Kernel (mathematics)
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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.
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[edit] In set theory
In set theory, the kernel of a function 
is an equivalence relation on X which is defined in terms of f:
The function f is injective if and only if the kernel is the diagonal in 
.
[edit] In abstract algebra
Let f be a homomorphism. The equivalence relation 
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:
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 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.
[edit] In 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:
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
is the function k.
[edit] In statistics
A stochastic kernel is the transition function of a stochastic process (usually discrete).
