Kernel (algebra)

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. 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.

In this article, we first survey kernels for some important types of algebraic structures; then we give general definitions from universal algebra for generic algebraic structures.

Read more about Kernel (algebra):  Algebras With Nonalgebraic Structure, Kernels in Category Theory

Famous quotes containing the word kernel:

    We should never stand upon ceremony with sincerity. We should never cheat and insult and banish one another by our meanness, if there were present the kernel of worth and friendliness. We should not meet thus in haste.
    Henry David Thoreau (1817–1862)