線性代數(shù)發(fā)展簡(jiǎn)英文版

1、線性代數(shù)發(fā)展簡(jiǎn)介線性代數(shù)發(fā)展簡(jiǎn)介中文版英文版ONONTHETHEHISTYHISTYOFOFDETERMINANTSDETERMINANTSByG.A.MILLERUniversityofIllinoisThehistyofdeterminantsisanunusuallyinterestingpartofthehistyofelementarymathematicsinviewofthefactthatitillustratesver

2、yclearlysomeofthedifficultiesinthishistywhichresultfromtheuseoftechnicaltermsthereinwithoutexhibitingthedefinitemeaningwhichistobegiventotheseterms.Manymodernwritershavebasedtheirdefinitionsofadeterminantontheexistenceof

3、asquarematrix.ThiswasdonefinstanceinthewidelyusedWeberWellsteinEncyklopadiederElementarmathematikvolume14thedition1922page304.Fromthispointofviewadeterminantdoesnotexistwithoutitssquarematrixjudgingfrommanyofthetextbooks

4、onelementarymathematicsitislikelythatmanystudentsconsiderthesquarematrixasanessentialpartofadeterminantsothatthetermdeterminantconveystothemadualconceptcomposedofasquarematrixacertainpolynomialassociatedtherewith.Whenthe

5、yspeakoftherowscolumnsofadeterminanttheynaturallyarethinkingofitsmatrixwhentheyspeakofthevaluethereoftheyarenaturallythinkingofthepolynomialimpliedbythetermdeterminant.Whenastudentwhoisfamiliarwithnodefinitionofthetermde

6、terminantexceptthedualonenotedintheprecedingparagraphmeetswiththecommonstatementthatthediscoveryofdeterminantsisusuallyribedtoG.W.LeibnizhenaturallyconcludesthatasquarematrixapolynomialwereassociatedbyG.W.Leibnizinaboutt

7、hesamewayastheyareassociatedatthepresenttime.Thisishowevernotthecase.InfactG.W.Leibnizassociatedapolynomialwithtwosquarematriceshederivedthispolynomialtherefrominawaywhichdifferswidelyfromtheonenowfollowedinexpingadeterm

8、inant.HencethequestionariseswhetheritisdesirabletoassociatethenameofG.W.Leibnizwiththediscoveryofdeterminants.Tothrowsomelightonthisquestionitmaybedesirabletoconsiderherethemotiveswhichledtosomeoftheearlydevelopmentswhic

9、harenowcommonlyassociatedwiththebeginningsofthetheyofdeterminants.Thethreesubjectswhicharecommonlyassociatedwiththeearlyhistyofdeterminantsare:Thesolutionofasystemofnlinearequationsinnunknownstheeliminationoftheunknownsf

10、romasystemofn1linearequationsinnunknownslineartransfmations.Thefirstofthesesubjectsisnaturallyoneoftheoldestinthehistyofmathematicswhengeneralmethodsthepolynomialswhicharenowusedtoexhibitthegeneralsolutionofsuchasystemof

11、equations.HencesomemathematicalhistianshavebeeninclinedtotracethehistyofdeterminantstomethodsusedbytheancientChinesetheancientJapaneseinregardtothesolutionofasystemoflinearequations.Inviewofthegreatdisparitybetweentheirm

12、ethodsthosenowcommonlyemployeditisdifficulttoexhibitanydefinitecontactbetweentheirmethodsourmodernideasofdeterminants.Whilethemodernstudentofmathematicsusuallybecomesfamiliarwiththetheyofdeterminantsfthefirsttimeinconnec

13、tionwiththesolutionofasystemofnlinearequationsinnunknownsitisprobablethatthesubjectoflineartransfmationsinfluencedmostprofoundlytheearlydevelopmentofdeterminantsifweadoptthedualmeaningofthetermdeterminantnotedabove.Infac

14、titappearsthatthefirstassociationofasquarematrixthecrespondingpolynomialinthemodernsenseisduetoC.F.Gausswasemployedinconnectionwithlineartransfmations.Meoverthefundamentalsubjectofmultiplicationofdeterminantswasdeveloped

15、inconnectionwithlineartransfmations.Thedateatwhichdeterminantswerefirstmultipliedisalsogreatlyaffectedbyourdefinitionofthetermdeterminantsincethepolynomialswhichresultfromsuccessivelineartransfmationswerestudiedbefethemu

16、ltiplicationofdeterminantsasthetermisnowcommonlyunderstoodwasconsidered.InthespecialcasewhenthedeterminantsareofthethirdderthetheemfmultiplicationcaneasilybeinferredfromthewkofC.F.GaussasgiveninthefifthchapterofhisnotedD

17、isquisitionesArithmeticae.Themainpointswhichweaimedtoexhibitinthepresentarticlearethatunlessthecontraryisexplicitlystatedthetermdeterminantshouldbeusedinthehistiesofelementarymathematicswiththedualmeaningimplyingbothasqu

18、arematrixacertainpolynomialassociatedtherewiththatthehistyofdeterminantswouldtherebybegreatlysimplified.Whilethegeneralfmulasinvolvedinthesolutionofnlinearequationsinnunknownsarecloselyrelatedtothedeterminantsitisundesir

19、abletoregardtheefftstodevelopsuchfmulasasdevelopmentsinthetheyofdeterminants.Theseparationofthesedevelopmentsthetheyofdeterminantsenablesustodistinguishsharplybetweenthedevelopmentofageneralalgebraicnotationwhichmadeitpo

20、ssibletousethemodernmathematicalfmulasthespecialnotationrelatingonlytothesubjectofdeterminants.Justastheuseofourdinarycomplexnumberswasfirstgreatlystimulatedbytheirappearanceinwkrelationtothesolutionofthecubicequationnot

21、bytheirappearanceinthecloselysothedevelopmentofthetheyofdeterminantswasfirstgreatlystimulatedbylineartransfmationsnotbytheusefulnessofdeterminantsinthesolutionofasystemoflinearequations.Theuseofdoublesubsmayservethesamep

22、urposeasthatofrowscolumnsinthesquarematrixofadeterminantitisofsomeinteresttonotethatG.W.Leibnizusednumberpairsascoefficientswhichservedthesamepurposeasdoublesubs.Thecreditcommonlygivenhimfthediscoveryofdeterminantsispart

23、lybasedonthisfact.InviewofthefactthatWebstersNewInternationalDictionaryiswidelyusedbyteachersofmathematicsitmaybedesirabletoreferheretoaslighterrwhichappearsthereinundertheterm“determinants“.Itiscrectlystatedthereunderth

24、atJacobiemployeddeterminants“powerfully“butthedate1823assignedfthisemploymentisnotquitecrect.AsJacobiwasbnonDecember101804hebecame19yearsoldlatein1823henceifthegivendatewerecrectitwouldimplythathewasayouthfulprodigyhence

25、hisbiographywouldbeofespecialinteresttoteachersofoursubject.Asamatteroffacthoweverhisimptantcontributionstodeterminantswerepublishedmuchlaterafterhehadmadeveryvaluablecontributionstowardtheadvancementofmathematicsalongot