In mathematics, a linear algebraic group is a subgroup of the group of invertible n × n {\displaystyle n\times n} matrices (under matrix multiplication)...

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...

orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally...

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...

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...

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...

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...

linear algebraic problems like solving linear systems of equations, locating eigenvalues, or least squares optimisation. Numerical linear algebra's central...

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...

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...

commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled...

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...

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...

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...

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...

is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector...

( 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...

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...

(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...

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...

In linear algebra and functional analysis, a projection is a linear transformation P {\displaystyle P} from a vector space to itself (an endomorphism)...

abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures...

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...

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...

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...

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}...

empirical sciences. Algebra is the branch of mathematics that studies algebraic operations and algebraic structures. An algebraic structure is a non-empty...

The Poincaré algebra is the Lie algebra of the Poincaré group. It is a Lie algebra extension of the Lie algebra of the Lorentz group. More specifically...

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...

of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of the general linear group GL(n,...