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.

$\blacksquare$

Sources

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