Linear system/Superposition principle/Section

Theorem

Let ${}M={\left(a_{ij}\right)}_{1\leq i\leq m,1\leq j\leq n}$ denote a matrix over a field ${}K$. Let ${}c={\left(c_{1},\ldots ,c_{m}\right)}$ and ${}d={\left(d_{1},\ldots ,d_{m}\right)}$ denote two ${}m$-tuples, and let ${}y={\left(y_{1},\ldots ,y_{n}\right)}\in K^{n}$ be a solution of the linear system

${}Mx=c\,,$

and ${}z={\left(z_{1},\ldots ,z_{n}\right)}\in K^{n}$ a solution of the system

${}Mx=d\,.$

Then

${}y+z={\left(y_{1}+z_{1},\ldots ,y_{n}+z_{n}\right)}$ is a solution of the system

${}Mx=c+d\,.$

Proof

See exercise.

$\Box$

Corollary

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

This follows immediately from fact.

$\Box$

In particular, this means that when ${}L$ is the solution space of a homogeneous linear system, and when ${}y$ is one (particular) solution of an inhomogeneous linear system, then the mapping

$L\longrightarrow L',z\longmapsto y+z,$

gives a bijection between ${}L$ and the solution set ${}L'$ of the inhomogeneous linear system.