The left null space of a matrix is a subspace of R^m (or C^m).

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)).