The word kernel has several meanings in mathematics, some related to each other and some not. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...

Kernels in algebra

Kernels in abstract algebra are general constructions which measure the failure of a homomorphism or function to be injective. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... This word should not be confused with homeomorphism. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...

In set theory, the kernel of a function f : X → Y is an equivalence relation on X which is defined in terms of f. Specifically, Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, the kernel of a function f may be taken to be either the equivalence relation on the functions domain that roughly expresses the idea of equivalent as far as the function f can tell, or the corresponding partition of the domain. ... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...

ker(f) = {(x₁, x₂) ∈ X × X | f(x₁) = f(x₂)}.

The function f is injective iff the kernel is the diagonal in X × X. ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Although P iff Q is most standard, common alternative phrases include P is necessary and sufficient for Q and P...

This idea generalizes readily to the algebraic setting where f is now a homomorphism. The equivalence relation ker(f) becomes a congruence relation on X (i.e. the equivalence relation is compatible with the algebraic structure). This word should not be confused with homeomorphism. ... In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s). ...

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. In these cases the kernel is defined as the preimage of the neutral element in Y. In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...

ker(f) = {x ∈ X | f(x) = 0 }

The congruence relation is now 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 also goes by the name of null space. In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g-1ng is still in N. The statement N is a normal subgroup of G is written: . Another way... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers like even number or multiple of 3. For instance, in rings one studies prime ideals instead of prime numbers, one defines coprime ideals as a generalization of... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... The null space (also nullspace) of a matrix A is the set of all vectors v which solve the equation Av = 0. ...

For more on the kernel of a homomorphism, see 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. ...

Kernels in category theory

There exists several notions in category theory which seek to generalize the concept of a kernel in algebra. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...

• In categories with zero morphisms one can define the kernel of a morphism f as the equalizer of f and the parallel zero morphism. For more on this see kernel (category theory).
• A kernel pair is a categorical notion which is more closely related to the notion of a congruence relation in algebra. The kernel pair of a morphism f is defined as a pullback of f with itself. In the category of sets this just gives the familiar kernel of a function defined above.
• 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: eq(f,g) = ker(f − g). Difference kernels, however, make sense in arbitrary categories and are often used in conjunction with kernel pairs.

In category theory, a zero morphism is a special kind of trivial morphism. ... In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ... This article is about equalisers in mathematics. ... In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. ... In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ... In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ... This article is about equalisers in mathematics. ... This article is about equalisers in mathematics. ... A preadditive category is a category that is enriched over the monoidal category of abelian groups. ...

Kernels in graph theory

Unrelated to the meanings in algebra and category theory, there is the notion of a kernel of a digraph.

Kernels of operators

Unrelated to the meanings in algebra, category theory and graph theory, there is the notion of a kernel of an integral operator. In analysis, consider an integral transform T which transforms a function f into a function Tf given by the integral formula The function k(x,y) that appears in this formula is the kernel of the operator T. See also: Dirichlet kernel convolution kernel trick Categories: Stub | Mathematical analysis ...

Kernel in statistics

A stochastic kernel is the transition function of a (usually discrete) stochastic process. A stochastic kernel is the transition function of a (usually discrete) stochastic process. ... In the mathematics of probability, a stochastic process can be thought of as a random function. ...