Definition
System of linear equations (or linear system), is defined as a finite set of linear equation.
Remark
The variables in linear system are called unknowns
A linear system of equations in unknowns is written as
a_{11}x_{1} & + & a_{12}x_{2} & + & \dots & + & a_{1n}x_{n} & = & b_{1} \\ a_{21}x_{1} & + & a_{22}x_{2} & + & \dots & + & a_{2n}x_{n} & = & b_{2} \\ \vdots & & \vdots & && & \vdots & & \vdots \\ a_{m1}x_{1} & + & a_{m2}x_{2} & + & \dots & + & a_{mn}x_{n} & = & b_{m} \\ \end{matrix} $$ A linear system may also be written in [[3 Reference/Def-matrix\|matrix]] form as $$ \mathbf{Ax}=\mathbf{b} $$ , where: - $\mathbf{A} = [a_{ij}]$ is the $m \times n$ coefficient matrix - $\mathbf{x} = [x_1, x_2, \ldots, x_n]^T$ is the variable vector - $\mathbf{b} = [b_1, b_2, \ldots, b_m]^T$ is the constant vector ## Related theorems - [[3 Reference/Def-solution-of-linear-system\|Solution of Linear System]] - [[3 Reference/theorem-elementary-row-operation_202510021405\|Elementary Row Operation]] - [[3 Reference/1.1 Introduction to Systems of Linear Equations#theorem-solutions-of-linear-system|Theorem Solutions of linear system]] - [[3 Reference/1.1 Introduction to Systems of Linear Equations#theorem-elementary-row-operations|Theorem Elementary row operations]]