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1、 art I (必做题,共5题，70分) 第1题（15分）得分Let enote te etof al rea olynoils odegre essha 3 with doai（定义域). Thdtion ad salar mltilicatnaredefid nteual wa. fine in proct o by .(1) Cnstruct northonormal asis for frothe basis b sing the Gram-Schmtorthonalizao rocess (2) Lt. Fn theprjectionof nto th subsac spand bo
2、luon: (1) ，, , , ， - (2) -第2题（1分）得分 et be theiearnsfrmatn n (te vector ce rl pynmias o degre less han 3） defid by . （1) nd the marix reesenin with espct toterered asis o . (2) Fn basis fr uc that whrespect o his bas,he atrix represtin isdiagoa. (3） ind he knl(核) ad range (值域)of this trnsfomaio.oto:
3、() - (2） (Tecolumn ector f re h ignvectos ） hecorrespndingeienector n ae (T diaonalizes A） . ithrespect to ths nw bi , t rpreetin atrix idiaonal. - （3)Th kere i thesubspae nsistgof al constant polynomas The ag isthe bspace pnnedby the vetor -第题(20分）得分et (1) Fid alldetrminat divisors and elemenry dvo
4、rs of. (2) F a Jordan cnonical m f .(3) Cmue . (ie th detais of you coputations.) oluo: (1),（特性多项式 .Eigenvales re1,2, 2)terminantdisor of order， , Eetary ivisosar . - () Th ordancanoical fm is - (3) Foreigelue 1, , An einvector is Foreigene2, ， An eienectr is Solve , we obn that , - 第题(10分）得分upps th
5、at and .(1) Wat re the possle miniml olynmialof? xpin.(2) In ech cae fpat （1), whatare the posble charctstic pynoils o? ExplinSoutio: （1） An annihilatingplynomial is . e miiml plynomial f Adiisan annatn lynomal of A. The possle mini polnmils are，, an . -(2) Theminimal olynoma o A dvide the caatritc
6、polynomial of A. Sine A is marixf oder 3,thchararist olynomialofis of degree .Th minal polynmia f A d he chrcteristic polynoal oA have th se linea factors. Case , thecharcteritic polynomil s Cae , he charcteristic ploial is Cae , te characterstic olynmil is o -第5题(10分)得分Le Find th ore-Prosinerseof .
7、 Sotion： , 也可以用SVD求 -PrtII (选做题, 每题10分) 请在如下题目中(第6至第9题)选择三题解答. 如果你做了四题,请在题号上画圈标明需要批改的三题. 否则，阅卷者会随意挑选三题批改,这也许影响你的成绩.第6题 Lt be th eco ae consisng al eal polnmals of egre less thn wth sua dditio d scalar mtipication. L be three iinct ealnmber Fo each arf lynomias and n, efine .tei weher define n innerprout o or Explan. 第7题 et. Sow that i ithe orthogoal projectionfom t, hen A isymetrindth eig
