Matrix Form of a System of Linear Equations

Definition: Matrix Form of a System of Linear Equations

Every system of linear equations

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

can be represented as a single matrix equation, called a matrix form, where $A$ is the coefficient matrix:

$$A\vec{x}=\vec{b}$$