单选题 （共 6 题 ），每题只有一个选项正确
$n$ 维向量组 ${\alpha}_{1}, {\alpha}_{2}, \cdots, {\alpha}_{s}(3 \leqslant s \leqslant n)$ 线性无关的充分必要条件是 （ ）
$\text{A.}$ 存在一组不全为零的数 $k_{1}, k_{2}, \cdots, k_{s}$, 使 $k_{1} {\alpha}_{1}+k_{2} {\alpha}_{2}+\cdots+k_{s} {\alpha}_{s} \neq \mathbf{0}$.
$\text{B.}$ ${\alpha}_{1}, {\alpha}_{2}, \cdots, {\alpha}_{s}$ 中任意两个向量都线性无关.
$\text{C.}$ ${\alpha}_{1}, {\alpha}_{2}, \cdots, {\alpha}_{s}$ 中存在一个向量,它不能用其余向量线性表出.
$\text{D.}$ ${\alpha}_{1}, {\alpha}_{2}, \cdots, {\alpha}_{s}$ 中任意一个向量都不能用其余向量线性表出.
已知向量组 $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{4}$ 线性无关, 则向量组
$\text{A.}$ $ {\alpha}_{1}+ {\alpha}_{2}, {\alpha}_{2}+ {\alpha}_{3}, {\alpha}_{3}+ {\alpha}_{4}, {\alpha}_{4}+ {\alpha}_{1}$ 线性无关.
$\text{B.}$ $ {\alpha}_{1}- {\alpha}_{2}, {\alpha}_{2}- {\alpha}_{3}, {\alpha}_{3}- {\alpha}_{4}, {\alpha}_{4}- {\alpha}_{1}$ 线性无关.
$\text{C.}$ $ {\alpha}_{1}+ {\alpha}_{2}, {\alpha}_{2}+ {\alpha}_{3}, {\alpha}_{3}+ {\alpha}_{4}, {\alpha}_{4}- {\alpha}_{1}$ 线性无关.
$\text{D.}$ $ {\alpha}_{1}+ {\alpha}_{2}, {\alpha}_{2}+ {\alpha}_{3}, {\alpha}_{3}- {\alpha}_{4}, {\alpha}_{4}- {\alpha}_{1}$ 线性无关.
向量组 $\alpha_1, \alpha_2 \cdots, \alpha_s$ 线性无关的充分条件是
$\text{A.}$ $\alpha_1, \alpha_2 \cdots, \alpha_s$ 均不为零向量
$\text{B.}$ $\alpha_1, \alpha_2 \cdots, \alpha_s$ 中任意两个向量的分量不成比例
$\text{C.}$ $\alpha_1, \alpha_2 \cdots, \alpha_s$ 中任意一个向量均不能由其余 $s-1$ 个向量线性表示
$\text{D.}$ $\alpha_1, \alpha_2 \cdots, \alpha_s$ 中有一部分向量线性无关
设 $\alpha_1, \alpha_2, \cdots, \alpha_m$ 均为 $n$ 维向量，那么下列结论正确的是
$\text{A.}$ 若 $k_1 \alpha_1+k_2 \alpha_2+\cdots+k_m \alpha_m=0$ ，则 $\alpha_1, \alpha_2, \cdots, \alpha_m$线性相关
$\text{B.}$ 若对任意一组不全为零的数 $k_1, k_2, \cdots, k_m$ ，都有$k_1 \alpha_1+k_2 \alpha_2+\cdots+k_m \alpha_m \neq 0 \text { ， }$ 则 $\alpha_1, \alpha_2, \cdots, \alpha_m$ 线性无关
$\text{C.}$ 若 $\alpha_1, \alpha_2, \cdots, \alpha_m$ 线性相关，则对任意一组不全为零的数 $k_1, k_2, \cdots, k_m$ ，都有 $k_1 \alpha_1+k_2 \alpha_2+\cdots+k_m \alpha_m=0$
$\text{D.}$ 若 $0 \alpha_1+0 \alpha_2+\ldots+0 \alpha_m=0$ ，则 $\alpha_1, \alpha_2, \cdots, \alpha_m$ 线性无关
设向量组 $\alpha_1=(1,-1,2,4), \alpha_2=(0,3,1,2), \alpha_3=(3,0,7,14)$, $\alpha_4=(1,-2,2,4), \alpha_5=(2,1,5,10)$ ，则该向量组的极大线性无关组是
$\text{A.}$ $\alpha_1, \alpha_2, \alpha_3$
$\text{B.}$ $\alpha_1, \alpha_2, \alpha_4$
$\text{C.}$ $\alpha_1, \alpha_2, \alpha_5$
$\text{D.}$ $\alpha_1, \alpha_2, \alpha_4, \alpha_5$
设 $n$ 维行向量 $\alpha=\left(\frac{1}{2}, 0 \cdots, 0 \frac{1}{2}\right)$ ，矩阵 $A=E-\alpha^T \alpha, B=E+2 \alpha^T \alpha,$ 其中 $E$ 为 $n$ 阶单位矩阵，则 $A B$ 等于
$\text{A.}$ 0
$\text{B.}$ $-{E}$
$\text{C.}$ $E$
$\text{D.}$ $E+\alpha^T \alpha$