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Elements taken to zero by a homomorphism
In algebra, the kernel of a homomorphism is the relation describing how elements in the domain of the homomorphism become related in the image. A homomorphism
Kernel_(algebra)
Vectors mapped to 0 by a linear map
Numerical Linear Algebra, SIAM, ISBN 978-0-89871-361-9. Wikibooks has a book on the topic of: Linear Algebra/Null Spaces "Kernel of a matrix", Encyclopedia
Kernel_(linear_algebra)
Generalization of the kernel of a homomorphism
mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra. Intuitively
Kernel_(category_theory)
Topics referred to by the same term
system Kernel (algebra), a general concept that includes: Kernel (linear algebra) or null space, a set of vectors mapped to the zero vector Kernel (category
Kernel
Routines for performing common linear algebra operations
would have three nested loops. Linear algebra programs have many common low-level operations (the so-called "kernel" operations, not related to operating
Basic Linear Algebra Subprograms
Basic_Linear_Algebra_Subprograms
Class of algorithms for pattern analysis
In machine learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). These
Kernel_method
Branch of mathematics
Linear algebra is the branch of mathematics concerning linear equations such as a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b
Linear_algebra
Mathematical concept
inequivalent irreducible representations with kernel K(H) or with kernel {0}. Suppose A is a finite-dimensional C*-algebra. It is known A is isomorphic to a finite
Spectrum_of_a_C*-algebra
Algebraic variety with a group structure
a group homomorphism. Its kernel is an algebraic subgroup of G {\displaystyle \mathrm {G} } , and its image is an algebraic subgroup of G ′ {\displaystyle
Algebraic_group
In mathematics, a Malcev algebra (or Maltsev algebra or Moufang–Lie algebra) over a field is a nonassociative algebra that is antisymmetric, so that x
Malcev_algebra
Algebraic structure used in analysis
isomorphism of Lie algebras is a bijective homomorphism. As with normal subgroups in groups, ideals in Lie algebras are precisely the kernels of homomorphisms
Lie_algebra
Algebra based on a vector space with a quadratic form
mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure
Clifford_algebra
Group of mathematical theorems
modules, Lie algebras, and other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences
Isomorphism_theorems
Reduction of a ring by one of its ideals
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite
Quotient_ring
Concept in probability theory
In probability theory, a Markov kernel (also known as a stochastic kernel or probability kernel) is a map that in the general theory of Markov processes
Markov_kernel
Optimized math routines developed by Intel
Intel oneAPI Math Kernel Library (Intel oneMKL), formerly known as Intel Math Kernel Library, is a library of optimized math routines for science, engineering
Math_Kernel_Library
Idempotent linear transformation from a vector space to itself
In linear algebra and functional analysis, a projection is a linear transformation P {\displaystyle P} from a vector space to itself (an endomorphism)
Projection_(linear_algebra)
In mathematics, a restricted Lie algebra (or p-Lie algebra) is a Lie algebra over a field of characteristic p>0 together with an additional "pth power"
Restricted_Lie_algebra
Branch of mathematics that studies algebraic structures
preserving maps called homomorphisms are vital in the study of algebraic objects. Homomorphisms Kernels and cokernels Image and coimage Epimorphisms and monomorphisms
List of abstract algebra topics
List_of_abstract_algebra_topics
Set of arguments where two or more functions have the same value
context of abstract algebra: The difference kernel of f and g is simply the kernel of the difference f − g. Furthermore, the kernel of a single function
Equaliser_(mathematics)
library of algorithms, efficient data structures, and a fast kernel. These computer algebra systems are sometimes combined with "front end" programs that
List of computer algebra systems
List_of_computer_algebra_systems
Equivalence relation expressing that two elements have the same image under a function
In set theory, the kernel of a function f {\displaystyle f} (or equivalence kernel) may be taken to be either the equivalence relation on the function's
Kernel_(set_theory)
Tool in homological algebra
In homological algebra, the mapping cone is a construction on a map of chain complexes inspired by the analogous construction in topology. In the theory
Mapping cone (homological algebra)
Mapping_cone_(homological_algebra)
Scientific area at the interface between computer science and mathematics
