Definition of trivial solution to a homogeneous linear system of equations

Definition: The trivial solution to a homogeneous system $Ax = 0$ is the zero vector:
$$x = \mathbf{0} = (0, 0, \ldots, 0)$$

The trivial solution always exists for any homogeneous system because substituting all zeros satisfies every equation: $a_{i1} \cdot 0 + \cdots + a_{in} \cdot 0 = 0$.

Terminology: The trivial solution is called "trivial" because it is obvious and uninteresting. The focus is usually on whether nontrivial solutions exist, which reveals structural properties of the system (e.g., whether columns of $A$ are linearly dependent).