> [!Definition] Definition (Linear system of equations) > Let $m,n\in \mathbb{N}^{+}.$ A *linear system of equations* is a set of $m$ equations in $n$ variables $x_{1},\dots,x_{n}$ which are of the form $\begin{array}{cc} a_{11} x_1+a_{12} x_2+\cdots+a_{1 n} x_n= & b_1 \\a_{21} x_1+a_{22} x_2+\cdots+a_{2 n} x_n= & b_2 \\\vdots & \vdots \\a_{m 1} x_1+a_{m 2} x_2+\cdots+a_{m n} x_n & =b_m\end{array} \tag{*}$where the $a_{ij}$ and $b_{k}$ are real numbers. # Properties