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is the study and development of algorithms
Computer_algebra
Algebraic structure with addition and multiplication
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations typically called addition and multiplication and denoted
Ring_(mathematics)
Ring that is also a vector space or a module
In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center
Associative_algebra
Mathematical structure in abstract algebra
mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra; read as "star-algebra") is a mathematical structure consisting of
*-algebra
Branch of mathematics
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins
Homological_algebra
Generalization of vector spaces from fields to rings
central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. In a vector space, the
Module_(mathematics)
Generalization of topological interior
In functional analysis, a branch of mathematics, the algebraic interior or radial kernel of a subset of a vector space is a refinement of the concept
Algebraic_interior
Mathematical representation in functional analysis
representing commutative Banach algebras as algebras of continuous functions; the fact that for commutative C*-algebras, this representation is an isometric
Gelfand_representation
Particular kind of algebraic structure
trivial kernel. An important example of such an algebra is a commutative C*-algebra. In fact, when A {\displaystyle A} is a commutative unital C*-algebra, the
Banach_algebra
Type of group in mathematics
matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension
Orthogonal_group
Construction in algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a (unital associative) algebra and a (counital coassociative)
Hopf_algebra
In linear algebra, relation between 3 dimensions
The rank–nullity theorem is a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity
Rank–nullity_theorem
Algebra associated to any vector space
In mathematics, the exterior algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle
Exterior_algebra
Integral expressing the amount of overlap of one function as it is shifted over another
have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters
Convolution
Branch of algebra that studies commutative rings
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both
Commutative_algebra
Algebraic structure in linear algebra
also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrices, which allows computing in vector
Vector_space
Mathematical function, in linear algebra
In mathematics, and more specifically in linear algebra, a linear map, linear mapping, or linear operator is a particular kind of function between vector
Linear_map
Mapping involving integration between function spaces
kernels correspond to self-adjoint operators. There are many classes of problems that are difficult to solve—or at least quite unwieldy algebraically—in
Integral_transform
Algebraic structure used in logic
In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with
Heyting_algebra
Group of unitary complex matrices with determinant of 1
rotation group SO(3) whose kernel is {+I, −I}. Since the quaternions can be identified as the even subalgebra of the Clifford algebra Cl(3), SU(2) is identical
Special_unitary_group
Dimension of the column space of a matrix
In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. This corresponds to the maximal
Rank_(linear_algebra)
Structure-preserving function between two rings
of S. The kernel of f, defined as ker(f) = {a in R | f(a) = 0S}, is a two-sided ideal in R. Every two-sided ideal in a ring R is the kernel of some ring
Ring_homomorphism
Concept in mathematics
enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal
Universal_enveloping_algebra
248-dimensional exceptional simple Lie group
several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding
E8_(mathematics)
Commutative group (mathematics)
abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally
Abelian_group
Commutative ring with no zero divisors other than zero
unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields An integral domain is a nonzero commutative ring in which
Integral_domain
R-algebra. Such an algebra comes equipped with a ring homomorphism to R. The kernel of this homomorphism is the augmentation ideal of the algebra. The
Augmentation_ideal
Equivalence relation in algebra
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector
Congruence_relation
Quotient space of a codomain of a linear map by the map's image
called the corank of f. Cokernels are dual to the kernels of category theory, hence the name: the kernel is a subobject of the domain (it maps to the domain)
Cokernel
Branch of functional analysis
