# 2012年考研数二第07题解析

## 题目

$$A. \alpha_{1}, \alpha_{2}, \alpha_{3}$$

$$B. \alpha_{1}, \alpha_{2}, \alpha_{4}$$

$$C. \alpha_{1}, \alpha_{3}, \alpha_{4}$$

$$D. \alpha_{2}, \alpha_{3}, \alpha_{4}$$

## 解析

### 方法一

$$c_{1} = c_{2} = c_{3} = c_{4} = 0.$$

$$\alpha_{1} = \begin{pmatrix} 0\\ 0\\ 0 \end{pmatrix}$$

$$\alpha_{2} = \begin{pmatrix} 0\\ 1\\ 0 \end{pmatrix}$$

$$\alpha_{3} = \begin{pmatrix} 1\\ -1\\ 0 \end{pmatrix}$$

$$\alpha_{4} = \begin{pmatrix} -1\\ 1\\ 0 \end{pmatrix}$$

$$\alpha_{1} = 0 \times \alpha_{2};$$

$$\alpha_{1} = 0 \times \alpha_{3};$$

$$\alpha_{1} = 0 \times \alpha_{4}.$$

$$(-1) \times \alpha_{3} = \alpha_{4};$$

$$(-1) \times \alpha_{4} = \alpha_{3}.$$

$$\alpha_{3} + \alpha_{4} + \alpha_{1} = 0.$$

### 方法二

$$|\alpha_{1}, \alpha_{3}, \alpha_{4}| =$$

$$\begin{vmatrix} 0 & 1& -1\\ 0 & -1& 1\\ c_{1}& c_{3}& c_{4} \end{vmatrix} =$$

$$\begin{vmatrix} 0 & 0& 0\\ 0 & -1& 1\\ c_{1}& c_{3}& c_{4} \end{vmatrix} = 0.$$

### 方法三

“一个向量组线性相关的充分必要条件是，在这个向量组中至少存在一个向量可有【其余】向量线性表示。”

$$\alpha_{3} + \alpha_{4} =$$

$$\begin{pmatrix} 0\\ 0\\ c_{3} + c_{4} \end{pmatrix}$$

$$c_{1} = k(c_{3} + c_{4}).$$

$$\alpha_{1} = k(\alpha_{3} + \alpha_{4}) =$$

$$\alpha_{1} = k \alpha_{3} + k \alpha_{4}.$$

EOF