$$ \left\{ \begin{array}{ll} \vec{x}, \vec{y} \in \mathbb{R}^n, & f(\vec{x}+\vec{y})=f(\vec{x})+f(\vec{y}) \\ \vec{x} \in \mathbb{R}^n, k \in \mathbb{R}^1, & f(k \vec{x})=k f(\vec{x}) \end{array} \right. \Leftrightarrow f(\vec{x})=A\vec{x} $$

$$\vec{y}=f(\vec{x})$$

$$\vec{x}=x_1\vec{e}_1+x_2\vec{e}_2+\cdots$$

$$\vec{y}=y_1\vec{e}_1+y_2\vec{e}_2+\cdots$$

$$ \begin{array}{rcl} f(\vec{x})&=&f(x_1\vec{e}_1+x_2\vec{e}_2+\cdots) \\ &=&f(x_1\vec{e}_1)+f(x_2\vec{e}_2)+\cdots \\ &=&x_1f(\vec{e}_1)+x_2f(\vec{e}_2)+\cdots \\ \end{array} $$

$$ \begin{array}{rcl} f(\vec{e}_1)&=&a_{11}\vec{e}_1+a_{21}\vec{e}_2+\cdots\\ f(\vec{e}_2)&=&a_{12}\vec{e}_1+a_{22}\vec{e}_2+\cdots \\ &\vdots& \\ \end{array} $$

$$ \begin{array}{rcl} f(\vec{x})&=&x_1f(\vec{e}_1)+x_2f(\vec{e}_2)+\cdots \\ &=&x_1(a_{11}\vec{e}_1+a_{21}\vec{e}_2+\cdots) \\ & &+x_2(a_{12}\vec{e}_1+a_{22}\vec{e}_2+\cdots) \\ & &+\cdots \\ &=&(a_{11}x_1+a_{12}x_2+\cdots)\vec{e}_1 \\ & &+(a_{21}x_1+a_{22}x_2+\cdots)\vec{e}_2 \\ & &+\cdots \\ \end{array} $$

$$ \begin{array}{rcl} y_1\vec{e}_1+y_2\vec{e}_2+\cdots &=&(a_{11}x_1+a_{12}x_2+\cdots)\vec{e}_1 \\ & &+(a_{21}x_1+a_{22}x_2+\cdots)\vec{e}_2 \\ & &+\cdots \\ \end{array} $$

$$ \left\{\begin{array}{rcl} y_1&=&a_{11}x_1+a_{12}x_2+\cdots\\ y_2&=&a_{21}x_1+a_{22}x_2+\cdots\\ &\vdots & \\ \end{array} \right. $$

$$ \left[\begin{array}{c} y_1\\ y_2\\ \vdots\\ \end{array} \right] =\left[\begin{array}{ccc} a_{11}&a_{12}&\cdots\\ a_{21}&a_{22}&\cdots\\ \vdots & & \\ \end{array} \right] \left[\begin{array}{c} x_{1}\\ x_{2}\\ \vdots\\ \end{array} \right] $$