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方程组的向量表示

Kamimika...小于 1 分钟学习笔记

方程组的向量表示

线性方程组

{a11x1+a12x2++a1nxn=b1,a21x1+a22x2++a2nxn=b2,an1x1+an2x2++annxn=bn, \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

A=(aij)m×n=(a11a12a1na21a22a2nan1an2ann),x=(x1x2xn),β=(b1b2bn) \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}

Ax=β \mathbf A \mathbf x = \boldsymbol \beta

引入向量后

A=(α1α2αn) \mathbf A = \begin{pmatrix} \boldsymbol \alpha_1 & \boldsymbol \alpha_2 & \cdots & \boldsymbol \alpha_n \end{pmatrix}

则方程可表示为

Ax=x1α1+x2α2++αn \mathbf A \mathbf x = x_1 \boldsymbol \alpha_1 + x_2 \boldsymbol \alpha_2 + \cdots + \boldsymbol \alpha_n

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贡献者: wzh
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