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:

$$\dim (\ker (A))+\text{rank}(A)=n$$

Relation to Homomorphism

In abstract algebra, the kernel of a homomorphism $\phi$ is $\{x\mid \phi (x)=0\}$. 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.