Linear system/Superposition principle/Homogeneous and inhomogeneous/Fact

Superposition principle

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

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

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

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

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

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

Proof, Write another proof