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Determinant

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In linear algebra, the determinant is a value that can be computed from the elements of a square matrix. [1]

190 relations: Absolute value, Academic Press, Adjugate matrix, Alexandre-Théophile Vandermonde, Algebraic group, Alice's Adventures in Wonderland, American Mathematical Monthly, Analytic function, Area, Arthur Cayley, Augustin-Louis Cauchy, Étienne Bézout, Bareiss algorithm, Basis (linear algebra), Bell polynomials, Berezinian, Big O notation, Bivector, Calculus, Capelli's identity, Carl Friedrich Gauss, Carl Gustav Jacob Jacobi, Categories for the Working Mathematician, Category theory, Cauchy matrix, Cauchy–Binet formula, Chain complex, Change of variables, Character theory, Characteristic polynomial, Cholesky decomposition, Circulant matrix, Coefficient, Commutative property, Commutative ring, Complex conjugate, Computational geometry, Context of computational complexity, Coppersmith–Winograd algorithm, Cramer's rule, Crelle's Journal, Cross product, Derivative, Determinant identities, Dieudonné determinant, Differentiable function, Differential equation, Dimension, Discriminant, Dodgson condensation, ..., Dot product, Eigenvalues and eigenvectors, Elimination theory, Elwin Bruno Christoffel, Equiareal map, Euclidean space, Eugène Charles Catalan, Exterior algebra, Factorial, Faddeev–LeVerrier algorithm, Ferdinand Georg Frobenius, Field (mathematics), Fredholm determinant, Functional analysis, Functional determinant, Gabriel Cramer, Gaussian elimination, Gerolamo Cardano, Gottfried Wilhelm Leibniz, Graded ring, Griffith Baley Price, Group homomorphism, Henri Lebesgue, Hermann Hankel, Hermitian matrix, Hessian matrix, Homogeneous function, Homogeneous polynomial, Hypercube, Identity matrix, Immanant, Injective function, Integration by substitution, Inverse function theorem, Invertible matrix, Jacobi's formula, Jacobian matrix and determinant, Jacques Philippe Marie Binet, James Joseph Sylvester, James Whitbread Lee Glaisher, Joseph-Louis Lagrange, Laplace expansion, Lebesgue measure, Leibniz formula for determinants, Levi-Civita symbol, Lewis Carroll, Lie group, Linear algebra, Linear independence, Linear map, Linear span, Logarithm of a matrix, LU decomposition, Main diagonal, Manin matrix, Mathematics of Computation, Matrix determinant lemma, Matrix exponential, Matrix group, Matrix multiplication, Matrix similarity, Mercator series, Minor (linear algebra), Modular arithmetic, Multilinear map, Multiplicative group, Multivariate normal distribution, Natural transformation, Number theory, Numerical linear algebra, Open set, Orientation (vector space), Orthogonal group, Orthogonal matrix, Orthogonal transformation, Orthonormal basis, Otto Hesse, Parallelepiped, Parallelogram, Parity of a permutation, Permanent (mathematics), Permutation, Permutation matrix, Persymmetric matrix, Pfaffian, Pierre-Simon Laplace, Plane (geometry), Polynomial, Positive-definite matrix, Pseudo-determinant, QR decomposition, Quasideterminant, Rank (linear algebra), Rank–nullity theorem, Real number, Ring homomorphism, Rotation (mathematics), Rotation matrix, Row and column vectors, Rule of Sarrus, Seki Takakazu, Sequence, Serge Lang, Similarity invariance, Sine, Singular-value decomposition, Skew lines, Slater determinant, Spanning tree, Special linear group, Special unitary group, Square matrix, Standard basis, Subset, Superalgebra, Surjective function, Sylvester's criterion, Sylvester's determinant identity, Symmetric group, System of linear equations, Tangent space, Tetrahedron, The Nine Chapters on the Mathematical Art, Thomas Muir (mathematician), Trace (linear algebra), Trace class, Transpose, Triangular matrix, Trigonometric functions, Trudi, Unimodular matrix, Unit (ring theory), Unit square, Vandermonde's identity, Vector bundle, Vector space, William Spottiswoode, Wronskian, YouTube, Zero divisor. Expand index (140 more) »

Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

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Academic Press

Academic Press is an academic book publisher.

