# System of linear equations/Set of variables/Homogeneous and inhomogeneous/Definition

System of linear equations

Let ${\displaystyle {}K}$ denote a field, and let ${\displaystyle {}a_{ij}\in K}$ for ${\displaystyle {}1\leq i\leq m}$ and ${\displaystyle {}1\leq j\leq n}$. We call

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

a (homogeneous) system of linear equations in the variables ${\displaystyle {}x_{1},\ldots ,x_{n}}$. A tuple ${\displaystyle {}(\xi _{1},\ldots ,\xi _{n})\in K^{n}}$ is called a solution of the linear system, if ${\displaystyle {}\sum _{j=1}^{n}a_{ij}\xi _{j}=0}$ holds for all ${\displaystyle {}i=1,\ldots ,m}$.

If ${\displaystyle {}(c_{1},\ldots ,c_{m})\in K^{m}}$ is given,[1] then

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

is called an inhomogeneous system of linear equations. A tuple ${\displaystyle {}(\zeta _{1},\ldots ,\zeta _{n})\in K^{n}}$ is called a solution to the inhomogeneous linear system, if ${\displaystyle {}\sum _{j=1}^{n}a_{ij}\zeta _{j}=c_{i}}$ holds for all ${\displaystyle {}i}$.

1. Such a vector is sometimes called a disturbance vector of the system.