Null Space of Reduced Echelon Form

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.

The null space of $\mathbf A$ is the same as that of the null space of the reduced row echelon form of $\mathbf A$:


 * $\map {\mathrm N} {\mathbf A} = \map {\mathrm N} {\map {\mathrm {rref} } {\mathbf A} }$

Proof
By the definition of null space:


 * $\mathbf x \in \map {\mathrm N} {\mathbf A} \iff \mathbf A \mathbf x = \mathbf 0$

From the corollary to Row Equivalent Matrix for Homogeneous System has same Solutions:


 * $\mathbf A \mathbf x = \mathbf 0 \iff \map {\mathrm {rref} } {\mathbf A} \mathbf x = \mathbf 0$

Hence the result, by the definition of set equality.