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Adjugate matrix

In linear algebra, the adjugate, classical adjoint, or adjunct of a square matrix is the transpose of its cofactor matrix.

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Alexandre-Théophile Vandermonde

Alexandre-Théophile Vandermonde (28 February 1735 – 1 January 1796) was a French mathematician, musician and chemist who worked with Bézout and Lavoisier; his name is now principally associated with determinant theory in mathematics.

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Algebraic group

In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety.

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Alice's Adventures in Wonderland

Alice's Adventures in Wonderland (commonly shortened to Alice in Wonderland) is an 1865 novel written by English author Charles Lutwidge Dodgson under the pseudonym Lewis Carroll.

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American Mathematical Monthly

The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.

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Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series.

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Area

Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane.

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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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Augustin-Louis Cauchy

Baron Augustin-Louis Cauchy FRS FRSE (21 August 178923 May 1857) was a French mathematician, engineer and physicist who made pioneering contributions to several branches of mathematics, including: mathematical analysis and continuum mechanics.

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Étienne Bézout

Étienne Bézout (31 March 1730 – 27 September 1783) was a French mathematician who was born in Nemours, Seine-et-Marne, France, and died in Avon (near Fontainebleau), France.

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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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Bell polynomials

In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions.

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Berezinian

In mathematics and theoretical physics, the Berezinian or superdeterminant is a generalization of the determinant to the case of supermatrices.

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Big O notation

Big O notation is a mathematical notation that describes the limiting behaviour of a function when the argument tends towards a particular value or infinity.

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Bivector

In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors.

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Calculus

Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

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Capelli's identity

In mathematics, Capelli's identity, named after, is an analogue of the formula det(AB).

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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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Carl Gustav Jacob Jacobi

Carl Gustav Jacob Jacobi (10 December 1804 – 18 February 1851) was a German mathematician, who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory.

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Categories for the Working Mathematician

Categories for the Working Mathematician (CWM) is a textbook in category theory written by American mathematician Saunders Mac Lane, who cofounded the subject together with Samuel Eilenberg.

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Category theory

Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).

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Cauchy matrix

In mathematics, a Cauchy matrix, named after Augustin Louis Cauchy, is an m×n matrix with elements aij in the form a_.

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Cauchy–Binet formula

In linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so that the product is well-defined and square).

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Chain complex

In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next.

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Change of variables

In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables.

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Character theory

In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix.

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Characteristic polynomial

In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots.

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Cholesky decomposition

In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced /ʃ-/) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g. Monte Carlo simulations.

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Circulant matrix

In linear algebra, a circulant matrix is a special kind of Toeplitz matrix where each row vector is rotated one element to the right relative to the preceding row vector.

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Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but may be any expression.

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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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Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.

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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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Computational geometry

Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry.

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Context of computational complexity

In computational complexity theory and analysis of algorithms, a number of metrics are defined describing the resources, such as time or space, that a machine needs to solve a particular problem.

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Coppersmith–Winograd algorithm

In linear algebra, the Coppersmith–Winograd algorithm, named after Don Coppersmith and Shmuel Winograd, was the asymptotically fastest known matrix multiplication algorithm until 2010.

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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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Crelle's Journal

Crelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik (in English: Journal for Pure and Applied Mathematics).

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Cross product

In mathematics and vector algebra, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space \left(\mathbb^3\right) and is denoted by the symbol \times.

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Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

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Determinant identities

In mathematics the determinant is an operator which has certain useful identities.

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Dieudonné determinant

In linear algebra, the Dieudonné determinant is a generalization of the determinant of a matrix to matrices over division rings and local rings.

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Differentiable function

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

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Differential equation

A differential equation is a mathematical equation that relates some function with its derivatives.

