Definition of homogeneous linear system of equations

**Definition:** A **homogeneous linear system** is a system of the form $Ax = 0$, where every constant term on the right-hand side is zero.

Written out:
$$\begin{cases}
 a_{11} x_1 + a_{12} x_2 + \cdots + a_{1n} x_n = 0 \\
 a_{21} x_1 + a_{22} x_2 + \cdots + a_{2n} x_n = 0 \\
 \quad\vdots \\
 a_{m1} x_1 + a_{m2} x_2 + \cdots + a_{mn} x_n = 0\n\end{cases}$$

Every homogeneous system is consistent because $x = 0$ is always a solution (the **trivial solution**).

**Example:**
$$\begin{cases} x + y - z = 0 \\ 2x - y + 3z = 0 \end{cases}$$Copy