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1、LinearAlgebraJimHefferon¡¢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,~0Vzerovect
2、or,zerovectorofVB;DbasesE=h~e;:::;~eistandardbasisforRnn1n¯;~~±basisvectorsRepB(~v)matrixrepresentingthevectorPnsetofn-thdegreepolynomialsMn£msetofn£mmatrices[S]spanofthesetSM©NdirectsumofsubspacesV»=Wisomorphicspacesh;ghomomorphisms,linearmapsH;Gmatricest;stran
3、sformations;mapsfromaspacetoitselfT;SsquarematricesRepB;D(h)matrixrepresentingthemaphhi;jmatrixentryfromrowi,columnjjTjdeterminantofthematrixTR(h);N(h)rangespaceandnullspaceofthemaphR1(h);N1(h)generalizedrangespaceandnullspaceLowercaseGreekalphabetnamecharactern
4、amecharacternamecharacteralpha®iota¶rho½beta¯kappa·sigma¾gamma°lambda¸tau¿delta±mu¹upsilonÀepsilon²nuºphiÁzeta³xi»chiÂeta´omicronopsiÃthetaµpi¼omega!Cover.ThisisCramer’sRuleforthesystemx+2y=6,3x+y=8.Thesizeofthefirstboxisthedeterminantshown(theabsolutevalueofthes
5、izeisthearea).Thesizeofthesecondboxisxtimesthat,andequalsthesizeofthefinalbox.Hence,xisthefinaldeterminantdividedbythefirstdeterminant.PrefaceInmostmathematicsprogramslinearalgebracomesinthefirstorsecondyear,followingoralongwithatleastonecourseincalculus.Whiletheloc
6、ationofthiscourseisstable,latelythecontenthasbeenunderdiscussion.Someinstructorshaveexperimentedwithvaryingthetraditionaltopicsandothershavetriedcoursesfocusedonapplicationsoroncomputers.Despitethishealthydebate,mostinstructorsarestillconvinced,Ithink,thattherig
7、htcorematerialisvectorspaces,linearmaps,determinants,andeigenvaluesandeigenvectors.Applicationsandcodehaveaparttoplay,butthethemesofthecourseshouldremainunchanged.Notthatallisfinewiththetraditionalcourse.Manyofusbelievethatthestandardtexttypecoulddowithachange.In
8、troductorytextshavetraditionallystartedwithextensivecomputationsoflinearreduction,matrixmultiplication,anddeterminants,whichtakeuphalfofthecourse.Then,whenvectorspace