$A.$ ${\alpha}_{1}+ {\alpha}_{2}, {\alpha}_{2}+ {\alpha}_{3}, {\alpha}_{3}+ {\alpha}_{4}, {\alpha}_{4}+ {\alpha}_{1}$ 线性无关. $B.$ ${\alpha}_{1}- {\alpha}_{2}, {\alpha}_{2}- {\alpha}_{3}, {\alpha}_{3}- {\alpha}_{4}, {\alpha}_{4}- {\alpha}_{1}$ 线性无关. $C.$ ${\alpha}_{1}+ {\alpha}_{2}, {\alpha}_{2}+ {\alpha}_{3}, {\alpha}_{3}+ {\alpha}_{4}, {\alpha}_{4}- {\alpha}_{1}$ 线性无关. $D.$ ${\alpha}_{1}+ {\alpha}_{2}, {\alpha}_{2}+ {\alpha}_{3}, {\alpha}_{3}- {\alpha}_{4}, {\alpha}_{4}- {\alpha}_{1}$ 线性无关.

C

#### 解析：

(A): 由于 $\left(\alpha_{1}+\alpha_{2}\right)-\left(\alpha_{2}+\alpha_{3}\right)+\left(\alpha_{3}+\alpha_{4}\right)-\left(\alpha_{4}+\alpha_{1}\right)=0$, 所以 (A) 线性相关.
(B) : 由于 $\left(\alpha_{1}-\alpha_{2}\right)+\left(\alpha_{2}-\alpha_{3}\right)+\left(\alpha_{3}-\alpha_{4}\right)+\left(\alpha_{4}-\alpha_{1}\right)=0$, 所以 (B) 线性相关. 对于 (C), 实验几组数据不能得到 0 时, 应立即计算由 $\alpha$ 的系数构成的行列式, 即
$$\left|\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{array}\right|=2 \neq 0,$$

$$\left(\alpha_{1}+\alpha_{2}\right)-\left(\alpha_{2}+\alpha_{3}\right)+\left(\alpha_{3}-\alpha_{4}\right)+\left(\alpha_{4}-\alpha_{1}\right)=0 \text {, }$$

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