\boldsymbol{\alpha}_{2}-\boldsymbol{\alpha}_{4}$, $\boldsymbol{\beta}_{3}=\boldsymbol{\alpha}_{3}+\boldsymbol{\alpha}_{4}$, $\boldsymbol{\beta}_{4}=\boldsymbol{\alpha}_{2}+\boldsymbol{\alpha}_{3}$, $\boldsymbol{\beta}_{5}=2 \boldsymbol{\alpha}_{1}+\boldsymbol{\alpha}_{2}+\boldsymbol{\alpha}_{3}$. 则 $r\left(\boldsymbol{\beta}_{1}, \boldsymbol{\beta}_{2}, \boldsymbol{\beta}_{3}, \boldsymbol{\beta}_{4}, \boldsymbol{\beta}_{5}\right)=?$

难度评级：

二、解析

已知：

$$
\left(\beta_{1}, \beta_{2}, \beta_{3}, \beta_{4}\right)=\left(\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}\right)\left[\begin{array}{cccc}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & -1 & 1 & 0\end{array}\right]
$$

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：

$$
r\left(\boldsymbol{\beta}_{1}, \boldsymbol{\beta}_{2}, \boldsymbol{\beta}_{3}, \boldsymbol{\beta}_{4}, \boldsymbol{\beta}_{5}\right) = 3
$$

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