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with
Operator_algebra
Index of articles associated with the same name
describe an algebraic structure which in some sense cannot be divided by a smaller structure of the same type. Put another way, an algebraic structure is
Simple_(abstract_algebra)
Submodule of a mathematical ring
ideal as its kernel. Conversely, the kernel of a ring homomorphism is a two-sided ideal. Therefore, the two-sided ideals are exactly the kernels of ring homomorphisms
Ideal_(ring_theory)
A\to k} , typically denoted by ε. An algebra together with an augmentation is called an augmented algebra. The kernel of the augmentation is a two-sided
Augmentation_(algebra)
Group in group theory and physics
constants forms a Lie algebra under the Poisson bracket. This Lie algebra is a one-dimensional central extension of the commutative Lie algebra R 2 n {\displaystyle
Heisenberg_group
Mathematical function between groups that preserves multiplication structure
directly gives that there is a unique element in the kernel, and, conversely, a unique element in the kernel gives injectivity: h ( g 1 ) = h ( g 2 ) ⇔ h (
Group_homomorphism
Algebraic structure
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more
Polynomial_ring
Branch of number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations
Algebraic_number_theory
Getting better now but I'm still waiting for the time
special train algebras, gametic algebras, Bernstein algebras, copular algebras, zygotic algebras, and baric algebras (also called weighted algebra). The study
Genetic_algebra
Subgroup invariant under conjugation
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation
Normal_subgroup
Mathematical concept
kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel
Poisson_kernel
Algebra over a field where binary multiplication is not necessarily associative
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative
Non-associative_algebra
In algebra, the principal factor of a J {\displaystyle {\mathcal {J}}} -class J of a semigroup S is equal to J if J is the kernel of S, and to J ∪ { 0
Principal_factor
Concept in mathematics
central simple algebra A over k determines a reductive group G = SL(1,A), the kernel of the reduced norm on the group of units A* (as an algebraic group over
Reductive_group
Set without nontrivial polynomial equalities
In abstract algebra, a subset S {\displaystyle S} of a field L {\displaystyle L} is algebraically independent over a subfield K {\displaystyle K} if the
Algebraic_independence
Category with direct sums and certain types of kernels and cokernels
is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical
Abelian_category
Lie group of complex numbers of unit modulus; topologically a circle
can also be viewed as an algebraic torus. The field norm from ℂ to ℝ is Nℂ/ℝ(z)=zz¯=|z|2, and the circle group is the kernel of the norm homomorphism
Circle_group
Most widely known generalized inverse of a matrix
In mathematics, and in particular linear algebra, the Moore–Penrose inverse A + {\displaystyle A^{+}} of a matrix A {\displaystyle A} , often called
Moore–Penrose_inverse
Net in a normed algebra
= meλ. Mollifier Nascent delta function Summability kernel Dales, H. Garth (2000). Banach Algebras and Automatic Continuity. Clarendon Press (London Mathematical
Approximate_identity
Mathematical method in functional analysis
algebra, given by modular automorphisms. The kernel of δ is an important invariant of the algebra. For simplicity assume that the von Neumann algebra
Tomita–Takesaki_theory
Mathematical function
mathematics of probability, a transition kernel or kernel is a function in mathematics that has different applications. Kernels can for example be used to define
Transition_kernel
Branch of algebra
In algebra, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those
Ring_theory
Mathematical term
the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is G L ( n , R ) {\displaystyle
Adjoint_representation
Specific algebraic group
{\displaystyle F} -rank is zero. An isogeny between algebraic groups is a surjective morphism with finite kernel; two tori are said to be isogenous if there exists
Algebraic_torus
Polynomial associated with a matrix
give ever larger kernels, but further increasing the exponent beyond m will just give the same kernel. If the field F is not algebraically closed, then the
Minimal polynomial (linear algebra)
Minimal_polynomial_(linear_algebra)
*-algebra of bounded operators on a Hilbert space
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology
Von_Neumann_algebra
In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations (specifically in the representation
Weyl's theorem on complete reducibility
