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- General_linear_group abstract "In mathematics, the 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 the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible. The group is so named because the columns 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 R (the set of real numbers) is the group of n×n invertible matrices of real numbers, and is denoted by GLn(R) or GL(n, R).More generally, the general linear group of degree n over any field F (such as the complex numbers), or a ring R (such as the ring of integers), is the set of n×n invertible matrices with entries from F (or R), again with matrix multiplication as the group operation. Typical notation is GLn(F) or GL(n, F), or simply GL(n) if the field is understood.More generally still, the general linear group of a vector space GL(V) is the abstract automorphism group, not necessarily written as matrices.The special linear group, written SL(n, F) or SLn(F), is the subgroup of GL(n, F) consisting of matrices with a determinant of 1.The group GL(n, F) and its subgroups are often called linear groups or matrix groups (the abstract group GL(V) is a linear group but not a matrix group). These groups are important in the theory of group representations, and also arise in the study of spatial symmetries and symmetries of vector spaces in general, as well as the study of polynomials. The modular group may be realised as a quotient of the special linear group SL(2, Z).If n ≥ 2, then the group GL(n, F) is not abelian.".
- General_linear_group wikiPageExternalLink GL2PAndGL33ActingOnPoints.
- General_linear_group wikiPageID "113564".
- General_linear_group wikiPageRevisionID "600763720".
- General_linear_group b "2".
- General_linear_group b "p".
- General_linear_group hasPhotoCollection General_linear_group.
- General_linear_group id "p/g043680".
- General_linear_group p "3".
- General_linear_group p "n".
- General_linear_group title "General linear group".
- General_linear_group subject Category:Abstract_algebra.
- General_linear_group subject Category:Lie_groups.
- General_linear_group subject Category:Linear_algebra.
- General_linear_group type Abstraction100002137.
- General_linear_group type Group100031264.
- General_linear_group type LieGroups.
- General_linear_group comment "In mathematics, the 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 the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible.".
- General_linear_group label "Algemene lineaire groep".
- General_linear_group label "Allgemeine lineare Gruppe".
- General_linear_group label "General linear group".
- General_linear_group label "Groupe général linéaire".
- General_linear_group label "Grupo lineal general".
- General_linear_group label "Grupo linear geral".
- General_linear_group label "Gruppo generale lineare".
- General_linear_group label "Pełna grupa liniowa".
- General_linear_group label "Полная линейная группа".
- General_linear_group label "一般线性群".
- General_linear_group sameAs Allgemeine_lineare_Gruppe.
- General_linear_group sameAs Grupo_lineal_general.
- General_linear_group sameAs Groupe_général_linéaire.
- General_linear_group sameAs Gruppo_generale_lineare.
- General_linear_group sameAs 一般線型群.
- General_linear_group sameAs Algemene_lineaire_groep.
- General_linear_group sameAs Pełna_grupa_liniowa.
- General_linear_group sameAs Grupo_linear_geral.
- General_linear_group sameAs m.0sy6r.
- General_linear_group sameAs Q524607.
- General_linear_group sameAs Q524607.
- General_linear_group sameAs General_linear_group.
- General_linear_group wasDerivedFrom General_linear_group?oldid=600763720.
- General_linear_group isPrimaryTopicOf General_linear_group.