Homogeneous System has Zero Vector as Solution

Theorem

Every homogeneous system of linear equations has the zero vector as a solution.

Corollary

Every homogeneous system of linear equations is consistent.

Proof

By the definition of null space, $\mathbf 0$ is a solution if and only if the null space contains the zero vector.

The result follows from Null Space Contains Zero Vector.

$\blacksquare$