Let $𝒮$ be a system of linear equation in the set of variables $𝒳=\{x_1,x_2,\dots ,x_n\}$ given with equations

$$\begin{array}{ccc}{a}_{11}{x}_{11}+a_{12}{x}_{12}+\cdots +a_{1n}{x}_{1n}& =& b_1\\ a_{21}{x}_{21}+a_{22}{x}_{22}+\cdots +a_{2n}{x}_{2n}& =& b_2\\ & ⋮& \\ a_{m1}{x}_{m1}+a_{m2}{x}_{m2}+\cdots +a_{mn}{x}_{mn}& =& b_m\end{array}$$

Let $𝒜$ be the $m\times n$ matrix where entry $ij$ equals $a_{ij}$ and $𝒷$ be the $1\times n$ column matrix whose entry $1j$ equals $b_j$. The matrix $𝒜$ is coefficient matrix of the system of linear equations. The augmented matrix of the system $(𝒜\mid 𝒷)$ is the $m\times n+1$ matrix where entry $ij$ equals $a_{ij}$ if $j\le n$ and $b_j$ otherwise.That is the coefficient matrix is

$$\left(\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ & & \ddots & \\ a_{m1}& a_{m2}& \cdots & a_{mn}\end{array}\right)$$

and the augmented matrix is

$$\left(\begin{array}{ccccc}{a}_{11}& a_{12}& \cdots & a_{1n}& b_1\\ a_{21}& a_{22}& \cdots & a_{2n}& b_2\\ & & \ddots & & \\ a_{m1}& a_{m2}& \cdots & a_{mn}& b_m\end{array}\right)$$

Let $𝒮$ be a system of linear equation in the set of variables $𝒳=\{x_1,x_2,\dots ,x_n\}$ given with equations

$$\begin{array}{ccc}{a}_{11}{x}_{11}+a_{12}{x}_{12}+\cdots +a_{1n}{x}_{1n}& =& b_1\\ a_{21}{x}_{21}+a_{22}{x}_{22}+\cdots +a_{2n}{x}_{2n}& =& b_2\\ & ⋮& \\ a_{m1}{x}_{m1}+a_{m2}{x}_{m2}+\cdots +a_{mn}{x}_{mn}& =& b_m\end{array}$$

the vector representation of the system of linear equations is

$$\left(\begin{array}{c}{a}_{11}\\ a_{21}\\ ⋮\\ a_{m1}\end{array}\right)x_1+\left(\begin{array}{c}{a}_{12}\\ a_{22}\\ ⋮\\ a_{m2}\end{array}\right)x_2+\cdots +\left(\begin{array}{c}{a}_{1n}\\ a_{2n}\\ ⋮\\ a_{mn}\end{array}\right)x_n=\left(\begin{array}{c}{b}_1\\ b_2\\ ⋮\\ b_n\end{array}\right)$$

Let $𝒮$ be a system of linear equation in the set of variables $𝒳=\{x_1,x_2,\dots ,x_n\}$ given with equations

$$\begin{array}{ccc}{a}_{11}{x}_{11}+a_{12}{x}_{12}+\cdots +a_{1n}{x}_{1n}& =& b_1\\ a_{21}{x}_{21}+a_{22}{x}_{22}+\cdots +a_{2n}{x}_{2n}& =& b_2\\ & ⋮& \\ a_{m1}{x}_{m1}+a_{m2}{x}_{m2}+\cdots +a_{mn}{x}_{mn}& =& b_m\end{array}$$

the matrix representation of the system of linear equations is

$$\left(\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ & & \ddots & \\ a_{m1}& a_{m2}& \cdots & a_{mn}\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right)=\left(\begin{array}{c}{b}_1\\ b_2\\ ⋮\\ b_n\end{array}\right)$$

or

$$\left(\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ & & \ddots & \\ a_{m1}& a_{m2}& \cdots & a_{mn}\end{array}\right)\overrightarrow{x}=\left(\begin{array}{c}{b}_1\\ b_2\\ ⋮\\ b_n\end{array}\right)$$

for short.