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1、LinearAlgebra¯¯¯12¯¯¯¯31¯¯¯¯x¢12¯¯¯¯x¢31¯¯¯¯62¯¯¯¯81¯JimHefferonNotationRrealnumbersNnaturalnumbers:f0;1;2;:::g¯Ccomplexnumbersf:::¯:::gsetof:::suchthat:::h:::isequence;likeasetbutordermattersV;W;Uvectorspaces~v;~wvectors~0,~0Vzerovector,zerovectorofVB;DbasesE=h~e;:::;~eistandardbasisforRnn1n¯;~~
2、±basisvectorsRepB(~v)matrixrepresentingthevectorPnsetofn-thdegreepolynomialsMn£msetofn£mmatrices[S]spanofthesetSM©NdirectsumofsubspacesV»=Wisomorphicspacesh;ghomomorphismsH;Gmatricest;stransformations;mapsfromaspacetoitselfT;SsquarematricesRepB;D(h)matrixrepresentingthemaphhi;jmatrixentryfromrowi
3、,columnjjTjdeterminantofthematrixTR(h);N(h)rangespaceandnullspaceofthemaphR1(h);N1(h)generalizedrangespaceandnullspaceLowercaseGreekalphabetnamesymbolnamesymbolnamesymbolalpha®iota¶rho½beta¯kappa·sigma¾gamma°lambda¸tau¿delta±mu¹upsilonÀepsilon²nuºphiÁzeta³xi»chiÂeta´omicronopsiÃthetaµpi¼omega!Cov
4、er.ThisisCramer'sRuleappliedtothesystemx+2y=6,3x+y=8.Theareaofthe¯rstboxisthedeterminantshown.Theareaofthesecondboxisxtimesthat,andequalstheareaofthe¯nalbox.Hence,xisthe¯naldeterminantdividedbythe¯rstdeterminant.PrefaceInmostmathematicsprogramslinearalgebraistakeninthe¯rstorsecondyear,followingor
5、alongwithatleastonecourseincalculus.Whilethelocationofthiscourseisstable,latelythecontenthasbeenunderdiscussion.Somein-structorshaveexperimentedwithvaryingthetraditionaltopics,tryingcoursesfocusedonapplications,oronthecomputer.Despitethis(entirelyhealthy)debate,mostinstructorsarestillconvinced,It
6、hink,thattherightcorematerialisvectorspaces,linearmaps,determinants,andeigenvaluesandeigenvectors.Applicationsandcomputationscertainlycanhaveaparttoplaybutmostmath-ematiciansagreethatthethemesofthecourseshouldremainunchanged.Notthatallis¯newiththetraditionalcourse.Mostofusdothinkthatthestandardte
7、xttypeforthiscourseneedstobereexamined.Elementarytextshavetraditionallystartedwithextensivecomputationsoflinearreduction,matrixmultiplication,anddeterminants.Thesetakeuphalfofthecourse.Finally,whenvectorspacesandlinear
