Difference between revisions of "Kernel"

Latest revision as of 19:53, 10 May 2015

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:

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

Potential generalisation

Given a function [math]f:X\rightarrow Y[/math] 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:

• [math]\text{Ker}(f)=\{x\in X|f(x)=e\}[/math]

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:

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

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:

• [math]\text{Ker}(f)=\{x\in X|f(x)=0\}[/math] where [math]0[/math] 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:

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

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:

• [math]\text{Ker}(f)=\{x\in X|f(x)=e\}[/math]

Notes

1. ↑ Ambiguous for fields as they have two identities.

References

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