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

System of linear equations

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

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

a (homogeneous) system of linear equations in the variables ${}x_{1},\ldots ,x_{n}$. A tuple ${}(\xi _{1},\ldots ,\xi _{n})\in K^{n}$ is called a solution of the linear system, if Failed to parse (syntax error): {\displaystyle {{}} \sum_{j <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|System of linear equations/Set of variables/Homogeneous and inhomogeneous/Definition]] __NOINDEX__ 1}^n a_{ij } \xi_j = 0} holds for all ${}i=1,\ldots ,m$.

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

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

is called an inhomogeneous system of linear equations. A tuple ${}(\zeta _{1},\ldots ,\zeta _{n})\in K^{n}$ is called a solution to the inhomogeneous linear system, if Failed to parse (syntax error): {\displaystyle {{}} \sum_{j <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|System of linear equations/Set of variables/Homogeneous and inhomogeneous/Definition]] __NOINDEX__ 1}^n a_{ij} \zeta_j = c_i} holds for all ${}i$.