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Kernel (linear algebra)

Index Kernel (linear algebra)

In mathematics, and more specifically in linear algebra and functional analysis, the kernel (also known as null space or nullspace) of a linear map between two vector spaces V and W, is the set of all elements v of V for which, where 0 denotes the zero vector in W. That is, in set-builder notation,. [1]

70 relations: Analysis of algorithms, Augmented matrix, Bareiss algorithm, Basis (linear algebra), Closed set, Cokernel, Complex number, Computer hardware, Condition number, Constant function, Continuous linear operator, Cryptography, Differential operator, Dimension (vector space), Direct product, Domain of a function, Dot product, Field (mathematics), Finite field, Flat (geometry), Floating-point arithmetic, Fredholm alternative, Free variables and bound variables, Function space, Functional analysis, Fundamental theorem of linear algebra, Gaussian elimination, Gilbert Strang, Gröbner basis, Homomorphism, Identity matrix, Image (mathematics), Inner product space, Isomorphism, Kernel (algebra), Khan Academy, LAPACK, Line (geometry), Linear algebra, Linear map, Linear span, Linear subspace, Mathematics, MIT OpenCourseWare, Modular arithmetic, Module (mathematics), Orthogonal complement, Orthogonality, Projection (linear algebra), Quotient space (linear algebra), ..., Rank (linear algebra), Rank–nullity theorem, Real coordinate space, Real number, Ring (mathematics), Round-off error, Row and column spaces, Row echelon form, Scalar (mathematics), Set-builder notation, Shift operator, System of linear equations, Topological vector space, Translation (geometry), Transpose, Vector space, Well-posed problem, Zero element, Zero matrix, Zero of a function. Expand index (20 more) »

Analysis of algorithms

In computer science, the analysis of algorithms is the determination of the computational complexity of algorithms, that is the amount of time, storage and/or other resources necessary to execute them.

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Augmented matrix

In linear algebra, an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices.

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Bareiss algorithm

In mathematics, the Bareiss algorithm, named after Erwin Bareiss, is an algorithm to calculate the determinant or the echelon form of a matrix with integer entries using only integer arithmetic; any divisions that are performed are guaranteed to be exact (there is no remainder).

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Basis (linear algebra)

In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.

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Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

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Cokernel

In mathematics, the cokernel of a linear mapping of vector spaces f: X → Y is the quotient space Y/im(f) of the codomain of f by the image of f. The dimension of the cokernel is 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), while the cokernel is a quotient object of the codomain (it maps from the codomain).

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Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

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Computer hardware

Computer hardware includes the physical parts or components of a computer, such as the central processing unit, monitor, keyboard, computer data storage, graphic card, sound card and motherboard.

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Condition number

In the field of numerical analysis, the condition number of a function with respect to an argument measures how much the output value of the function can change for a small change in the input argument.

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Constant function

In mathematics, a constant function is a function whose (output) value is the same for every input value.

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Continuous linear operator

In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.

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Cryptography

Cryptography or cryptology (from κρυπτός|translit.

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Differential operator

In mathematics, a differential operator is an operator defined as a function of the differentiation operator.

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Dimension (vector space)

In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field.

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Direct product

In mathematics, one can often define a direct product of objects already known, giving a new one.

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Domain of a function

In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

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Dot product

In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.

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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

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Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.

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Flat (geometry)

In geometry, a flat is a subset of n-dimensional space that is congruent to a Euclidean space of lower dimension.

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Floating-point arithmetic

In computing, floating-point arithmetic is arithmetic using formulaic representation of real numbers as an approximation so as to support a trade-off between range and precision.

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Fredholm alternative

In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory.

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Free variables and bound variables

In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation that specifies places in an expression where substitution may take place.

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Function space

In mathematics, a function space is a set of functions between two fixed sets.

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Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.

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Fundamental theorem of linear algebra

In mathematics, the fundamental theorem of linear algebra makes several statements regarding vector spaces.

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Gaussian elimination

In linear algebra, Gaussian elimination (also known as row reduction) is an algorithm for solving systems of linear equations.

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Gilbert Strang

William Gilbert Strang (born November 27, 1934), usually known as simply Gilbert Strang or Gil Strang, is an American mathematician, with contributions to finite element theory, the calculus of variations, wavelet analysis and linear algebra.

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Gröbner basis

In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field.

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Homomorphism

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 spaces).

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Identity matrix

In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere.

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Image (mathematics)

In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.

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Inner product space

In linear algebra, an inner product space is a vector space with an additional structure called an inner product.

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Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

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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.

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Khan Academy

Khan Academy is a non-profit educational organization created in 2006 by educator Salman Khan with a goal of creating a set of online tools that help educate students.

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LAPACK

LAPACK (Linear Algebra Package) is a standard software library for numerical linear algebra.

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Line (geometry)

The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.

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Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.

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Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

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Linear span

In linear algebra, the linear span (also called the linear hull or just span) of a set of vectors in a vector space is the intersection of all subspaces containing that set.

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Linear subspace

In linear algebra and related fields of mathematics, a linear subspace, also known as a vector subspace, or, in the older literature, a linear manifold, is a vector space that is a subset of some other (higher-dimension) vector space.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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MIT OpenCourseWare

MIT OpenCourseWare (MIT OCW) is an initiative of the Massachusetts Institute of Technology (MIT) to publish all of the educational materials from its undergraduateand graduate-level courses online, freely and openly available to anyone, anywhere.

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Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).

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Module (mathematics)

In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.

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Orthogonal complement

In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W⊥ of all vectors in V that are orthogonal to every vector in W. Informally, it is called the perp, short for perpendicular complement.

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Orthogonality

In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.

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Projection (linear algebra)

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that.

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Quotient space (linear algebra)

In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero.

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Rank (linear algebra)

In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns.

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Rank–nullity theorem

In mathematics, the rank–nullity theorem of linear algebra, in its simplest form, states that the rank and the nullity of a matrix add up to the number of columns of the matrix.

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Real coordinate space

In mathematics, real coordinate space of dimensions, written R (also written with blackboard bold) is a coordinate space that allows several (''n'') real variables to be treated as a single variable.

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Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

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Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

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Round-off error

A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value due to rounding.

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Row and column spaces

In linear algebra, the column space (also called the range or '''image''') of a matrix A is the span (set of all possible linear combinations) of its column vectors.

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Row echelon form

In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination.

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Scalar (mathematics)

A scalar is an element of a field which is used to define a vector space.

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Set-builder notation

In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements or stating the properties that its members must satisfy.

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Shift operator

In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function to its translation.

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System of linear equations

In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same set of variables.

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Topological vector space

In mathematics, a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis.

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Translation (geometry)

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction.

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Transpose

In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as AT (also written A′, Atr, tA or At).

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Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

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Well-posed problem

The mathematical term well-posed problem stems from a definition given by Jacques Hadamard.

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Zero element

In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures.

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Zero matrix

In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero.

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Zero of a function

In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x).

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Redirects here:

Kernel (functional analysis), Kernel (linear operator), Kernel (linear), Kernel (matrix), Kernel of a linear mapping, Kernel of a linear operator, Kernel of a linear transformation, Kernel of a matrix, Left null space, Left nullspace, Matrix kernel, NS(A), Null Space, Null space, Null space (matrix), Nullspace, Nullspace (linear algebra), Nullspace (linear operator), Nullspace (matrix), Right null space.

References

[1] https://en.wikipedia.org/wiki/Kernel_(linear_algebra)

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