The left null space of a matrix is a subspace of R^m (or C^m).
Created almost 9 years ago, updated 5 months ago
ID24111
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)).
**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:** \(\dim( ext{left-null}(A)) = m - ext{rank}(A)\).