Definition of solution to a linear equation

Definition: A system of linear equations (or linear system) is a finite collection of linear equations involving the same set of variables $x_1, x_2, \ldots, x_n$:

$$\begin{cases}\n a_{11} x_1 + a_{12} x_2 + \cdots + a_{1n} x_n = b_1 \\\n a_{21} x_1 + a_{22} x_2 + \cdots + a_{2n} x_n = b_2 \\\n \quad\vdots \\\n a_{m1} x_1 + a_{m2} x_2 + \cdots + a_{mn} x_n = b_m\n\end{cases}$$

This can be written compactly in matrix form as $Ax = b$, where:

• $A$ is the $m \times n$ coefficient matrix
• $x$ is the column vector of variables
• $b$ is the column vector of constants

Example:
$$\begin{cases}\n2x + y = 5 \\\nx - 3y = -1\n\end{cases}$$

is a system of 2 equations in 2 unknowns.