B站刘老师开讲《线性代数B》第七套期末模拟考试-Kmath科数- 中考、高考、考验数学题库

$\alpha_1=\left(\begin{array}{r}2 \\ 2 \\ -1\end{array}\right), \alpha_2=\left(\begin{array}{l}0 \\ 4 \\ 8\end{array}\right), \alpha_3=\left(\begin{array}{r}-1 \\ a \\ 3\end{array}\right), \beta=\left(\begin{array}{l}1 \\ 1 \\ 2\end{array}\right), \beta$ 不能由 $\alpha_1, \alpha_2, \alpha_3$ 线性表示, 求 $a$ 的值.

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答案：

解线性方程组无解 $x_1 \boldsymbol{\alpha}_1+x_2 \boldsymbol{\alpha}_2+x_3 \boldsymbol{\alpha}_3=\boldsymbol{\beta}$, 把向量组 $\boldsymbol{\alpha}_1, \boldsymbol{\alpha}_2, \boldsymbol{\alpha}_3, \boldsymbol{\beta}$ 按列排为矩阵, 并进行初等行变换:

$$
\left(\boldsymbol{\alpha}_1, \boldsymbol{\alpha}_2, \boldsymbol{\alpha}_3, \boldsymbol{\beta}\right)=\left(\begin{array}{rrrr}
2 & 0 & -1 & 1 \\
2 & 4 & a & 1 \\
-1 & 8 & 3 & 2
\end{array}\right)
$$
=
$$
\left(\begin{array}{cccc}
-1 & 0 & 1-2 a & 2 \\
0 & -4 & -1-a & 0 \\
0 & 0 & 1-4 a & 5
\end{array}\right)
$$

当 $1-4 a=0$, 即 $a=\frac{1}{4}$ 时, $\mathrm{r}\left(\alpha_1, \alpha_2, \alpha_3\right)=2, \mathrm{r}\left(\alpha_1, \alpha_2, \alpha_3, \beta\right)=3, \mathrm{r}\left(\alpha_1, \alpha_2, \alpha_3\right) \neq \mathrm{r}\left(\alpha_1, \alpha_2, \alpha_3, \beta\right)$,

线性方程组 $x_1 \boldsymbol{\alpha}_1+x_2 \alpha_2+x_3 \alpha_3=\boldsymbol{\beta}$ 无解, 此时 $\boldsymbol{\beta}$ 不能由 $\boldsymbol{\alpha}_1, \boldsymbol{\alpha}_2, \boldsymbol{\alpha}_3$ 线性表示.

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