Null Space

Definition

$\mathbf{N}(\mathbf{A})=\ker (f):=\{\mathbf{x}\in V:f(\mathbf{x})=0\}$

The vector space of $\mathbf{x}$ satisfying $\mathbf{Ax}=0$ for the matrix $\mathbf{A}$

Facts

${\mathbf{x}}_1+{\mathbf{x}}_2\in \mathbf{N}(\mathbf{A})\to \mathbf{A}({\mathbf{x}}_1+{\mathbf{x}}_2)=0$

$\mathbf{A}(k\mathbf{x})=k\mathbf{A}(\mathbf{x})=0$

$A0=0$

$\exists {\mathbf{A}}_{n\times n}^{-1}\iff \mathbf{N}(\mathbf{A})=\{0\}$

$\nexists {\mathbf{A}}^{-1}$ and there exists infinitely many solutions, $\Rightarrow \mathbf{x}={\mathbf{x}}_n+{\mathbf{x}}_p$ where ${\mathbf{x}}_n\in \mathbf{N}(\mathbf{A})$