Kernel and Image in linear algebra

Kernel

For a linear transformation $T$ $\mathrm{Ker}(T):=\{\mathbf{x}\in {\mathbb{R}}^n:T(x)=0\}\subseteq {\mathbb{R}}^n$

$$\mathrm{Ker}(T)=\mathrm{N}(A)$$

NOTE

The kernel of a matrix is the null space of it

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.Link to original

Image

For a linear transformation $T$ $\mathrm{Im}(T):=\{T(\mathbf{x}):\mathbf{x}\in {\mathbb{R}}^n\}\subseteq {\mathbb{R}}^m$

In matrix

$$\mathrm{Im}(T)=\mathrm{C}(A)$$