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Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

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Discriminant

In algebra, the discriminant of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them.

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Dodgson condensation

In mathematics, Dodgson condensation is a method of computing the determinants of square matrices.

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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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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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Elimination theory

In commutative algebra and algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating some variables between polynomials of several variables, in order to solve systems of polynomial equations.

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Elwin Bruno Christoffel

Elwin Bruno Christoffel (November 10, 1829 – March 15, 1900) was a German mathematician and physicist.

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Equiareal map

In differential geometry, an equiareal map is a smooth map from one surface to another that preserves the area of figures.

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Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

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Eugène Charles Catalan

Eugène Charles Catalan (30 May 1814 – 14 February 1894) was a French and Belgian mathematician who worked on continued fractions, descriptive geometry, number theory and combinatorics.

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Exterior algebra

In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs.

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Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, The value of 0! is 1, according to the convention for an empty product.

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Faddeev–LeVerrier algorithm

In mathematics (linear algebra), the Faddeev–LeVerrier algorithm is a recursive method to calculate the coefficients of the characteristic polynomial p(\lambda).

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Ferdinand Georg Frobenius

Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory.

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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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Fredholm determinant

In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant of a finite dimensional linear operator.

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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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Functional determinant

In functional analysis, a branch of mathematics, it is sometimes possible to generalize the notion of the determinant of a square matrix of finite order (representing a linear transformation from a finite-dimensional vector space to itself) to the infinite-dimensional case of a linear operator S mapping a function space V to itself.

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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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Gerolamo Cardano

Gerolamo (or Girolamo, or Geronimo) Cardano (Jérôme Cardan; Hieronymus Cardanus; 24 September 1501 – 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged from being a mathematician, physician, biologist, physicist, chemist, astrologer, astronomer, philosopher, writer, and gambler.

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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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Graded ring

In mathematics, in particular abstract algebra, a graded ring is a ring that is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_.

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Griffith Baley Price

G.

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Group homomorphism

In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h: G → H such that for all u and v in G it holds that where the group operation on the left hand side of the equation is that of G and on the right hand side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure".

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Henri Lebesgue

Henri Léon Lebesgue (June 28, 1875 – July 26, 1941) was a French mathematician most famous for his theory of integration, which was a generalization of the 17th century concept of integration—summing the area between an axis and the curve of a function defined for that axis.

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Hermann Hankel

Hermann Hankel (14 February 1839 – 29 August 1873) was a German mathematician who was born in Halle, Germany and died in Schramberg (near Tübingen), Imperial Germany.

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Hermitian matrix

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and: Hermitian matrices can be understood as the complex extension of real symmetric matrices.

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Hessian matrix

In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field.

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Homogeneous function

In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.

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Homogeneous polynomial

In mathematics, a homogeneous polynomial is a polynomial whose nonzero terms all have the same degree.

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Hypercube

In geometry, a hypercube is an ''n''-dimensional analogue of a square and a cube.

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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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Immanant

In mathematics, the immanant of a matrix was defined by Dudley E. Littlewood and Archibald Read Richardson as a generalisation of the concepts of determinant and permanent.

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Injective function

In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.

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Integration by substitution

In calculus, integration by substitution, also known as u-substitution, is a method for finding integrals.

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Inverse function theorem

In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point.

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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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Jacobi's formula

In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. If is a differentiable map from the real numbers to matrices, where is the trace of the matrix.

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Jacobian matrix and determinant

In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.

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Jacques Philippe Marie Binet

Jacques Philippe Marie Binet (2 February 1786 – 12 May 1856) was a French mathematician, physicist and astronomer born in Rennes; he died in Paris, France, in 1856.

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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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James Whitbread Lee Glaisher

James Whitbread Lee Glaisher FRS FRSE FRAS (5 November 1848, Lewisham – 7 December 1928, Cambridge), son of James Glaisher the meteorologist and Cecilia Glaisher the photographer, was a prolific English mathematician and astronomer.

