线性方程组

Linear Equations

The central problem of linear algebra is to solve a system of equations
"线性代数的核心问题就是求解线性方程组"

n 个未知数，n 个方程联立为线性方程组

{\begin{cases} a_{11} x_{1} + a_{12} x_{2} + \dots + a_{1 n} x_{n} = b_{1} \\ a_{21} x_{1} + a_{22} x_{2} + \dots + a_{2 n} x_{n} = b_{2} \\ \dots \\ a_{m 1} x_{1} + a_{m 2} x_{2} + \dots + a_{m n} x_{n} = b_{n} \end{cases}

将线性方程组改写为向量方程组的形式，本质上就是向量的线性组合 Linear Combination

\begin{array}{r} (\begin{array}{c} a_{11} \\ a_{21} \\ \dots \\ a_{n 1} \end{array}) x_{1} + (\begin{array}{c} a_{12} \\ a_{22} \\ \dots \\ a_{n 2} \end{array}) x_{2} + \dots + (\begin{array}{c} a_{1 n} \\ a_{2 n} \\ \dots \\ a_{n n} \end{array}) x_{n} = (\begin{array}{c} b_{1} \\ b_{2} \\ \dots \\ b_{n} \end{array}) \end{array}

进一步写为矩阵方程，引出矩阵的概念

\begin{aligned} A \vec{x} = \vec{b} \Leftrightarrow (\begin{array}{c} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{array}) (\begin{array}{c} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{array}) & = (\begin{array}{c} b_{1} \\ b_{2} \\ ⋮ \\ b_{m} \end{array}) \end{aligned}

$A$ 即为线性方程组的系数矩阵 Coefficient matrix

线性方程组 <--> 向量等式 (向量的线性组合) <--> 矩阵等式

\begin{matrix} {\begin{cases} x - 2 y = 1 \\ 3 x + 2 y = 11 \end{cases} \Leftrightarrow x (\begin{matrix} 1 \\ 3 \end{matrix}) + y (\begin{matrix} - 2 \\ 2 \end{matrix}) = (\begin{matrix} 1 \\ 11 \end{matrix}) \Leftrightarrow (\begin{matrix} 1 & - 2 \\ 3 & 2 \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} 1 \\ 11 \end{matrix}) \end{matrix}