Center of general linear group - Mathematics Stack Exchange

Given a (not necessarily finite dimensional) vector space V prove that the center of GL(V) is the set of all scalar transformations (i.e all ... MathematicsStackExchangeisaquestionandanswersiteforpeoplestudyingmathatanylevelandprofessionalsinrelatedfields.Itonlytakesaminutetosignup. Signuptojointhiscommunity Anybodycanaskaquestion Anybodycananswer Thebestanswersarevotedupandrisetothetop Home Public Questions Tags Users Unanswered Teams StackOverflowforTeams –Startcollaboratingandsharingorganizationalknowledge. CreateafreeTeam WhyTeams? Teams CreatefreeTeam Teams Q&Aforwork Connectandshareknowledgewithinasinglelocationthatisstructuredandeasytosearch. LearnmoreaboutTeams Centerofgenerallineargroup[duplicate] AskQuestion Asked 8yearsago Modified 1year,7monthsago Viewed 6ktimes 6 $\begingroup$ Thisquestionalreadyhasanswershere: Alinearoperatorcommutingwithallsuchoperatorsisascalarmultipleoftheidentity. (9answers) Closed7yearsago. Givena(notnecessarilyfinitedimensional)vectorspace$V$provethatthecenterof$\operatorname{GL}(V)$isthesetofallscalartransformations(i.ealltransformationsoftheform$a\operatorname{Id}$)? Iknowhowtoprovethisforgenerallineargroupofdegree$n$,pleasehelpmesolvethisforthecaseofagenerallinearmap. linear-algebra Share Cite Follow editedOct6,2014at7:05 hjhjhj57 4,03211goldbadge1818silverbadges3838bronzebadges askedOct6,2014at5:50 DopemanDopeman 6111silverbadge22bronzebadges $\endgroup$ 0 Addacomment  |  2Answers 2 Sortedby: Resettodefault Highestscore(default) Datemodified(newestfirst) Datecreated(oldestfirst) 3 $\begingroup$ Let$T$inthecenter.Forany$L$wehave$T\circL=L\circT$,thatis $$T(Lx)=L(Tx)$$forall$L$andall$x\inV$. Let$x$in$V$.Thereexists$L$linearmapsothatthesubspace$\{y\|\Ly=y\}$equals$\mathbb{F}\cdotx$(useabasisstartingfrom$x$). Weget $L(Tx)=T(Lx)=Tx$andso$Tx\in\mathbb{F}\cdotx$. So,forany$x\inV$wehave$T(x)$proportionalto$x$.It'seasynow. Share Cite Follow answeredOct6,2014at7:02 orangeskidorangeskid 48.8k33goldbadges3636silverbadges8989bronzebadges $\endgroup$ Addacomment  |  3 $\begingroup$ Letassumethatthereexistsanelement$A$inthecenterofagenerallineargroupoveranarbitraryvectorspace$V$suchthat$A$isnotascalartransformation.($\dimV$isinfinite) CLAIM:$\exists\,v$suchthat$v$isnotaneigenvectorof$A$. Letassumethateveryvectorin$V$isaneigenvectorof$A$.Picktwonon-parallelvectors$v_1$,$v_2$andlet$a_1$and$a_2$becorrespondingeigenvalues. Theassumptionimpliesthat$v_1+v_2$shouldbeaneigenvectorof$A$andlet$a$betheeigenvalueof$v_1+v_2$. $$A\cdotv_1+A\cdotv_2=a_1v_1+a_2v_2=av_1+av_2$$ $$(a_1-a)v_1=(a-a_2)v_2$$ $v_1$and$v_2$arenotparallel,so$a_1=a_2=a$forarbitraryvector$v_1$and$v_2$.So$A$shouldbeascalartransformation.Thisisacontradiction. Sowecanassumethatthereexistsavector$v$whichisnotaneigenvectorof$A$.Let$w=A\cdotv$.Then$w\notin\left$.Let$S$bethevectorsubspacegeneratedby$v$and$w$.Then$\left\{v,w\right\}$isabasisof$S$andthereisabasis$T$of$V$s.t.$\left\{v,w\right\}\subsetT$.Let$S'$bethevectorsubspaceof$V$generatedby$T\setminus\left\{v,w\right\}$. Wecanconstructtwogenerallinearmatrices$B_1$and$B_2$s.t. $$B_1\cdotv=w,\B_1\cdotw=v$$ $$B_2\cdotv=\frac{1}{2}w,\B_2\cdotw=v$$ $$B_1|_{S'}=B_2|_{S'}=\mathrm{id}|_{S'}.$$ Thenwegetfollowingresults: $$A\cdotw=A\cdotB_1\cdotv=B_1\cdotA\cdotv=B_1\cdotw=v$$ $$A\cdotw=A\cdotB_2\cdot2v=B_2\cdotA\cdot2v=B_2\cdot2w=2v$$ Thisisacontradiction.So$A$shouldbeascalartransformation. Share Cite Follow editedFeb20,2021at22:46 user26857 1 answeredOct6,2014at7:58 a--a-- 13355bronzebadges $\endgroup$ 2 $\begingroup$ thankyousomuch,butwillthemapsB1,B2belinear $\endgroup$ – Dopeman Oct6,2014at10:23 $\begingroup$ Foreveryvectorspace$V$andit'sbasis$B$,weknowthatevery$f:B\toV$canbeextendedtoalinearmap$F:V\toV$.Sowecanassumethat$B_1$and$B_2$arelinear. $\endgroup$ – a-- Oct7,2014at1:31 Addacomment  |  Nottheansweryou'relookingfor?Browseotherquestionstaggedlinear-algebraoraskyourownquestion. FeaturedonMeta BookmarkshaveevolvedintoSaves Inboximprovements:markingnotificationsasread/unread,andafiltered... Linked 58 Alinearoperatorcommutingwithallsuchoperatorsisascalarmultipleoftheidentity. 2 Determinethecenterof$GL_n(\mathbb{R})$[Artin2.5.7](similartoAxler3.D.16) 0 Whatexactlydoes${\rmPGL}(n,K)$represent? 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