Editing General linear group (section)

{{Short description|Group of ''n'' × ''n'' invertible matrices}}
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{{Use American English|date=January 2019}}
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In [[mathematics]], the '''general linear group''' of degree <math>n</math> is the set of <math>n\times n</math> [[invertible matrix|invertible matrices]], together with the operation of ordinary [[matrix multiplication]]. This forms a [[group (mathematics)|group]], because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with the identity matrix as the identity element of the group. The group is so named because the columns (and also the rows) of an invertible matrix are [[linearly independent]], hence the vectors/points they define are in [[general linear position]], and matrices in the general linear group take points in general linear position to points in general linear position.

To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over <math>\R</math> (the set of [[real numbers]]) is the group of <math>n\times n</math> invertible matrices of real numbers, and is denoted by <math>\operatorname{GL}_n(\R)</math> or <math>\operatorname{GL}(n,\R)</math>.

More generally, the general linear group of degree <math>n</math> over any [[field (mathematics)|field]] <math>F</math> (such as the [[complex number]]s), or a [[ring (mathematics)|ring]] <math>R</math> (such as the ring of [[integer]]s), is the set of <math>n\times n</math> invertible matrices with entries from <math>F</math> (or <math>R</math>), again with matrix multiplication as the group operation.<ref name="ring">Here rings are assumed to be [[Ring (mathematics)#Notes on the definition|associative and unital]].</ref> Typical notation is <math>\operatorname{GL}(n,F)</math> or <math>\operatorname{GL}_n(F)</math>, or simply <math>\operatorname{GL}(n)</math> if the field is understood.

More generally still, the [[#General linear group of a vector space|general linear group of a vector space]] <math>\operatorname{GL}(V)</math> is the [[automorphism group]], not necessarily written as matrices.

The '''[[#Special linear group|special linear group]]''', written <math>\operatorname{SL}(n,F)</math> or <math>\operatorname{SL}_n(F)</math>, is the [[subgroup]] of <math>\operatorname{GL}(n,F)</math> consisting of matrices with a [[determinant]] of 1.

The group <math>\operatorname{GL}(n,F)</math> and its [[subgroup]]s are often called '''linear groups''' or '''matrix groups''' (the automorphism group <math>\operatorname{GL}(V)</math> is a linear group but not a matrix group). These groups are important in the theory of [[group representation]]s, and also arise in the study of spatial [[symmetry|symmetries]] and symmetries of [[vector space]]s in general, as well as the study of [[polynomials]]. The [[modular group]] may be realised as a quotient of the special linear group <math>\operatorname{SL}(2,\Z)</math>.

If <math>n\geq 2</math>, then the group <math>\operatorname{GL}(n,F)</math> is not [[abelian group|abelian]].