General Linear Group of 2x2 real matrices

URI: https://gptkb.org/entity/General_Linear_Group_of_2x2_real_matrices

GPTKB entity

Statements (49)

Predicate	Object
gptkbp:instanceOf	gptkb:group_of_people gptkb:Lie_group
gptkbp:actsOn	real 2-dimensional vector space
gptkbp:centralTo	scalar multiples of identity matrix with nonzero real scalars
gptkbp:compact	true
gptkbp:component	{determinant > 0, determinant < 0}
gptkbp:contains	orthogonal group O(2) diagonal matrices with nonzero entries lower triangular invertible matrices special linear group SL(2, ℝ) upper triangular invertible matrices
gptkbp:containsElement	invertible 2x2 real matrices
gptkbp:definedIn	real numbers
gptkbp:determinant	GL(2, ℝ) → ℝ\\{0}
gptkbp:determinantCondition	determinant not zero
gptkbp:dimensions	4
gptkbp:field	ℝ
gptkbp:hasApplication	gptkb:geometry partial differential equations physics linear algebra
gptkbp:hasConnection	false
gptkbp:hasDeterminantMap	true
gptkbp:hasSubgroup	gptkb:Special_Linear_Group_of_2x2_real_matrices
gptkbp:hasTwoConnectedComponents	true
https://www.w3.org/2000/01/rdf-schema#label	General Linear Group of 2x2 real matrices
gptkbp:identityElement	matrix inverse 2x2 identity matrix
gptkbp:isAlgebraicGroup	true
gptkbp:isDenseIn	M(2, ℝ)
gptkbp:isGroupUnder	matrix multiplication
gptkbp:isMatrixGroup	true
gptkbp:isNonAbelian	true
gptkbp:isNonConnected	true
gptkbp:isOpenSubsetOf	M(2, ℝ)
gptkbp:isReductive	true
gptkbp:isSimple	false
gptkbp:isTopologicalGroup	true
gptkbp:kernelOfDeterminantMap	gptkb:SL(2,_ℝ)
gptkbp:Lie_algebra	gl(2, ℝ)
gptkbp:LieAlgebraDimension	4
gptkbp:notation	GL(2, ℝ)
gptkbp:operator	matrix multiplication
gptkbp:order	infinite
gptkbp:quotientBySL2R	isomorphic to ℝ^×
gptkbp:realForm	gptkb:GL(2,_ℂ)
gptkbp:symbol	GL(2, ℝ)
gptkbp:bfsParent	gptkb:Special_Linear_Group_of_2x2_real_matrices
gptkbp:bfsLayer	7