Let $K$ be a [[field]]. Let $A=\left(a_{i j}\right)\in K^{m\times n}$ be a [[matrix]] called the [[matrix of coefficients]] Let $B=\left(b_{i j}\right)\in K^{m\times 1}$ be the [[column vector]] of constants Let $X=(x_i)\in K^{n\times1}$ be the *solution vector* of unknowns A **system of linear equations** is collection of equations $\begin{cases} a_{11} x_{1}&+&\cdots&+&a_{1 n} x_{n}&&=b_{1}\\ a_{21} x_{1}&+&\cdots&+&a_{2 n} x_{n}&&=b_{2}\\ &&\ \ \vdots&&&&\ \vdots\\ a_{m 1} x_{1}&+&\cdots&+&a_{m n} x_{n}&&=b_{m}\end{cases}$ Which can be rewritten $x_{1}\left(\begin{array}{c}a_{11} \\ \vdots \\ a_{m 1}\end{array}\right)+\cdots+x_{n}\left(\begin{array}{c}a_{1 n} \\ \vdots \\ a_{m n}\end{array}\right)=\left(\begin{array}{c}b_{1} \\ \vdots \\ b_{m}\end{array}\right)$