南航双语矩阵论 matrix theory第三章部分题解.doc

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1、SolutionKeytoSomeExercisesinChapter3#5.Determinethekernelandrangeofeachofthefollowinglineartransformationson(a)(b)(c)Solution(a)Let..ifandonlyififandonlyif.Thus,Therangeofis(b)Let..ifandonlyififandonlyifand.Thus,Therangeofis(c)Let..ifandonlyififandonlyifand.Thus,Therangeofis备注:映射的核以及映射的像都是集合

2、,应该以集合的记号来表达或者用文字来叙述.#7.LetbethelinearmappingthatmapsintodefinedbyFindamatrixAsuchthat.SolutionHence,#10.Letbethetransformationondefinedbya)FindthematrixArepresentingwithrespecttob)FindthematrixBrepresentingwithrespecttoc)FindthematrixSsuchthatd)If,calculate.Solution(a)4(b)(c)Thetransitionma

3、trixfromtois,(d)#11.LetAandBbematrices.ShowthatifAissimilartoBthenthereexistmatricesSandT,withSnonsingular,suchthatand.ProofThereexistsanonsingularmatrixPsuchthat.Let,.Thenand.#12.LetbealineartransformationonthevectorspaceVofdimensionn.Ifthereexistavectorvsuchthatand,showthat(a)arelinearlyin

4、dependent.(b)thereexistsabasisEforVsuchthatthematrixrepresentingwithrespecttothebasisEisProof4(a)SupposethatThenThatis,Thus,mustbezerosince.Thiswillimplythatmustbezerosince.Byrepeatingtheprocessabove,weobtainthatmustbeallzero.Thisprovesthatarelinearlyindependent.(b)Sincearenlinearlyindepende

5、nt,theyformabasisforV.Denote…….#13.IfAisanonzerosquarematrixandforsomepositiveintegerk,showthatAcannotbesimilartoadiagonalmatrix.ProofSupposethatAissimilartoadiagonalmatrix.Thenforeach,thereexistsanonzerovectorsuchthatsince.Thiswillimplythatfor.Thus,matrixAissimilartothezeromatrix.Therefore,

6、sinceamatrixthatissimilartothezeromatrixmustbethezeromatrix,whichcontradictstheassumption.ThiscontradictionshowsthatAcannotbesimilartoadiagonalmatrix.OrIfthen.4impliesthatfor.Hence,.Thiswillimplythat.Contradiction!4