linear algebra answers hefferon

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1、AnswerstoExercisesLinearAlgebraJimHefferon¡¢13¡¢21¯¯¯12¯¯¯¯31¯¡¢1x1¢3¡¢21¯¯¯x¢12¯¯¯¯x¢31¯¡¢68¡¢21¯¯¯62¯¯¯¯81¯NotationRrealnumbersNnaturalnumbers:f0;1;2;:::g¯Ccomplexnumbersf:::¯:::gsetof...suchthat...h:::isequence;likeasetbutordermattersV;W;Uvectorspaces~v;~wvectors~0,~0Vzerovecto

2、r,zerovectorofVB;DbasesE=h~e;:::;~eistandardbasisforRnn1n¯;~~±basisvectorsRepB(~v)matrixrepresentingthevectorPnsetofn-thdegreepolynomialsMn£msetofn£mmatrices[S]spanofthesetSM©NdirectsumofsubspacesV»=Wisomorphicspacesh;ghomomorphisms,linearmapsH;Gmatricest;stransformations;mapsfrom

3、aspacetoitselfT;SsquarematricesRepB;D(h)matrixrepresentingthemaphhi;jmatrixentryfromrowi,columnjjTjdeterminantofthematrixTR(h);N(h)rangespaceandnullspaceofthemaphR1(h);N1(h)generalizedrangespaceandnullspaceLowercaseGreekalphabetnamecharacternamecharacternamecharacteralpha®iota¶rho

4、½beta¯kappa·sigma¾gamma°lambda¸tau¿delta±mu¹upsilonÀepsilon²nuºphiÁzeta³xi»chiÂeta´omicronopsiÃthetaµpi¼omega!Cover.ThisisCramer’sRuleforthesystemx1+2x2=6,3x1+x2=8.Thesizeofthefirstboxisthedeterminantshown(theabsolutevalueofthesizeisthearea).Thesizeofthesecondboxisx1timesthat,andeq

5、ualsthesizeofthefinalbox.Hence,x1isthefinaldeterminantdividedbythefirstdeterminant.TheseareanswerstotheexercisesinLinearAlgebrabyJ.Hefferon.Correctionsorcommentsareverywelcome,emailtojimjoshua.smcvt.eduAnanswerlabeledhereas,forinstance,One.II.3.4,matchesthequestionnumbered4fromthefirs

6、tchapter,secondsection,andthirdsubsection.TheTopicsarenumberedseparately.ContentsChapterOne:LinearSystems4SubsectionOne.I.1:Gauss’Method..............................5SubsectionOne.I.2:DescribingtheSolutionSet.......................10SubsectionOne.I.3:General=Particular+Homogeneou

7、s.................14SubsectionOne.II.1:VectorsinSpace............................17SubsectionOne.II.2:LengthandAngleMeasures......................20SubsectionOne.III.1:Gauss-JordanReduction.......................25SubsectionOne.III.2:RowEquivalence............................27Top

8、ic:ComputerAlgebraSystems........