Definition of nontrivial solution to a homogeneous linear system of equations

Definition: A nontrivial solution to a homogeneous system $Ax = 0$ is any solution $x \neq \mathbf{0}$.

In other words, a nontrivial solution is a nonzero vector $x$ such that $Ax = 0$.

Example: For the system:
$$\begin{cases} x + y - z = 0 \\ 2x + 2y - 2z = 0 \end{cases}$$

$(1, 0, 1)$ is a nontrivial solution because:

• $1 + 0 - 1 = 0$ ✓
• $2(1) + 2(0) - 2(1) = 0$ ✓
• and $(1, 0, 1) \neq (0, 0, 0)$

The existence of a nontrivial solution is equivalent to the existence of a free variable in the system.