Null Space (Kernel of a Matrix)

Definition

For an $m\times n$ matrix $A$ representing a linear map

$$A:{\mathbb{R}}^n\to {\mathbb{R}}^m,x\mapsto Ax,$$

the kernel (or null space) is

$$\ker (A)=\{x\in {\mathbb{R}}^n\mid Ax=0\}.$$

Properties

• $\ker (A)$ is a subspace of ${\mathbb{R}}^n$.
• The nullity is $\dim (\ker (A))$.
• Rank–Nullity Theorem: $$ \dim\bigl(\ker(A)\bigr) + rank(A) = n.

## Relation to Homomorphism In abstract algebra, the [[Section 13 Homomorphisms#1313-definition-kernel|kernel of a homomorphism]] $\varphi$ is $\{x\mid \varphi(x)=0\}$. A matrix $A$ is a linear map, so its kernel matches this definition:

\ker(A) = {,x \mid A(x)=0}.

## Geometric Interpretation The kernel describes the directions that are “flattened” to $0$ by the linear map. A larger nullity means more information is collapsed.