南航双语矩阵论matrixtheory第三章部分题解之欧阳学创编

欧阳学创编2020.03.03欧阳学创编2020.03.03SolutionKeytoSomeExercisesinChapter3时间：2021.03.03创作：欧阳学#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,应该以集合的记号来表达或者用文字来叙述.#7.LetbethelinearmappingthatmapsintodefinedbyFindamatrixAsuchthat.SolutionHence,#10.LetbethetransformationondefinedbyFindthematrixArepresentingwithrespecttoFindthematrixBrepresentingwithrespecttoFindthematrixSsuchthatIf,calculate.Solution(a)(b)(c)Thetransitionmatrixfromtois,(d)#11.LetAandBbematrices.ShowthatifAissimilartoBthenthereexistmatricesSandT,withSnonsingular,suchthatand.ProofThereexistsanonsingularmatrixPsuchthat.Let,.Thenand.#12.LetbealineartransformationonthevectorspaceVofdimensionn.Ifthereexistavectorvsuchthatand,showthat(a)arelinearlyindependent.(b)thereexistsabasisEforVsuchthatthematrixrepresentingwithrespecttothebasisEisProofSupposethatThenThatis,Thus,mustbezerosince.Thiswillimplythatmustbezerosince.Byrepeatingtheprocessabove,weobtainthatmustbeallzero.Thisprovesthatarelinearlyindependent.Sincearenlinearlyindependent,theyformabasisforV.Denote…….#13.IfAisanonzerosquarematrixandforsomepositiveintegerk,showthatAcannotbesimilartoadiagonalmatrix.ProofSupposethatAissimilartoadiagonalmatrix.Thenforeach,thereexistsanonzerovectorsuchthatsince.Thiswillimplythatfor.Thus,matrixAissimilartothezeromatrix.Therefore,sinceamatrixthatissimilartothezeromatrixmustbethezeromatrix,whichcontradictstheassumption.ThiscontradictionshowsthatAcannotbesimilartoadiag