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1、ElementsofAbstractandLinearAlgebraE.H.ConnelliiE.H.ConnellDepartmentofMathematicsUniversityofMiamiP.O.Box249085CoralGables,Florida33124USAec@math.miami.eduMathematicalSubjectClassi¯cations(1991):12-01,13-01,15-01,16-01,20-01°c1999E.H.ConnellNovember30,2000[http://www.math.miami.edu
2、/»ec/book/]iiiIntroductionIn1965I¯rsttaughtanundergraduatecourseinabstractalgebra.Itwasfuntoteachbecausethematerialwasinterestingandtheclasswasoutstanding.FiveofthosestudentslaterearnedaPh.D.inmathematics.SincethenIhavetaughtthecourseaboutadozentimesfromvarioustexts.OvertheyearsIde
3、velopedasetoflecturenotesandin1985Ihadthemtypedsotheycouldbeusedasatext.Theynowappear(inmodi¯edform)asthe¯rst¯vechaptersofthisbook.Hereweresomeofmymotivesatthetime.1)Tohavesomethingasshortandinexpensiveaspossible.Inmyexperience,studentslikeshortbooks.2)Toavoidallinnovation.Toorgani
4、zethematerialinthemostsimple-mindedstraightforwardmanner.3)Toorderthemateriallinearly.Totheextentpossible,eachsectionshouldusetheprevioussectionsandbeusedinthefollowingsections.4)Toomitasmanytopicsaspossible.Thisisafoundationalcourse,notatopicscourse.Ifatopicisnotusedlater,itshould
5、notbeincluded.Therearethreegoodreasonsforthis.First,linearalgebrahastoppriority.Itisbettertogoforwardanddomorelinearalgebrathantostopanddomoregroupandringtheory.Second,itismoreimportantthatstudentslearntoorganizeandwriteproofsthemselvesthantocovermoresubjectmatter.Algebraisaperfect
6、placetogetstartedbecausetherearemanyeasy"theoremstoprove.Therearemanyroutinetheoremsstatedherewithoutproofs,andtheymaybeconsideredasexercisesforthestudents.Third,thematerialshouldbesofundamentalthatitbeappropriateforstudentsinthephysicalsciencesandincomputerscience.Zillionsofstude
7、ntstakecalculusandcookbooklinearalgebra,butfewtakeabstractalgebracourses.Somethingiswronghere,andonethingwrongisthatthecoursestrytodotoomuchgroupandringtheoryandnotenoughmatrixtheoryandlinearalgebra.5)Too®eranalternativeforcomputersciencemajorstothestandarddiscretemathematicscourse
8、s.Mostofthematerialinthe¯r