In mathematics, a linear algebraic group is a subgroup of the group of invertible n × n {\displaystyle n\times n} matrices (under matrix multiplication)...
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orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally...
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general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because...
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mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field...
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Linear algebra is the branch of mathematics concerning linear equations such as: a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b...
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multiplicative group of R (that is, R excluding 0). These elements are "special" in that they form an algebraic subvariety of the general linear group – they...
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In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group G over a number field K, and the adele ring A...
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commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled...
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group of rational points of a reductive linear algebraic group with values in a finite field. The phrase group of Lie type does not have a widely accepted...
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In algebraic geometry, given a linear algebraic group G over a field k, a distribution on it is a linear functional k [ G ] → k {\displaystyle k[G]\to...
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linear algebraic problems like solving linear systems of equations, locating eigenvalues, or least squares optimisation. Numerical linear algebra's central...
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the unitary group is a linear algebraic group. The unitary group of a quadratic module is a generalisation of the linear algebraic group U just defined...
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and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have...
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In linear algebra, the trace of a square matrix A, denoted tr(A), is defined to be the sum of elements on the main diagonal (from the upper left to the...
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equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension n has two connected components...
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is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector...
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and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have...
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these. The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linear algebraic group, etc. is the Weyl group of the root system...
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( b ) × ( a , c ) {\displaystyle (b)\times (a,c)} in the group. For a linear algebraic group G {\displaystyle G} , a Borel subgroup is defined as a subgroup...
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Representation theory (redirect from Linear representation)
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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(2002). The so-called classical groups generalize the examples 1 and 2 above. They arise as linear algebraic groups, that is, as subgroups of GLn defined...
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In linear algebra and functional analysis, a projection is a linear transformation P {\displaystyle P} from a vector space to itself (an endomorphism)...
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chapter of linear algebra. A group of Lie type is a group closely related to the group G(k) of rational points of a reductive linear algebraic group G with...
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Algebras], translated by Jones, G. A., Springer, ISBN 978-3-540-67827-4. Affine Weyl group Finite Coxeter group Hasse diagram Linear algebraic group Nilpotent...
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even a linear algebraic group: see below). If G is a Lie group with Lie algebra g {\displaystyle {\mathfrak {g}}} , then the automorphism group of G has...
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especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced...
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may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components...
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schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily...
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empirical sciences. Algebra is the branch of mathematics that studies algebraic operations and algebraic structures. An algebraic structure is a non-empty...
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Jordan–Chevalley decomposition (redirect from Jordan decomposition in a Lie algebra)
decomposition was motivated by the applications to the theory of Lie algebras and linear algebraic groups, as described in sections below. Let K {\displaystyle K}...
41 KB (5,902 words) - 17:34, 17 March 2024