Characterization of Left Null Space

Definition
Let $\mathbf A_{m \times n}$ be a matrix in the matrix space $\map {\MM_{m, n} } \R$.

Let $\map {\operatorname {N^\gets} } {\mathbf A}$ be used to denote the left null space of $\mathbf A$.

Then:
 * $\map {\operatorname {N^\gets} } {\mathbf A} = \set {\mathbf x \in \R^n: \mathbf x^\intercal \mathbf A = \mathbf 0^\intercal}$

where $\mathbf X^\intercal$ is the transpose of $\mathbf X$.

Proof
Let $\mathbf x = \begin {bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end {bmatrix} \in \R^n$.

We have that $\mathbf A^\intercal \mathbf x = \mathbf 0$ is equivalent to $\mathbf x^\intercal \mathbf A = \mathbf 0^\intercal$.

This implies that $\mathbf x \in \map {\operatorname N} {\mathbf A^\intercal} \iff \mathbf x^\intercal \mathbf A = \mathbf 0^\intercal$.

Recall that:
 * $\mathbf x \in \map {\operatorname N} {\mathbf A^\intercal} \iff \mathbf x \in \map {\operatorname {N^\gets} } {\mathbf A}$

Hence the result, by definition of set equality.