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Linear algebra

Index 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. [1]

137 relations: Abelian group, Abstract algebra, Additive inverse, Algebra over a field, Algebraic geometry, Algebraically closed field, Algorithm, Angular momentum, Arthur Cayley, Associative property, Axiom, Axiom of choice, Basis (linear algebra), Bijection, Bilinear form, Binary operation, Cardinality, Carl Friedrich Gauss, Cauchy–Schwarz inequality, Commutative property, Complex conjugate, Complex number, Computer vision, Coordinate system, Cramer's rule, Definite quadratic form, Determinant, Diagonal matrix, Diagonalizable matrix, Differential calculus, Dimension (vector space), Dimension theorem for vector spaces, Dirichlet conditions, Distributive property, Dual space, Eigenvalues and eigenvectors, Element (mathematics), Energy, Engineering, Euclidean vector, Field (mathematics), Finitely generated module, Fourier series, Free module, Function (mathematics), Functional analysis, Fundamental matrix (computer vision), Gabriel Cramer, Gaussian elimination, Geodesy, ..., Geometry, Giuseppe Peano, Gottfried Wilhelm Leibniz, Graduate Texts in Mathematics, Gram–Schmidt process, Hamiltonian (quantum mechanics), Hüseyin Tevfik Pasha, Hermann Grassmann, Hermitian adjoint, Homogeneous coordinates, Identity element, Identity matrix, Inner product space, Inverse element, Invertible matrix, Isomorphism, James Joseph Sylvester, Kernel (linear algebra), Line (geometry), Linear combination, Linear equation, Linear equation over a ring, Linear form, Linear independence, Linear least squares (mathematics), Linear map, Linear programming, Linear regression, Linear span, Linear subspace, Linearity, Lipschitz continuity, List of linear algebra topics, Logical equivalence, Lp space, Map (mathematics), Martha Siegel, Mathematical analysis, Mathematical model, Mathematics, MathWorld, Matrix (mathematics), Matrix similarity, Module (mathematics), Momentum, Multilinear algebra, Nonlinear system, Normal matrix, Numerical linear algebra, Observable, Origin (mathematics), Partial differential equation, Plane (geometry), Polynomial, Polynomial ring, Potential energy, Quantum mechanics, Range (mathematics), Rational number, Real number, Representation theory, Rotation (mathematics), Scalar multiplication, School Mathematics Study Group, Schrödinger equation, Secondary school, Sequence, Set (mathematics), Simplex algorithm, Singular-value decomposition, Special relativity, Springer Science+Business Media, Standard basis, Statistics, System of linear equations, System of polynomial equations, Tensor product, Transformation matrix, Triangular matrix, Undergraduate Texts in Mathematics, Vector space, Wave function, Well-defined, Well-order, Zero element, 1, 2 × 2 real matrices. Expand index (87 more) »

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

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Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

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Additive inverse

In mathematics, the additive inverse of a number is the number that, when added to, yields zero.

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Algebra over a field

In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.

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Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.

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Algebraically closed field

In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.

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In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.

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Angular momentum

In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum.

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Arthur Cayley

Arthur Cayley F.R.S. (16 August 1821 – 26 January 1895) was a British mathematician.

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Associative property

In mathematics, the associative property is a property of some binary operations.

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An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

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Axiom of choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.

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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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In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

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Bilinear form

In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map, where K is the field of scalars.

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Binary operation

In mathematics, a binary operation on a set is a calculation that combines two elements of the set (called operands) to produce another element of the set.

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In mathematics, the cardinality of a set is a measure of the "number of elements of the set".

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Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

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Cauchy–Schwarz inequality

In mathematics, the Cauchy–Schwarz inequality, also known as the Cauchy–Bunyakovsky–Schwarz inequality, is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, vector algebra and other areas.

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Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

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Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.

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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 vision

Computer vision is a field that deals with how computers can be made for gaining high-level understanding from digital images or videos.

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Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.

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Cramer's rule

In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution.

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Definite quadratic form

In mathematics, a definite quadratic form is a quadratic form over some real vector space that has the same sign (always positive or always negative) for every nonzero vector of.

