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 is consistent.

Proof
By the definition of null space, $\mathbf 0$ is a solution iff the null space contains the zero vector.

The result follows from Null Space Contains Zero Vector.

Proof of Corollary
Follows from the definition of consistency.