System of Linear Equations

Definition: System of Linear Equations

A system of $m$ linear equations with $n$ unknowns $x_1,\cdots ,x_n$ over some field $F$ is a set of $m$ equations which can be expressed in the following way:

$$|\begin{array}{rl}{a}_{11}{x}_1+\cdots +a_{1n}{x}_n& =b_1\\ a_{21}{x}_1+\cdots +a_{2n}{x}_n& =b_2\\ ⋮⋮& \\ a_{m1}{x}_1+\cdots +a_{mn}{x}_n& =b_n\end{array}\text{ where }{a}_{ij},b_i\in F$$