general linear group GL(n,R)

E593509
Lie group matrix group real algebraic group topological group

The general linear group GL(n,ℝ) is the Lie group consisting of all invertible n×n real matrices under matrix multiplication, fundamental in linear algebra and differential geometry.

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Observed surface forms (1)

Surface form Occurrences
GL(n,ℝ) 0

Statements (48)

Predicate Object
instanceOf Lie group ⓘ
matrix group ⓘ
real algebraic group ⓘ
topological group ⓘ
actsFaithfullyOn ℝ^n ⓘ
actsOn ℝ^n by left multiplication ⓘ
actsTransitivelyOn set of ordered bases of ℝ^n ⓘ
containsSubgroup O(n) ⓘ
SL(n,ℝ) NERFINISHED ⓘ
SO(n) NERFINISHED ⓘ
definedOver ℝ ⓘ
hasConnectedComponentOfIdentity GL^+(n,ℝ) ⓘ
hasDeformationRetractionTo O(n) ⓘ
hasDeterminantMapTo ℝ\{0} ⓘ
hasDeterminantSignHomomorphismTo {−1,1} ⓘ
hasDimension n^2 ⓘ
hasIdentityElement I_n ⓘ
hasInverseOperation matrix inversion ⓘ
hasLieAlgebra gl(n,ℝ) NERFINISHED ⓘ
hasMaximalCompactSubgroup O(n) ⓘ
hasNeutralElement I_n ⓘ
hasOperation matrix multiplication ⓘ
hasRealAnalyticStructure true ⓘ
hasTwoConnectedComponentsFor n ≥ 1 ⓘ
hasUnderlyingSet set of all invertible n×n real matrices ⓘ
isAbelianFor n = 1 ⓘ
isDenseIn M_n(ℝ) ⓘ
isDisconnected true ⓘ
isFundamentalIn differential geometry ⓘ
linear algebra ⓘ
isGroupUnder matrix multiplication ⓘ
isLinearLieGroup true ⓘ
isNonAbelianFor n ≥ 2 ⓘ
isNonCompact true ⓘ
isNonSimple true ⓘ
isOpenIn M_n(ℝ) with standard topology ⓘ
isOpenSubsetOf M_n(ℝ) ⓘ
isParacompactManifold true ⓘ
isRealPointsOf algebraic group GL_n over ℝ NERFINISHED ⓘ
isReductiveGroup true ⓘ
isSmoothManifold true ⓘ
isStructureGroupOf frame bundle of an n-dimensional real manifold ⓘ
isSubsetOf M_n(ℝ) ⓘ
isZariskiOpenIn M_n(ℝ) ⓘ
kernelOfDeterminant SL(n,ℝ) ⓘ
LieAlgebraDescription all n×n real matrices ⓘ
LieAlgebraDimension n^2 ⓘ
quotientBySL(n,ℝ) ℝ\{0} via determinant ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Lie group → hasExample → general linear group GL(n,R) ⓘ