; TeX output 2001.11.06:1419ǭ7o][o]N cmbx12MathematischesInstitut 8WS2001/02derUniversitXatM`unchenSet3Prof.Dr.B.Pareigis;KrNff cmbx12ProblemffsetforAdvancedffAlgebraKXQ cmr12(9)%(TVensors9inphrysics:R#9,10,11)Letg cmmi12VbSea nitedimensionalvectorspace%orver+Wthe eld( msbm10KandletVp2 K cmsy8 ˹bSeitsdualspace.LettbeatensorinVs"!", cmsy10 ֲ:::x %VG Vp2  ::: Vp2 ƹ=URVp2 2cmmi8ru (Vp2\t)2 s .1Shorw>9thatforeachbasisB~=(b|{Ycmr81;:::ʜ;bnP)anddualbasisB2 >=(b21;:::ʜ;b2nP)%there0isauniquelydeterminedscrheme(afamilyoran(r?+s)-dimensionalB%matrix)ofcoSecienrts(a(B)OiqAacmr61*;:::\;i;cmmi6rájq1*;:::\;jsZ)witha(B)Oiq1*;:::\;irájq1*;:::\;js2URKsucrhthat,㍍(1)'i tUR= )n"u cmex10X ㇍iq1*=1AM:::-nn'X"X ㇍' ir,p=1An;X ㇍:jq1*=1Op:::ekn_X ㇍_;pjs=1sa(B)Oiq1*;:::\;irájq1*;:::\;jsZbiq1 :) ::: bir ; b jq1 Q ::: b js:(1) 0 (10)%Shorw[thatforeachchangeofbasesLUR:BX !_7C8with[cj\=P2iRAjf bi¹(withinrverse*b%matrix(2iRAjf ))thefollorwing+@ cmti12trffansformation35formulaholds(2)24a(B)Oiq1*;:::\;irájq1*;:::\;js= {BndX'؍URkq1*=1ɀ:::/n)oX'؍(kr,p=1D>=n>'X'؍> lq1*=1Q:::g nanX'؍aeQls=1tniq1*kq1 :::Pߍirykr }nlq1Qٍjq1 :::Eߍls5hjsa(Cܞ)Okq1*;:::\;kr lq1*;:::\;ls]ፍ (11)%ShorwthateveryfamilyofschemesofcoSecientse-(a(B)jBbasisof.Vp)্%witha(B)پ=(a(B)Oiq1*;:::\;irájq1*;:::\;jsZ)anda(B)Oiq1*;:::\;irájq1*;:::\;js 4t2پKsatisfyingthetransformation%formrulaK(2)de nesauniquetensor(indepSendentofthechoiceofthebasis)%tUR2Vp2 ru (Vp2\t)2 s\sucrhthat(1)holds.%R2ulel:forphysicists:tAktensorisacffollectionl:ofschemesofcffoecientsl:that%trffansform35accordingtothetransformationformulafortensors. (12)%Shorwthat(A;r\:A A\u!ѓA;<:\Ku!ѓA)isaK-algebraifandonlyifAx%〹withthemrultiplicationAYA2ɪ p $!/A A28rp $!Aݹandtheunitn9(1)isaring%and`:'K!=۹Cenrt/u(A)isaringhomomorphisminrtothecffenterofA,Uwhere%Cenrt>^(A)UR:=fa2Aj8b2A:ab=bag.;! Duedate:8TVuesdary,06.11.2001,16:15inLectureHall138*;ǭ +@ cmti12( msbm10"u cmex10 K cmsy8!", cmsy10;cmmi62cmmi8g cmmi12Aacmr6|{Ycmr8Nff cmbx12N cmbx12XQ cmr12