Difference between revisions of "Kernel (algebra)"
Karl Jones (Talk | contribs) (Created page with "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 homom...") |
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== Description == | == 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. | + | An important special case is the [[Kernel (linear algebra)|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 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. | ||
Latest revision as of 12:50, 21 September 2016
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.
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
== External links
- Kernel (algebra) @ Wikipedia