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Joseph-Louis Lagrange

Joseph-Louis Lagrange (or;; born Giuseppe Lodovico Lagrangia, Encyclopædia Britannica or Giuseppe Ludovico De la Grange Tournier, Turin, 25 January 1736 – Paris, 10 April 1813; also reported as Giuseppe Luigi Lagrange or Lagrangia) was an Italian Enlightenment Era mathematician and astronomer.

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Laplace expansion

In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression for the determinant |B| of an n × n matrix B that is a weighted sum of the determinants of n sub-matrices of B, each of size (n−1) × (n−1).

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Lebesgue measure

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space.

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Leibniz formula for determinants

In algebra, the Leibniz formula, named in honor of Gottfried Leibniz, expresses the determinant of a square matrix in terms of permutations of the matrix elements.

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Levi-Civita symbol

In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol represents a collection of numbers; defined from the sign of a permutation of the natural numbers, for some positive integer.

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Lewis Carroll

Charles Lutwidge Dodgson (27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English writer, mathematician, logician, Anglican deacon, and photographer.

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Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

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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 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 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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Logarithm of a matrix

In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix.

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LU decomposition

In numerical analysis and linear algebra, LU decomposition (where "LU" stands for "lower–upper", and also called LU factorization) factors a matrix as the product of a lower triangular matrix and an upper triangular matrix.

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Main diagonal

In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, or major diagonal) of a matrix A is the collection of entries A_ where i.

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Manin matrix

In mathematics, Manin matrices, named after Yuri Manin who introduced them around 1987–88, are a class of matrices with elements in a not-necessarily commutative ring, which in a certain sense behave like matrices whose elements commute.

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Mathematics of Computation

Mathematics of Computation is a bimonthly mathematics journal focused on computational mathematics.

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Matrix determinant lemma

In mathematics, in particular linear algebra, the matrix determinant lemma computes the determinant of the sum of an invertible matrix A and the dyadic product, u vT, of a column vector u and a row vector vT.

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Matrix exponential

In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function.

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Matrix group

In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication.

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Matrix multiplication

In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring.

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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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Mercator series

In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm: In summation notation, The series converges to the natural logarithm (shifted by 1) whenever -1.

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Minor (linear algebra)

In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns.

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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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Multilinear map

In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable.

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Multiplicative group

In mathematics and group theory, the term multiplicative group refers to one of the following concepts.

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Multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution or multivariate Gaussian distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions.

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Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved.

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Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

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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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Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

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Orientation (vector space)

In mathematics, orientation is a geometric notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed.

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Orthogonal group

In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.

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Orthogonal matrix

In linear algebra, an orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e. where I is the identity matrix.

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Orthogonal transformation

In linear algebra, an orthogonal transformation is a linear transformation T: V → V on a real inner product space V, that preserves the inner product.

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Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.

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Otto Hesse

Ludwig Otto Hesse (22 April 1811 – 4 August 1874) was a German mathematician.

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Parallelepiped

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term rhomboid is also sometimes used with this meaning).

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Parallelogram

In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides.

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Parity of a permutation

In mathematics, when X is a finite set of at least two elements, the permutations of X (i.e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations.

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Permanent (mathematics)

In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant.

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Permutation

In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting.

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Permutation matrix

\pi.

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Persymmetric matrix

In mathematics, persymmetric matrix may refer to.

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Pfaffian

In mathematics, the determinant of a skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries, a polynomial with integer coefficients that only depend on the size of the matrix.

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Pierre-Simon Laplace

Pierre-Simon, marquis de Laplace (23 March 1749 – 5 March 1827) was a French scholar whose work was important to the development of mathematics, statistics, physics and astronomy.

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Plane (geometry)

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

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Polynomial

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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Positive-definite matrix

In linear algebra, a symmetric real matrix M is said to be positive definite if the scalar z^Mz is strictly positive for every non-zero column vector z of n real numbers.

