方程组的向量表示

线性方程组

$$\begin{cases} a_{11} x_1 + a_{12} x_2 + \cdots + a_{1n} x_n = b_1, \\ a_{21} x_1 + a_{22} x_2 + \cdots + a_{2n} x_n = b_2, \\ \vdots \\ a_{n1} x_1 + a_{n2} x_2 + \cdots + a_{nn} x_n = b_n, \\ \end{cases}$$

矩阵表示为

$$\mathbf A = (a_{ij})_{m \times n} = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \\ \end{pmatrix}, \mathbf x = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\x_n \end{pmatrix}, \boldsymbol \beta = \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\b_n \end{pmatrix}$$

$$\mathbf A \mathbf x = \boldsymbol \beta$$

引入向量后

$$\mathbf A = \begin{pmatrix} \boldsymbol \alpha_1 & \boldsymbol \alpha_2 & \cdots & \boldsymbol \alpha_n \end{pmatrix}$$

则方程可表示为

$$\mathbf A \mathbf x = x_1 \boldsymbol \alpha_1 + x_2 \boldsymbol \alpha_2 + \cdots + \boldsymbol \alpha_n$$