View source for Kernel (algebra)

In the various branches of [[mathematics]] that fall under the heading of '''abstract algebra''', the '''kernel''' of a [[homomorphism]] measures the degree to which the homomorphism fails to be [[Injective function|injective]].

== Description ==

An important special case is the kernel of a linear map. The kernel of a matrix, also called the null space, is the kernel of the linear map defined by the matrix.

The definition of kernel takes various forms in various contexts. But in all of them, the kernel of a homomorphism is trivial (in a sense relevant to that context) if and only if the homomorphism is injective.

The fundamental theorem on homomorphisms (or first isomorphism theorem) is a theorem, again taking various forms, that applies to the [[quotient algebra]] defined by the kernel.

== See also ==

* [[Homomorphism]]
* [[Injective function]]
* [[Kernel (linear algebra)]]
* [[Quotient algebra]]
* [[Zero set]]

== External links

* [https://en.wikipedia.org/wiki/Kernel_(algebra) Kernel (algebra)] @ Wikipedia

[[Category:Mathematics]]