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Pseudo-determinant

In linear algebra and statistics, the pseudo-determinant is the product of all non-zero eigenvalues of a square matrix.

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QR decomposition

In linear algebra, a QR decomposition (also called a QR factorization) of a matrix is a decomposition of a matrix A into a product A.

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Quasideterminant

In mathematics, the quasideterminant is a replacement for the determinant for matrices with noncommutative entries.

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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 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 homomorphism

In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the structure.

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Rotation (mathematics)

Rotation in mathematics is a concept originating in geometry.

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Rotation matrix

In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space.

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Row and column vectors

In linear algebra, a column vector or column matrix is an m × 1 matrix, that is, a matrix consisting of a single column of m elements, Similarly, a row vector or row matrix is a 1 × m matrix, that is, a matrix consisting of a single row of m elements Throughout, boldface is used for the row and column vectors.

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Rule of Sarrus

Sarrus' rule or Sarrus' scheme is a method and a memorization scheme to compute the determinant of a 3×3 matrix.

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Seki Takakazu

, also known as,Selin, was a Japanese mathematician and author of the Edo period.

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Sequence

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

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Serge Lang

Serge Lang (May 19, 1927 – September 12, 2005) was a French-born American mathematician and activist.

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Similarity invariance

In linear algebra, similarity invariance is a property exhibited by a function whose value is unchanged under similarities of its domain.

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Sine

In mathematics, the sine is a trigonometric function of an angle.

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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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Skew lines

In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel.

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Slater determinant

In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system that satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electrons (or other fermions).

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Spanning tree

In the mathematical field of graph theory, a spanning tree T of an undirected graph G is a subgraph that is a tree which includes all of the vertices of G, with minimum possible number of edges.

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Special linear group

In mathematics, the special linear group of degree n over a field F is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion.

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Special unitary group

In mathematics, the special unitary group of degree, denoted, is the Lie group of unitary matrices with determinant 1.

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Square matrix

In mathematics, a square matrix is a matrix with the same number of rows and columns.

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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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Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

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Superalgebra

In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra.

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Surjective function

In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).

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Sylvester's criterion

In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite.

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Sylvester's determinant identity

In matrix theory, Sylvester's determinant identity is an identity useful for evaluating certain types of determinants.

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Symmetric group

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions.

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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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Tangent space

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the one point from the other.

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Tetrahedron

In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners.

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The Nine Chapters on the Mathematical Art

The Nine Chapters on the Mathematical Art is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE.

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Thomas Muir (mathematician)

Sir Thomas Muir FRS FRSE CMG LLD (25 August 1844 – 21 March 1934) was a Scottish mathematician, remembered as an authority on determinants.

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Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e., where aii denotes the entry on the ith row and ith column of A. The trace of a matrix is the sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis.

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Trace class

In mathematics, a trace class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis.

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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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Triangular matrix

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

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Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.

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Trudi

Trudi may refer to.

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Unimodular matrix

In mathematics, a unimodular matrix M is a square integer matrix having determinant +1 or −1.

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Unit (ring theory)

In mathematics, an invertible element or a unit in a (unital) ring is any element that has an inverse element in the multiplicative monoid of, i.e. an element such that The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation.

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Unit square

In mathematics, a unit square is a square whose sides have length.

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Vandermonde's identity

In combinatorics, Vandermonde's identity (or Vandermonde's convolution) is the following identity for binomial coefficients: for any nonnegative integers r, m, n. The identity is named after Alexandre-Théophile Vandermonde (1772), although it was already known in 1303 by the Chinese mathematician Zhu Shijie (Chu Shi-Chieh).

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Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X.

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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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William Spottiswoode

William H. Spottiswoode (11 January 1825 – 27 June 1883) was an English mathematician and physicist.

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Wronskian

In mathematics, the Wronskian (or Wrońskian) is a determinant introduced by and named by.

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YouTube

YouTube is an American video-sharing website headquartered in San Bruno, California.

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Zero divisor

In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero such that, or equivalently if the map from to that sends to is not injective.

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References

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

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