Kernel (mathematics)

In mathematics, the word kernel has several meanings. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In many cases it refers to a general construction which measures the failure of a function or homomorphism to be injective. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector

1 In set theory
2 In abstract algebra
3 In linear algebra and functional analysis
4 Of a matrix
5 In category theory
6 In integral calculus
7 In probability theory and statistics
8 See also

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 $X \times X$ . In Mathematics, the kernel of a function f may be taken to be either the Equivalence relation on the function's domain In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" ↔ A diagonal can refer to a line joining two nonconsecutive vertices of a Polygon or Polyhedron, or in contexts any upward or downward sloping line

In abstract algebra

Main article: Kernel (algebra)

Let $f$ be a homomorphism. 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 In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector The equivalence relation $\ker\left(f\right)$ defined in the previous section becomes a congruence relation on $X$ (that is, the equivalence relation is compatible with the algebraic structure). See Congruence (geometry for the term as used in elementary geometry 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. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added 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 In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage 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 In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. For linear operators between vector spaces, the kernel is also known as the null space. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that

In linear algebra and functional analysis

Main article: Kernel (linear operator)

The same definition is used in linear algebra as in abstract algebra: the kernel or nullspace of a linear operator T is the set of solutions to the equation Tx = 0. Linear algebra is the branch of Mathematics concerned with For functional analysis as used in psychology see the Functional analysis (psychology article See also Kernel (mathematics In Linear algebra and Functional analysis, the kernel of a linear Operator L is the set of all Linear algebra is the branch of Mathematics concerned with In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that

Of a matrix

Main article: Kernel (matrix)

The kernel, or nullspace, of a matrix A is the set of vectors that, when multiplied by A, give the zero vector. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Linear algebra, the kernel or null space (also nullspace) of a matrix A is the set of all vectors x for which In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally

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 Category theory and its applications to other branches of Mathematics, kernels are a generalization of the kernels of Group homomorphisms and the kernels In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In categories with zero morphisms, the kernel of a morphism $f$ is defined as the equalizer of $f$ and the parallel zero morphism. In Category theory, a zero morphism is a special kind of "trivial" Morphism. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and 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 Category theory, a branch of Mathematics, a pullback (also called a fibered product or Cartesian square) is the limit of a In the category of sets this is simply the kernel of a function. In Mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose Morphisms are

A difference kernel is another name for a binary equalizer. In Mathematics, an equaliser, or equalizer, is a set of arguments where two or more functions have equal values The name comes from preadditive categories, where one can define the equalizer of $f$ and $g$ as the kernel of the difference:

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

In integral calculus

In reference to a series, the kernel conveys the idea of the generating function. In Mathematics, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with In Mathematics a generating function is a Formal power series whose coefficients encode information about a Sequence a n 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

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

is the function k. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Mathematics, an integral transform is any transform T of the following form (Tf(u = \int_{t_1}^{t_2} K(t u\ f(t\ dt k is also called a kernel function.

In probability theory and statistics

Main article: Kernel (statistics)

A stochastic kernel is the transition function of a stochastic process (usually discrete). A kernel is a weighting function used in Non-parametric estimation techniques A stochastic process, or sometimes random process, is the counterpart to a deterministic process (or Deterministic system) in Probability theory. In a discrete time process with continuous probability distributions, it is the same thing as the kernel of the integral operator that advances the probability density function. In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability

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