Kernel

Kernel in group homomorphism

13.13 Definition: Kernel

Let $\varphi :G\to G^{\prime }$ $Ker(\varphi )$ is the kernel of $\varphi$, $Ker(\varphi )=\{x\in G|\varphi (x)=e^{\prime }\}$ In other words: The Elements in G that correspond to the identity element in G’

Example: Linear Transformation Let $\varphi :{\mathbb{R}}^n\to {\mathbb{R}}^m$ and A be an $m\times n$ matrix of real numbers: $\varphi (\mathbf{v})=A\mathbf{v}$ $\varphi$ is a homomorphism, because $\varphi (v+w)=A(v+w)=Av+Aw=\varphi (v)+\varphi (w)$ This is known as a linear transformation. $Ker(\varphi )$ is known as the Null Space (Kernel of a Matrix) of A. It consists all $\mathbf{v}\in {\mathbb{R}}^n$ such that $A\mathbf{v}=0$ (the zero vector)

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Kernel 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

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