Row Equivalent Matrix for Homogeneous System has same Solutions/Corollary

Theorem
Let $\mathbf A$ be a matrix in the matrix space $\map {\MM_\R} {m, n}$ such that:


 * $\mathbf A \mathbf x = \mathbf 0$

represents a homogeneous system of linear equations.

Then:
 * $\set {\mathbf x: \mathbf A \mathbf x = \mathbf 0} = \set {\mathbf x: \map {\mathrm {ref} } {\mathbf A} \mathbf x = \mathbf 0}$

where $\map {\mathrm {ref} } {\mathbf A}$ is the reduced echelon form of $\mathbf A$.

Proof
Follows from Row Equivalent Matrix for Homogeneous System has same Solutions and from Matrix is Row Equivalent to Reduced Echelon Matrix.