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In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

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Diagonal matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.

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Diagonalizable matrix

In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P−1AP is a diagonal matrix.

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Differential calculus

In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change.

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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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Dimension theorem for vector spaces

In mathematics, the dimension theorem for vector spaces states that all bases of a vector space have equally many elements.

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Dirichlet conditions

In mathematics, the Dirichlet conditions are sufficient conditions for a real-valued, periodic function f to be equal to the sum of its Fourier series at each point where f is continuous.

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Distributive property

In abstract algebra and formal logic, the distributive property of binary operations generalizes the distributive law from boolean algebra and elementary algebra.

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Dual space

In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

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Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.

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Element (mathematics)

In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.

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In physics, energy is the quantitative property that must be transferred to an object in order to perform work on, or to heat, the object.

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Engineering is the creative application of science, mathematical methods, and empirical evidence to the innovation, design, construction, operation and maintenance of structures, machines, materials, devices, systems, processes, and organizations.

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Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction.

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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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Finitely generated module

In mathematics, a finitely generated module is a module that has a finite generating set.

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Fourier series

In mathematics, a Fourier series is a way to represent a function as the sum of simple sine waves.

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Free module

In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements.

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Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

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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 matrix (computer vision)

In computer vision, the fundamental matrix \mathbf is a 3×3 matrix which relates corresponding points in stereo images.

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Gabriel Cramer

Gabriel Cramer (31 July 1704 – 4 January 1752) was a Genevan mathematician.

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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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Geodesy, also known as geodetics, is the earth science of accurately measuring and understanding three of Earth's fundamental properties: its geometric shape, orientation in space, and gravitational field.

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Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

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Giuseppe Peano

Giuseppe Peano (27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist.

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Gottfried Wilhelm Leibniz

Gottfried Wilhelm (von) Leibniz (or; Leibnitz; – 14 November 1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy.

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Graduate Texts in Mathematics

Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag.

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Gram–Schmidt process

In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalising a set of vectors in an inner product space, most commonly the Euclidean space Rn equipped with the standard inner product.

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Hamiltonian (quantum mechanics)

In quantum mechanics, a Hamiltonian is an operator corresponding to the total energy of the system in most of the cases.

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Hüseyin Tevfik Pasha

Hüseyin Tevfik Pasha (1832–1901) is credited as the inventor of Linear Algebra.

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Hermann Grassmann

Hermann Günther Grassmann (Graßmann; April 15, 1809 – September 26, 1877) was a German polymath, known in his day as a linguist and now also as a mathematician.

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Hermitian adjoint

In mathematics, specifically in functional analysis, each bounded linear operator on a complex Hilbert space has a corresponding adjoint operator.

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Homogeneous coordinates

In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry.

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Identity element

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, which leaves other elements unchanged when combined with them.

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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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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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Inverse element

In abstract algebra, the idea of an inverse element generalises concepts of a negation (sign reversal) in relation to addition, and a reciprocal in relation to multiplication.

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Invertible matrix

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate) if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.

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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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James Joseph Sylvester

James Joseph Sylvester FRS (3 September 1814 – 15 March 1897) was an English mathematician.

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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,.

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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 combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

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Linear equation

In mathematics, a linear equation is an equation that may be put in the form where x_1, \ldots, x_n are the variables or unknowns, and c, a_1, \ldots, a_n are coefficients, which are often real numbers, but may be parameters, or even any expression that does not contain the unknowns.

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Linear equation over a ring

In algebra, linear equations and systems of linear equations over a field are widely studied.

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Linear form

In linear algebra, a linear functional or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars.

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Linear independence

In the theory of vector spaces, a set of vectors is said to be if one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be.

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Linear least squares (mathematics)

In statistics and mathematics, linear least squares is an approach to fitting a mathematical or statistical model to data in cases where the idealized value provided by the model for any data point is expressed linearly in terms of the unknown parameters of the model.

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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 programming

Linear programming (LP, also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.

