题号:6639    题型:解答题    来源:B站刘老师开讲《线性代数B》第七套期末模拟考试
已知向量组: $\alpha_1=\left(\begin{array}{r}1 \\ -1 \\ 2 \\ 4\end{array}\right), \alpha_2=\left(\begin{array}{l}0 \\ 3 \\ 1 \\ 2\end{array}\right), \alpha_3=\left(\begin{array}{r}3 \\ 0 \\ 7 \\ 14\end{array}\right), \alpha_4=\left(\begin{array}{r}1 \\ -2 \\ 2 \\ 0\end{array}\right), \alpha_5=\left(\begin{array}{r}2 \\ 1 \\ 5 \\ 10\end{array}\right)$, 求该向量组的秩和一个最大无关组, 并把其余向量用最大无关组线性表示.
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答案：
答案：
解 :将 $\alpha_1, \alpha_2, \alpha_3, \boldsymbol{\alpha}_4, \boldsymbol{\alpha}_3$ 以列向量构成矩阵, 并对其进行初等行变换, 有

$$
\left(\alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5\right)=\left(\begin{array}{rrrrr}
1 & 0 & 3 & 1 & 2 \\
-1 & 3 & 0 & -2 & 1 \\
2 & 1 & 7 & 2 & 5 \\
4 & 2 & 14 & 0 & 10
\end{array}\right)
$$

$\sim$

$$
\left(\begin{array}{rrrrr}
1 & 0 & 3 & 1 & 2 \\
0 & 3 & 3 & -1 & 3 \\
0 & 1 & 1 & 0 & 1 \\
0 & 2 & 2 & -4 & 2
\end{array}\right)
$$
$\sim$
$$
\left(\begin{array}{rrrrr}
1 & 0 & 3 & 1 & 2 \\
0 & 1 & 1 & 0 & 1 \\
0 & 3 & 3 & -1 & 3 \\
0 & 2 & 2 & -4 & 2
\end{array}\right)
$$
$\sim$
$$
\left(\begin{array}{rrrrr}
1 & 0 & 3 & 1 & 2 \\
0 & 1 & 1 & 0 & 1 \\
0 & 0 & 0 & -1 & 0 \\
0 & 0 & 0 & -4 & 0
\end{array}\right)
$$
$\sim$
$$
\left(\begin{array}{rrrrr}
1 & 0 & 3 & 1 & 2 \\
0 & 1 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & -4 & 0
\end{array}\right)
$$

$\sim$
$$
\left(\begin{array}{ccccc}
\alpha_1 & \alpha_2 & \alpha_3 & \alpha_4 & \alpha_5 \\
1 & 0 & 3 & 0 & 2 \\
0 & 1 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0
\end{array}\right),
$$

从而该向量组的秩 $\mathrm{r}\left(\alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5\right)=3, \alpha_1, \alpha_2, \alpha_4$ 构成向量组的最大无关组, 且
$$
\alpha_3=3 \alpha_1+\alpha_2, \alpha_5=2 \alpha_1+\alpha_2 \text {. }
$$
点评 $\alpha_4$ 入选最大无关组!
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