Homogeneous System has Zero Vector as Solution

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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 iff the null space contains the zero vector.

The result follows from Null Space Contains Zero Vector.

$\blacksquare$