线性无关的向量组「乘以」线性相关的向量组会得到一个线性相关的向量组

一、题目

已知四维列向量组 ${\mathit{\alpha }}_1,{\mathit{\alpha }}_2,{\mathit{\alpha }}_3,{\mathit{\alpha }}_4$ 线性无关, 则下列向量组中线性无关的是哪个？

(A) ${\mathit{\alpha }}_1+{\mathit{\alpha }}_2,{\mathit{\alpha }}_2+{\mathit{\alpha }}_3,{\mathit{\alpha }}_3+{\mathit{\alpha }}_4,{\mathit{\alpha }}_4+{\mathit{\alpha }}_1$.

(B) ${\mathit{\alpha }}_1-{\mathit{\alpha }}_2,{\mathit{\alpha }}_2-{\mathit{\alpha }}_3,{\mathit{\alpha }}_3-{\mathit{\alpha }}_4,{\mathit{\alpha }}_4-{\mathit{\alpha }}_1$.

(C) ${\mathit{\alpha }}_1+{\mathit{\alpha }}_2,{\mathit{\alpha }}_2-{\mathit{\alpha }}_3,{\mathit{\alpha }}_3-{\mathit{\alpha }}_4,{\mathit{\alpha }}_4+{\mathit{\alpha }}_1$.

(D) ${\mathit{\alpha }}_1+{\mathit{\alpha }}_2,{\mathit{\alpha }}_2-{\mathit{\alpha }}_3,{\mathit{\alpha }}_3-{\mathit{\alpha }}_4,{\mathit{\alpha }}_4-{\mathit{\alpha }}_1$.

难度评级：

二、解析

由于：

$$({\mathit{\alpha }}_1+{\mathit{\alpha }}_2,{\mathit{\alpha }}_2-{\mathit{\alpha }}_3,{\mathit{\alpha }}_3-{\mathit{\alpha }}_4,{\mathit{\alpha }}_4-{\mathit{\alpha }}_1)=$$

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

$$\left|\begin{array}{cccc}1& 0& 0& -1\\ 1& 1& 0& 0\\ 0& -1& 1& 0\\ 0& 0& -1& 1\end{array}\right|=\left|\begin{array}{ccc}1& 1& 0\\ 0& -1& 1\\ 1& 0& 0\\ -1& 1& 0\\ 0& -1& 1\end{array}\right|=$$

$$1+1=2\ne 0$$

因此，D 选项中的向量组一定线性无关。