Characterization of Left Null Space

Definition
Let $\mathbf A_{m \times n}$ be a matrix.

Let the left null space of $\mathbf A$ be defined.

Then:


 * $\text{left null space}\left({\mathbf A}\right) = \left\{{\mathbf{x} \in \R^n : \mathbf x^T \mathbf A} = \mathbf 0 ^T\right\}$

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

Proof
For ease of presentation, let $\operatorname{N}^{\gets}\left({\mathbf A}\right)$ be the left null space of $\mathbf A$.

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

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

This implies that $\mathbf {x} \in \operatorname{N}\left({\mathbf A^T}\right) \iff \mathbf x^T \mathbf A = \mathbf 0^T$.

Recall $\mathbf {x} \in \operatorname{N}\left({\mathbf A^T}\right) \iff \mathbf {x} \in \operatorname{N}^{\gets} \left({\mathbf A}\right)$

Hence the result, by definition of set equality.