高等代数习题-欧式空间

AnnR12(,,,)Tnxxxa=L12(,,,)Tnyyyb=L,TAabab=nRnR,,abgk,(),TTTTTTAAAAabababbababa=====,(),,TTTAAAabgabgagbgagbg+=+=+=+,()(),TTkkAkAkabababab===,TAaaaa=Aa≠0,0TAaaaa=nR4R(1,1,1,1),(1,1,1,1),(2,1,1,3)TTT---111111112113A-⎛⎞⎜⎟=--⎜⎟⎜⎟⎝⎠4R1234(,,,)Txxxxx=(1,1,1,1),(1,1,1,1),(2,1,1,3)TTT---Ax=0(4,0,1,3)Tkx=-k∈R1(4,0,1,3)26T±-12,,,naaaLVVg,0iga=1,2,,in=Lg=0Va12,,gaga=12,Vgg∈12gg=12,,,naaaLVVa∈12,,,naaaL,0iga=1,2,,in=L,0ga=,0gg=g=0Va12,,gaga=12,0gga-=12gg-=012gg=ABn||1AB=-||||||1TTTTABABAB===-||1A=1-A||0AB+=||||||||||||1TTABABABAB====-||||1TTTABAB==-|||||()|||||||TTTAEAAAEAAEAAEA+=+=+=+=-+|+|0AE=1-AABnTAB||1AB=-||1TAB=-1-TAB||0TABE+=||||||||0TTABAAABAEAB+=+=+=123,,aaaV11231(22)3baaa=+-21231(22)3baaa=-+31231(22)3baaa=--123,,bbb123123(,,)(,,)Abbbaaa=22112123122A⎛⎞⎜⎟=--⎜⎟⎜⎟--⎝⎠TAAE=A123,,aaaV123,,bbbnV121,,,naaa-L12,bbia1,2,,1in=-L12,bb,0,1,2,,1,1,2ijinjab==-=L12121,((,,,))nLbbaaa⊥-∈L121dim((,,,))1nLaaa⊥-=L12,bb1V2VnV12dimdimVV2V1V11VVV⊥=⊕11dimdimdimVVVn⊥+==12dimdimVV21dimdimVVn⊥[phone]dim()dimdimdim()dimdimdim0VVVVVVVVV⊥⊥⊥⊥∩=+-+≥+-21VVa⊥∈∩2V1Va≠0nV1{|,0,}VVxxax==∈V1V1n-,0a=01V∈01V1V12,xx1212,,,0xxaxaxa+=+=121Vxx+∈1Vxk,,0kkxaxa==1kVx∈1VVa≠0V2,,,naaaL12(,,)nVLaa=L1V1n-1VnVVa∈Vbab-1V11VVV⊥=⊕12aaa=+1121,VVaa⊥∈∈1ba=121Vabaaa⊥-=-=∈1,0Vab-=ab-1VVb′∈1,0Vab′-=1Vab⊥′-∈()abab′′=+-12aa=+11VVV⊥=⊕12,baaba′′=-=bb′=bsnV*sV,Vab∈*(),,()sabasb=*sV*Ker()(())Vss⊥=,,,VkRabg∀∈∈*,()(),gsabsgab+=+=****(),(),,(),(),()()sgasgbgsagsbgsasb+=+=+***()()()sabsasb+=+*,()(),kkasbsab==(),ksab**,(),()kkasbasb==**()()kksbsb=*sV,Vab∈*(),,()sabasb=**Ker()()(),0,()0(())VVVassasaasas⊥∈⇔=⇔=⇔=⇔∈0*Ker()(())Vss⊥=AnARTARR=APTAPP=PPQR=QR()TTTTTAPPQRQRRQQPRR====ATARR=RAAVnsVsker()sIm()ssIm()sker()sIm()as∀∈Im()Vs⊆Va∈()Im()sas∈Im()ssker()bs∀∈()ker()sbs=∈0ker()ssker()as∀∈()sa=0Im()bs∀∈Vg∈()bsg=sV,,()(),0abasgsag===ker()sIm()sdimVn=dimKer()dimIm()nss+=Im()sker()s011101110A-⎛⎞⎜⎟=-⎜⎟⎜⎟⎝⎠UTUAU=Λ211||11(1)(2)11EAllllll--=-=-+--A1232,1lll=-==12l=-(2)AEx+=01111x-⎛⎞⎜⎟=-⎜⎟⎜⎟⎝⎠231ll==()AEx-=02110x-⎛⎞⎜⎟=⎜⎟⎜⎟⎝⎠3101x⎛⎞⎜⎟=⎜⎟⎜⎟⎝⎠1x23,xx1231111111,1,1326102hhh--⎛⎞⎛⎞⎛⎞⎜⎟⎜⎟⎜⎟=-==⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠()123,,Uhhh=U200010001TUAU-⎛⎞⎜⎟=⎜⎟⎜⎟⎝⎠nATAA=-An||0A≥n||0A=A||0A1A-A11,,,,sskikikiki--L1,,skk∈RLA||0A≥n||||||(1)||||TnAAAAA==-=-=-||0A=AA||0A1111()()()TTAAAA----==-=-1A-Vs,Vab∈(),,()sabasb=-nVssVWsW⊥sVs12,,,naaaLVs12,,,naaaL()ijnnAa=(),ijjiasaa=,(),(,1,,)ijijaijnasa==Ls(),,()jiijijijaasaaasa==-=-(,1,,)ijn=LA()ijnnAa=11,nniiiiiixyaaba====∑∑V1111(),(),,(),()nnnniijjiijjijijxyxysabsaaasaasb======-=-∑∑∑∑sWa⊥∈WbWs()Wsb∈,()0asb=(),,()0sabasb=-=()Wsa⊥∈W⊥ssVRl∈asl()sala=()a≠0,(),,(),laasaaasalaa==-=-,0laa=a≠0,0aa≠0l=