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).Copy