Weyl's_theorem_on_complete_reducibility
Simple Lie group; the automorphism group of the octonions
form and a split real form), their Lie algebras g 2 , {\displaystyle {\mathfrak {g}}_{2},} as well as some algebraic groups. They are the smallest of the
G2_(mathematics)
Finite extension of the rationals
In mathematics, an algebraic number field (or simply number field) is an extension field K {\displaystyle K} of the field of rational numbers Q {\displaystyle
Algebraic_number_field
Free object in the category of associative algebras
In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since
Free_algebra
Structure-preserving map between two algebraic structures of the same type
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector
Homomorphism
Sequence of homomorphisms such that each kernel equals the preceding image
kernel of the next. The sequence of groups and homomorphisms may be either finite or infinite. A similar definition can be made for other algebraic structures
Exact_sequence
Tensor product of algebras over a field; itself another algebra
the tensor product of two algebras over a commutative ring R is also an R-algebra. This gives the tensor product of algebras. When the ring is a field
Tensor_product_of_algebras
Computer system for solving algebra problems
a computer algebra system designed to solve problems in algebra, number theory, geometry and combinatorics. It is named after the algebraic structure magma
Magma (computer algebra system)
Magma_(computer_algebra_system)
Zero divisors in a module
R_{S}/R)} is the kernel of the localisation map of M. The symbol Tor denoting the functors reflects this relation with the algebraic torsion. This same
Torsion_(algebra)
Algebraic construction
In mathematics, the ring of integers of an algebraic number field K {\displaystyle K} (also sometimes called the number ring corresponding to number field
Ring_of_integers
Open-source software
Science, ISCAS. OpenBLAS adds optimized implementations of linear algebra kernels for several processor architectures, including Intel Sandy Bridge and
OpenBLAS
78-dimensional exceptional simple Lie group
(Gal(k), Z/2Z) with kernel H1(k, E6,ad). Over the field of real numbers, the real component of the identity of these algebraically twisted forms of E6
E6_(mathematics)
Algebraic ring without a multiplicative identity
In abstract algebra, a rng (pronounced "rung" /rʌŋ/) or non-unital ring or pseudo-ring is an alternative name for a ring that does not assume the existence
Rng_(algebra)
Mathematical category whose hom sets form Abelian groups
ring, this notion of kernel coincides with the ordinary notion of a kernel of a homomorphism, if one identifies the ordinary kernel K {\displaystyle K}
Preadditive_category
Algebra describing 2D conformal symmetry
mathematics, the Virasoro algebra is a complex Lie algebra and the unique nontrivial central extension of the Witt algebra. It is widely used in two-dimensional
Virasoro_algebra
Group that is also a differentiable manifold with group operations that are smooth
circle. Its Lie algebra is (more or less) the Witt algebra, whose central extension the Virasoro algebra (see Virasoro algebra from Witt algebra for a derivation
Lie_group
Category whose objects are rings and whose morphisms are ring homomorphisms
congruence relation is precisely the (ring-theoretic) kernel of f. Note that category-theoretic kernels do not make sense in Ring since there are no zero
Category_of_rings
Mathematical theorem
in the reproducing kernel Hilbert space theory where it characterizes a symmetric positive-definite kernel as a reproducing kernel. To explain Mercer's
Mercer's_theorem
Vector space consisting of affine subsets
In linear algebra, the quotient of a vector space V {\displaystyle V} by a subspace U {\displaystyle U} is a vector space obtained by "collapsing" U {\displaystyle
Quotient space (linear algebra)
Quotient_space_(linear_algebra)
Result of partitioning the elements of an algebraic structure using a congruence relation
a quotient algebra is the result of partitioning the elements of an algebraic structure using a congruence relation. Quotient algebras are also called
Quotient_(universal_algebra)
Group of flat spacetime symmetries
{Spin} (1,3)} . The Poincaré algebra is the Lie algebra of the Poincaré group. It is a Lie algebra extension of the Lie algebra of the Lorentz group. More
Poincaré_group
Linear function satisfying a support condition
construction for a Hopf algebra. Let A be a Hopf algebra. The finite dual of A is the space of linear functionals on A with kernels containing left ideals
Distribution on a linear algebraic group
Distribution_on_a_linear_algebraic_group
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