Linear system/Superposition principle/Homogeneous and inhomogeneous/Fact

Let ${\displaystyle {}K}$ be a field, and let

${\displaystyle {\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}&=&c_{1}\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}&=&c_{2}\\\vdots &\vdots &\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}&=&c_{m}\end{matrix}}}$

be an inhomogeneous linear system over ${\displaystyle {}K}$, and let

${\displaystyle {\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}&=&0\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}&=&0\\\vdots &\vdots &\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}&=&0\end{matrix}}}$

be the corresponding homogeneous linear system. If ${\displaystyle {}{\left(y_{1},\ldots ,y_{n}\right)}}$ is a solution of the inhomogeneous system and if ${\displaystyle {}{\left(z_{1},\ldots ,z_{n}\right)}}$ is a solution of the homogeneous system,

then ${\displaystyle {}{\left(y_{1}+z_{1},\ldots ,y_{n}+z_{n}\right)}}$ is a solution of the inhomogeneous system.