已知向量组overrightarrow(alpha_1), overrightarrow(alpha_2), overrightarrow(alpha_3), overrightarrow(alpha_4)线性无关，则下面向量组线性无关的是A overrightarrow(alpha_1) + overrightarrow(alpha_2), overrightarrow(alpha_2) + overrightarrow(alpha_3), overrightarrow(alpha_3) + overrightarrow(alpha_4), overrightarrow(alpha_4) + overrightarrow(alpha_1)B overrightarrow(alpha_1) - overrightarrow(alpha_2), overrightarrow(alpha_2) - overrightarrow(alpha_3), overrightarrow(alpha_3) - overrightarrow(alpha_4), overrightarrow(alpha_4) - overrightarrow(alpha_1)C overrightarrow(alpha_1) + overrightarrow(alpha_2), overrightarrow(alpha_2) + overrightarrow(alpha_3), overrightarrow(alpha_3) + overrightarrow(alpha_4), overrightarrow(alpha_4) - overrightarrow(alpha_1)D overrightarrow(alpha_1) + overrightarrow(alpha_2), overrightarrow(alpha_2) + overrightarrow(alpha_3), overrightarrow(alpha_3) - overrightarrow(alpha_4), overrightarrow(alpha_4) - overrightarrow(alpha_1)
已知向量组$\overrightarrow{\alpha_1}, \overrightarrow{\alpha_2}, \overrightarrow{\alpha_3}, \overrightarrow{\alpha_4}$线性无关，则下面向量组线性无关的是
A $\overrightarrow{\alpha_1} + \overrightarrow{\alpha_2}, \overrightarrow{\alpha_2} + \overrightarrow{\alpha_3}, \overrightarrow{\alpha_3} + \overrightarrow{\alpha_4}, \overrightarrow{\alpha_4} + \overrightarrow{\alpha_1}$
B $\overrightarrow{\alpha_1} - \overrightarrow{\alpha_2}, \overrightarrow{\alpha_2} - \overrightarrow{\alpha_3}, \overrightarrow{\alpha_3} - \overrightarrow{\alpha_4}, \overrightarrow{\alpha_4} - \overrightarrow{\alpha_1}$
C $\overrightarrow{\alpha_1} + \overrightarrow{\alpha_2}, \overrightarrow{\alpha_2} + \overrightarrow{\alpha_3}, \overrightarrow{\alpha_3} + \overrightarrow{\alpha_4}, \overrightarrow{\alpha_4} - \overrightarrow{\alpha_1}$
D $\overrightarrow{\alpha_1} + \overrightarrow{\alpha_2}, \overrightarrow{\alpha_2} + \overrightarrow{\alpha_3}, \overrightarrow{\alpha_3} - \overrightarrow{\alpha_4}, \overrightarrow{\alpha_4} - \overrightarrow{\alpha_1}$
题目解答
答案
答案：C
解析：
A. 考虑向量组 $\vec{\beta_1} = \vec{\alpha_1} + \vec{\alpha_2}$，$\vec{\beta_2} = \vec{\alpha_2} + \vec{\alpha_3}$，$\vec{\beta_3} = \vec{\alpha_3} + \vec{\alpha_4}$，$\vec{\beta_4} = \vec{\alpha_4} + \vec{\alpha_1}$。
线性组合 $k_1\vec{\beta_1} + k_2\vec{\beta_2} + k_3\vec{\beta_3} + k_4\vec{\beta_4} = \vec{0}$ 导出方程组：
$\begin{cases} k_1 + k_4 = 0 \\ k_1 + k_2 = 0 \\ k_2 + k_3 = 0 \\ k_3 + k_4 = 0 \end{cases}$
有非零解（如 $k_1 = 1, k_2 = -1, k_3 = 1, k_4 = -1$），故线性相关。
B. 同理，向量组 $\vec{\beta_1} = \vec{\alpha_1} - \vec{\alpha_2}$，$\vec{\beta_2} = \vec{\alpha_2} - \vec{\alpha_3}$，$\vec{\beta_3} = \vec{\alpha_3} - \vec{\alpha_4}$，$\vec{\beta_4} = \vec{\alpha_4} - \vec{\alpha_1}$ 也线性相关。
C. 向量组 $\vec{\beta_1} = \vec{\alpha_1} + \vec{\alpha_2}$，$\vec{\beta_2} = \vec{\alpha_2} + \vec{\alpha_3}$，$\vec{\beta_3} = \vec{\alpha_3} + \vec{\alpha_4}$，$\vec{\beta_4} = \vec{\alpha_4} - \vec{\alpha_1}$ 导出方程组：
$\begin{cases} k_1 - k_4 = 0 \\ k_1 + k_2 = 0 \\ k_2 + k_3 = 0 \\ k_3 + k_4 = 0 \end{cases}$
唯一解为 $k_1 = k_2 = k_3 = k_4 = 0$，故线性无关。
D. 同理，向量组 $\vec{\beta_1} = \vec{\alpha_1} + \vec{\alpha_2}$，$\vec{\beta_2} = \vec{\alpha_2} + \vec{\alpha_3}$，$\vec{\beta_3} = \vec{\alpha_3} - \vec{\alpha_4}$，$\vec{\beta_4} = \vec{\alpha_4} - \vec{\alpha_1}$ 线性相关。
结论： 线性无关的向量组是选项 C。
$\boxed{C}$