南航双语矩阵论 matrix theory第三章部分题解

1SolutionKeytoSomeExercisesinChapter3#5.Determinethekernelandrangeofeachofthefollowinglineartransformationson2P(a)(())'()pxxpx(b)(())()'()pxpxpx(c)(())(0)(1)pxpxpSolution(a)Let()pxaxb.(())pxax.(())0pxifandonlyif0axifandonlyif0a.Thus,ker(){|}bbRTherangeofis2()P{|}axaR(b)Let()pxaxb.(())pxaxba.(())0pxifandonlyif0axbaifandonlyif0aand0b.Thus,ker(){0}Therangeofis2()P2{|,}PaxbaabR(c)Let()pxaxb.(())pxbxab.(())0pxifandonlyif0bxabifandonlyif0aand0b.Thus,ker(){0}Therangeofis2()P2{|,}PbxababR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.Letbethelinearmappingthatmaps2Pinto2Rdefinedby10()(())(0)pxdxpxpFindamatrixAsuchthat()xA.Solution1(1)11/2()0x11/211/2()1010xHence,11/210A#10.Letbethetransformationon3Pdefinedby(())'()()pxxpxpxa)FindthematrixArepresentingwithrespectto2[1,,]xxb)FindthematrixBrepresentingwithrespectto2[1,,1]xxc)FindthematrixSsuchthat1BSASd)If2012()(1)pxaaxax,calculate(())npx.Solution(a)(1)0()xx222()22xx002010002A(b)(1)0()xx22(1)2(1)xx000010002B(c)2[1,,1]xx2[1,,]xx101010001Thetransitionmatrixfrom2[1,,]xxto2[1,,1]xxis101010001S,1BSAS(d)2201212((1))2(1)nnaaxaxaxax#11.LetAandBbennmatrices.ShowthatifAissimilartoBthenthereexistnnmatricesSandT,withSnonsingular,suchthatASTandBTS.ProofThereexistsanonsingularmatrixPsuchthat1APBP.Let1SP,TBP.ThenASTandBTS.#12.LetbealineartransformationonthevectorspaceVofdimensionn.Ifthereexistavectorvsuchthat1()v0nand()v0n,showthat(a)1,(),,()vvvnarelinearlyindependent.(b)thereexistsabasisEforVsuchthatthematrixrepresentingwithrespecttothebasisEis000010000010Proof3(a)Supposethat1011()()vvv0nnkkkThen11011(()())vvv0nnnkkkThatis,12210110()()())()vvvv0nnnnnkkkkThus,0kmustbezerosince1()v0n.211111(()())()vvv0nnnnkkkThiswillimplythat1kmustbezerosince1()v0n.Byrepeatingtheprocessabove,weobtainthat011,,,nkkkmustbeallzero.Thisprovesthat1,(),,()vvvnarelinearlyindependent.(b)Since1,(),,()vvvnarenlinearlyindependent,theyformabasisforV.Denote112,(),,()εvεvεvnn12()εε23()εε…….1()εεnn()ε0n12[(),(),,()]εεεn121[,,,,]εεεεnn000010000010#13.IfAisanonzerosquarematrixandkAOforsomepositiveintegerk,showthatAcannotbesimilartoadiagonalmatrix.ProofSupposethatAissimilartoadiagonalmatrix12diag(,,,)n.Thenforeachi,thereexistsanonzerovectorxisuchthatxxiiiAxxx0kkiiiiiAsincekAO.Thiswillimplythat0ifor1,2,,in.Thus,matrixAissimilartothezeromatrix.Therefore,AOsinceamatrixthatissimilartothezeromatrixmustbethezeromatrix,whichcontradictstheassumption.ThiscontradictionshowsthatAcannotbesimilartoadiagonalmatrix.OrIf112diag(,,,)nAPPthen112diag(,,,)kkkknAPP.4kAOimpliesthat0ifor1,2,,in.Hence,BO.ThiswillimplythatAO.Contradiction!