南航雙語矩陣論 matrix theory第三章部分題解

南航雙語矩陣論matrixtheory第三章部分題解南航雙語矩陣論matrixtheory第三章部分題解南航雙語矩陣論matrixtheory第三章部分題解南航雙語矩陣論matrixtheory第三章部分題解編制僅供參考審核批準生效日期地址：電話：傳真:郵編:SolutionKeytoSomeExercisesinChapter3#5.Determinethekernelandrangeofeachofthefollowinglineartransformationson(a)(b)(c)Solution(a)Let..ifandonlyififandonlyif.Thus,Therangeofis(b)Let..ifandonlyififandonlyifand.Thus,Therangeofis(c)Let..ifandonlyififandonlyifand.Thus,Therangeofis備注:映射的核以及映射的像都是集合,應該以集合的記號來表達或者用文字來敘述.#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.Thiscontradictionshowsthat