**Theorem:** The *left null space* of an \(m 	imes n\) matrix \(A\) is a subspace of \(\mathbb{F}^m\):
\[	ext{left-null}(A) = \{\mathbf{y} \in \mathbb{F}^m : A^T\mathbf{y} = \mathbf{0}\} = \{\mathbf{y} \in \mathbb{F}^m : \mathbf{y}^T A = \mathbf{0}^T\}\]

This is the null space of \(A^T\), hence a subspace of \(\mathbb{F}^m\).

**Geometric interpretation:** The left null space is the orthogonal complement of the column space:
\[	ext{left-null}(A) = (	ext{col}(A))^\perp\]

Every vector in the left null space is orthogonal to every column of \(A\).

**Dimension:** \(\dim(	ext{left-null}(A)) = m - 	ext{rank}(A)\).