Kernel (mathematics)

From Wikipedia, the free encyclopedia

You have new messages (last change).

In mathematics, especially abstract algebra, a kernel is a general construction which measures the failure of a homomorphism or function to be injective.

Contents

[edit] In set theory

Main article: Kernel (set theory)

In set theory, the kernel of a function f : X \to Y is an equivalence relation on X which is defined in terms of f:

\ker\left(f\right) = \{\left(x_1,x_2\right) \in X \times X : f\left(x_1\right) = f\left(x_2\right)\}

The function f is injective if and only if the kernel is the diagonal in X \times X.

[edit] In abstract algebra

Main article: Kernel (algebra)

In algebra, f is a homomorphism, and the equivalence relation \ker\left(f\right) becomes a congruence relation on X (i.e. the equivalence relation is compatible with the algebraic structure). For many algebraic structures, such as groups, rings, and vector spaces, there is a simpler definition of the kernel that is usually preferred; in these cases the equivalence relation is entirely determined by the equivalence class of the neutral element, and the kernel is defined as the preimage of the neutral element in Y:

\ker\left(f\right) = \{x \in X : f\left(x\right) = 0\}

The congruence relation is replaced with the notion of a normal subgroup, in the case of groups, or an ideal, in the case of rings. For linear operators between vector spaces, the kernel is also known as the null space.

[edit] In category theory

Main article: Kernel (category theory)

There exist several notions in category theory which seek to generalize the concept of a kernel in algebra. In categories with zero morphisms, the kernel of a morphism f is defined as the equalizer of f and the parallel zero morphism. Additionally, the kernel pair of a morphism f (similar to a congruence relation in algebra) is defined as the pullback of f with itself. In the category of sets this is simply the kernel of a function.

A difference kernel is another name for a binary equalizer. The name comes from preadditive categories, where one can define the equalizer of f and g as the kernel of the difference:

\mathrm{eq}\left(f, g\right) = \ker\left(f - g\right)

Difference kernels, however, make sense in arbitrary categories and are often used in conjunction with kernel pairs.

[edit] In integral calculus

In reference to a series, the kernel conveys the idea of the generating function. Similarly, in integral calculus, the kernel is the part of the integrand that defines the integral transform; specifically, the kernel of the operator Tk defined by

(T_k f)(x) = \int_X k(x, x') f(x') \, dx'

is the function k.

[edit] In statistics

Main article: Stochastic kernel

A stochastic kernel is the transition function of a stochastic process (usually discrete).

[edit] See also

Retrieved from "http://en.wikipedia.org../../../k/e/r/Kernel_%28mathematics%29.html"