KERNEL (MATHEMATICS)

In mathematics, the word 'kernel' has several meanings. In many cases it refers to a general construction which measures the failure of a function or homomorphism to be injective.

Contents

In set theory

In abstract algebra

In linear algebra

In category theory

In integral calculus

In statistics

See also

In set theory

Main articles: Kernel (set theory)

In set theory, the kernel of a function $f\; :\; X\; o\; Y$ is an equivalence relation on $X$ which is defined in terms of $f$:
:$kerleft(f$
ight) = {left(x_1,x_2
ight) in X imes X : fleft(x_1
ight) = fleft(x_2
ight)}.
The function ''f'' is injective if and only if the kernel is the diagonal in $X\; imes\; X$.

In abstract algebra

Main articles: Kernel (algebra)

Let $f$ be a homomorphism. The equivalence relation $kerleft(f$
ight) defined in the previous section 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 $Y$:
:$kerleft(f$
ight) = {x in X : fleft(x
ight) = 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.

In linear algebra

Main articles: Kernel (linear algebra)

The same definition is used in linear algebra as in abstract algebra: the kernel of a linear operator ''T'' is the set of solutions to the equation ''Tx'' = 0. Similarly, the kernel of a matrix ''A'' is the set of vectors that, when multiplied by ''A'', gives the zero vector.

In category theory

Main articles: 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:
:$mathrm\{eq\}left(f,\; g$
ight) = kerleft(f - g
ight).
Difference kernels, however, make sense in arbitrary categories and are often used in conjunction with kernel pairs.

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 $T\_k$ defined by
:$(T\_k\; f)(x)\; =\; int\_X\; k(x,\; x\text{'})\; f(x\text{'})\; ,\; dx\text{'}.$
is the function ''k''. ''k'' is also called a 'kernel function'.

In statistics

Main articles: Kernel (statistics)

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

See also

★ convolution kernel

★ heat kernel

★ kernel trick