Four Vector Subspaces of Matrix

Definition

For a matrix ${\mathbf{A}}_{m\times n}$

• Null Space: $\mathbf{N}(\mathbf{A})\subset {\mathbb{R}}^n$
• Row Space: $\mathbf{C}({\mathbf{A}}^{⊺})\subset {\mathbb{R}}^n$
• Left Null Space: $\mathbf{N}({\mathbf{A}}^{⊺})\subset {\mathbb{R}}^m$
• Column Space: $\mathbf{C}(\mathbf{A})\subset {\mathbb{R}}^m$

Example

For a matrix ${\mathbf{A}}_{2\times 3}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\end{array}\right]$, $\text{rank}(\mathbf{A})=1$

subspace	bais vector	demension of vectorspace
$\mathbf{C}(\mathbf{A})$	$k\left[\begin{array}{c}1\\ 0\end{array}\right]\subset {\mathbb{R}}^2$	$\dim (\mathbf{C}(\mathbf{A}))=1$
$\mathbf{C}({\mathbf{A}}^{⊺})$	$k\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]\subset {\mathbb{R}}^3$	$\dim (\mathbf{C}({\mathbf{A}}^{⊺}))=1$
$\mathbf{N}(\mathbf{A})$	$v\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]+w\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\subset {\mathbb{R}}^3$	$\dim (\mathbf{N}(\mathbf{A}))=2$
$\mathbf{N}({\mathbf{A}}^{⊺})$	$v\left[\begin{array}{c}0\\ 1\end{array}\right]\subset {\mathbb{R}}^2$	$\dim (\mathbf{N}({\mathbf{A}}^{⊺}))=2$

Visualization

Facts

$\dim (\mathbf{N}(\mathbf{A}))=n-r$ $\dim (\mathbf{C}({\mathbf{A}}^{⊺}))=r$ $\dim (\mathbf{N}({\mathbf{A}}^{⊺}))=m-r$ $\dim (\mathbf{C}(\mathbf{A}))=r$

$\mathbf{N}(\mathbf{A})\perp \mathbf{C}({\mathbf{A}}^{⊺})$: null space and row space are orthogonal complement $\mathbf{N}({\mathbf{A}}^{⊺})\perp \mathbf{C}(\mathbf{A})$: left null space and column space are orthogonal complement