The linear groups

The set of all invertible transformations (or equivalently of invertible matrices) with determinant 1 is then a subgroup of GL(n, R) called the Special Linear ... CourseMT3818TopicsinGeometry Previouspage (Foundations) Contents Nextpage (Examplesoflineargroups) Thelineargroups Weshallinvestigatethegroupswhichareassociatedwiththeusual(Pythagorean)metriconthevectorspaceRn.Thesearethegroupswhichpreservethisdistance. Itturnsoutthattheyinvolvelinearalgebra.Thereasonforthisisthatstraightlinesinthisgeometrycanbedefinedastheshortestpathsbetweenpointsinthemetricandsincethemetricispreservedbythetransformationstheymustthenmapstraightlinestostraightlinesandwewillsee(eventually)thatthismeanstheyinvolvelinearmaps(butratherspecialones). Definitions AmapffromRmtoRniscalledlinearifitmapsalinearcombinationofvectorstothesamelinearcombinationoftheimages. Thatis,ifu,v∈Rmandλ,μ∈Rthenf(λu+μv)=λf(u)+μf(v)∈Rn. Byfixingabasis{b1,b2,...,bn}ofthevectorspaceRn(forexample,{(1,0,0,...,0),(0,1,0,...,),...,(0,0,...,0,1)})wecandescribetheeffectofsuchamapbyitsmatrix.Mf=(aij)wherefmapstheithbasiselementbitoai1b1+ai2b2+...+ainbn. Suchatransformationisabijectionifithasaninversemapf-1orequivalentlyifthedeterminantofitsmatrixisnon-zero. ThesetofallsuchinvertiblelineartransformationsfromthevectorspaceRntoitselfiscalledtheGeneralLineargroupandisdenotedbyGL(n,R)orGLn(R)orGL(Rn) Notethatthedeterminantofamatrixsatisfiesdet(AB)=det(A)×det(B)andsoisagrouphomomorphismfromthegroupGL(n,R)tothegroupR-{0}underrealmultiplication.Thesetofallinvertibletransformations(orequivalentlyofinvertiblematrices)withdeterminant1isthenasubgroupofGL(n,R)calledtheSpecialLineargroupanddenotedbySL(n,R). Remarks Thesetofallrealn×nmatricescanberegardedasRn2andsincethedeterminantmapisapolynomialinthesen2entriesitiscontinuous.ItfollowsthatthesetGL(n,R)whichisdet-1(R-{0})isanopensetofRn2andcanbethoughtofashavingdimensionn2.ThesetSL(n,R)satisfiesoneequationandsohasone"degreeoffreedom"lessandsohasdimensionn2-1. Forexample,GL(1,R)isjustR-{0}andsohasdimension1.SL(1,R)istheset{1,-1}andsohasdimension0. GL(2,R)={(a,b,c,d)∈R4|ad-bc≠0}andisthesetofpointswhich"miss"thehypersurfacewithequationad=bcandsoisanopensetinR4.ThesubgroupSL(2,R)={(a,b,c,d)∈R4|ac-bd=1}andisa3-dimensionalsubsetofR4. IngeneralelementsofGLorSLdonotpreservethemetric. Alinearmapwillingeneralmaparectangletoaparallelogramandsoevenifitmanagestopreservethelengthsofthesidesoftherectangle,itwillingeneralstretchthediagonalsandsowillnotpreservealllengths. YoucanthinkofelementsofSLaspreservingthe"multi-dimensionalvolume"inRn(areainthecasen=2)buteventhesewillingeneralchangelengthsofsomevectors. TherearealsomapsfromRntoitselfwhichdopreservethemetricbutwhicharenotlinearmaps.Sinceanylineartransformationmapsthe0-vectortoitself,amaplikeatranslation:x↦x+aforafixedvectora,willpreservelengthbutisnotlinear. Wenowlookatlineartransformationswhichdopreservedistance. Definitions Wewilldenotethenormorlengthofavectorxby‖x‖Thisisd(x,0). AdistancepreservinglineartransformationTissaidtobeorthogonal.Thatis‖T(x)‖=‖x‖forallvectorsx. SuchtransformationsformtheorthogonalgroupO(n). Remarks Suchmapsareinvertible. Proof SinceTislengthpreservingitcan'tmapanon-zerovectorto0andsothenull-spacehasdimension0andTisone-one.Itisastandardresultaboutlineartransformationsfromndimensionalspacestondimensionalspacesthatdim(nullspace)+dim(range)=nandsothedimensionoftherangeofTisnanditishenceanontomap.Thusitisabijection. SinceTpreserveslengths,itpreservesanglesalso.Thisisbecausetheanglesofanytrianglearedeterminedbythelengthsofitssides. Withrespecttoanorthonormalbasis(onewhoseelementsareallofunitlengthandmutuallyperpendicular)thematrixofsuchatransformationhasorthonormalcolumnsandalsoorthonormalrows. Unlesswestateotherwisewewillusuallytakeourbasistobeorthonormalandwewillblurthedistinctionbetweenatransformationandthematrixrepresentingit. OnemayprovethatsuchamatrixsatisfiesAtA=AAt=Iwheretindicatesthetranspose.Itthenfollows[sincedet(A)=det(At)]that(detA)2=1andhencedetA=±1. Definition TheSpecialOrthogonalgroupSO(n)isthesubgroupofO(n)ofelementswhosematrixhasdeterminant1. Remarks ThegroupO(n)istheunionofSO(n)andthecosetK.SO(n)whereKisamatrix(say)whichisorthogonalwithdeterminant-1. SinceSO(n)hastwocosetsinO(n)itisanormalsubgroup. NotethatthesubsetH={I,K}isasubgroupofO(n).IngeneralthegroupO(n)isnotthedirectproductSO(n)×Halthoughitisequaltothecartesianproductasaset. Previouspage (Foundations) Contents Nextpage (Examplesoflineargroups) JOCFebruary2003