一、题目
已知 $n$ 维向量 $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{3}$ 线性无关, 则下列向量组中线性无关的是哪个？
(A) $\boldsymbol{\alpha}_{1}+\boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{2}+\boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{3}+\boldsymbol{\alpha}_{1}$.
(B) $\boldsymbol{\alpha}_{1}-\boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{2}-\boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{3}-\boldsymbol{\alpha}_{1}$.
(C) $\boldsymbol{\alpha}_{1}+\boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{2}-\boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{3}+\boldsymbol{\alpha}_{1}$.
(D) $\boldsymbol{\alpha}_{1}+\boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{2}+\boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{1}+2 \boldsymbol{\alpha}_{2}+\boldsymbol{\alpha}_{3}$.
难度评级：
二、解析 
由于：
$$
\left(\alpha_{1}+\alpha_{2}, \alpha_{2}+\alpha_{3}, \alpha_{3}+\alpha_{1}\right)=
$$
$$
\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right)\left[\begin{array}{lll}1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1\end{array}\right] \Rightarrow
$$
$$
\left|\begin{array}{lll}1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1\end{array}\right|=1+1=2 \neq 0
$$
因此，A 选项正确。