# Kernel

## Definition

Given a function between two spaces [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] of the same type of space (which is imbued with an identity element), the kernel of [ilmath]f:X\rightarrow Y[/ilmath] (where [ilmath]f[/ilmath] is a function) is defined as:

• $\text{Ker}(f)=\{x\in X|f(x)=e\}$ where $e$ denotes the identity of [ilmath]Y[/ilmath]

### Potential generalisation

Given a function $f:X\rightarrow Y$ the kernel where [ilmath]Y[/ilmath] is a space imbued with the concept of identity[footnotes 1] (this identity shall be denoted [ilmath]e[/ilmath]) written as follows:

• $\text{Ker}(f)=\{x\in X|f(x)=e\}$

### Vector spaces

Given a linear map [ilmath]T\in\mathcal{L}(V,W)[/ilmath] where [ilmath]V[/ilmath] and [ilmath]W[/ilmath] are vector spaces over the field [ilmath]F[/ilmath], that is [ilmath]T:U\rightarrow V[/ilmath], the kernel of [ilmath]T[/ilmath] is:

• $\text{Ker}(T)=\{v\in V|Tv=0\}$[1] using the notation $Tv=T(v)$

#### Potential generalisation

Given any function [ilmath]f:X\rightarrow V[/ilmath] where [ilmath]X[/ilmath] is any set and [ilmath]V[/ilmath] a vector space, we may define the kernel of [ilmath]f[/ilmath] as follows:

• $\text{Ker}(f)=\{x\in X|f(x)=0\}$ where $0$ denotes the additive identity of the vector space

### Groups

Given a homomorphism [ilmath]f:(G,\times_1)\rightarrow(H,\times_2)[/ilmath] between two groups, [ilmath]G[/ilmath] and [ilmath]H[/ilmath], we define the kernel of [ilmath]f[/ilmath] as follows:

• $\text{Ker}(f)=\{g\in G|f(g)=e_H\}$ where $e_H$ is the identity of $H$

#### Potential generalisation

I believe we can extend this definition to any map:

Given a group [ilmath](G,\times)[/ilmath] where [ilmath]e[/ilmath] denotes the identity, and a function [ilmath]f:X\rightarrow G[/ilmath] where [ilmath]X[/ilmath] is any set. The :kernel is defined as follows:
• $\text{Ker}(f)=\{x\in X|f(x)=e\}$

## Notes

1. Ambiguous for fields as they have two identities.

## References

1. Advanced Linear Algebra - Steven Roman - Third Edition - Springer GTM