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Linear regression

In statistics, linear regression is a linear approach to modelling the relationship between a scalar response (or dependent variable) and one or more explanatory variables (or independent variables).

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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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Linearity is the property of a mathematical relationship or function which means that it can be graphically represented as a straight line.

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Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions.

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List of linear algebra topics

This is a list of linear algebra topics.

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Logical equivalence

In logic, statements p and q are logically equivalent if they have the same logical content.

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Lp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces.

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Map (mathematics)

In mathematics, the term mapping, sometimes shortened to map, refers to either a function, often with some sort of special structure, or a morphism in category theory, which generalizes the idea of a function.

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Martha Siegel

Martha Jochnowitz Siegel is an American applied mathematician, probability theorist and mathematics educator who served as the editor of Mathematics Magazine from 1991 to 1996.

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Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

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Mathematical model

A mathematical model is a description of a system using mathematical concepts and language.

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

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MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein.

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Matrix (mathematics)

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

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Matrix similarity

In linear algebra, two n-by-n matrices and are called similar if for some invertible n-by-n matrix.

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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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In Newtonian mechanics, linear momentum, translational momentum, or simply momentum (pl. momenta) is the product of the mass and velocity of an object.

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Multilinear algebra

In mathematics, multilinear algebra extends the methods of linear algebra.

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Nonlinear system

In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input.

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Normal matrix

In mathematics, a complex square matrix is normal if where is the conjugate transpose of.

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Numerical linear algebra

Numerical linear algebra is the study of algorithms for performing linear algebra computations, most notably matrix operations, on computers.

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In physics, an observable is a dynamic variable that can be measured.

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Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space.

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Partial differential equation

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.

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Plane (geometry)

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.

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In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

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Polynomial ring

In mathematics, especially in the field of abstract algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.

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Potential energy

In physics, potential energy is the energy possessed by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.

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Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

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Range (mathematics)

In mathematics, and more specifically in naive set theory, the range of a function refers to either the codomain or the image of the function, depending upon usage.

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Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

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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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Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

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Rotation (mathematics)

Rotation in mathematics is a concept originating in geometry.

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Scalar multiplication

In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra).

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School Mathematics Study Group

The School Mathematics Study Group (SMSG) was an American academic think tank focused on the subject of reform in mathematics education.

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Schrödinger equation

In quantum mechanics, the Schrödinger equation is a mathematical equation that describes the changes over time of a physical system in which quantum effects, such as wave–particle duality, are significant.

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Secondary school

A secondary school is both an organization that provides secondary education and the building where this takes place.

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In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

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Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

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Simplex algorithm

In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming.

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Singular-value decomposition

In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix.

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Special relativity

In physics, special relativity (SR, also known as the special theory of relativity or STR) is the generally accepted and experimentally well-confirmed physical theory regarding the relationship between space and time.

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Springer Science+Business Media

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

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Standard basis

In mathematics, the standard basis (also called natural basis) for a Euclidean space is the set of unit vectors pointing in the direction of the axes of a Cartesian coordinate system.

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Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data.

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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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System of polynomial equations

A system of polynomial equations is a set of simultaneous equations f1.

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Tensor product

In mathematics, the tensor product of two vector spaces and (over the same field) is itself a vector space, together with an operation of bilinear composition, denoted by, from ordered pairs in the Cartesian product into, in a way that generalizes the outer product.

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Transformation matrix

In linear algebra, linear transformations can be represented by matrices.

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Triangular matrix

In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix.

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Undergraduate Texts in Mathematics

Undergraduate Texts in Mathematics (UTM) is a series of undergraduate-level textbooks in mathematics published by Springer-Verlag.

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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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Wave function

A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system.

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In mathematics, an expression is called well-defined or unambiguous if its definition assigns it a unique interpretation or value.

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In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.

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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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1 (one, also called unit, unity, and (multiplicative) identity) is a number, numeral, and glyph.

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2 × 2 real matrices

In mathematics, the associative algebra of real matrices is denoted by M(2, R).

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[1] https://en.wikipedia.org/wiki/Linear_algebra

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