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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.ConnellOctober5,2000[http://www.math.m
2、iami.edu/»ec/book/]iiiIntroductionIn1965I¯rsttaughtanundergraduatecourseinabstractalgebra.Itwasfuntoteachbecausethematerialwasinterestingandtheclasswasoutstanding.FiveofthosestudentslaterearnedaPh.D.inmathematics.SincethenIhavetaughtthecourseaboutadozentimesfromvarioustex
3、ts.OvertheyearsIdevelopedasetoflecturenotesandin1985Ihadthemtypedsotheycouldbeusedasatext.Theynowappear(inmodi¯edform)asthe¯rst¯vechaptersofthisbook.Hereweresomeofmymotivesatthetime.1)Tohavesomethingasshortandinexpensiveaspossible.Inmyexperience,studentslikeshortbooks.2)T
4、oavoidallinnovation.Toorganizethematerialinthemostsimple-mindedstraightforwardmanner.3)Toorderthemateriallinearly.Totheextentpossible,eachsectionshouldusetheprevioussectionsandbeusedinthefollowingsections.4)Toomitasmanytopicsaspossible.Thisisafoundationalcourse,notatopics
5、course.Ifatopicisnotusedlater,itshouldnotbeincluded.Therearethreegoodreasonsforthis.First,linearalgebrahastoppriority.Itisbettertogoforwardanddomorelinearalgebrathantostopanddomoregroupandringtheory.Second,itismoreimportantthatstudentslearntoorganizeandwriteproofsthemselv
6、esthantocovermoresubjectmatter.Algebraisaperfectplacetogetstartedbecausetherearemanyeasy"theoremstoprove.Therearemanyroutinetheoremsstatedherewithoutproofs,andtheymaybeconsideredasexercisesforthestudents.Third,thematerialshouldbesofundamentalthatitbeappropriateforstudent
7、sinthephysicalsciencesandincomputerscience.Zillionsofstudentstakecalculusandcookbooklinearalgebra,butfewtakeabstractalgebracourses.Somethingiswronghere,andonethingwrongisthatthecoursestrytodotoomuchgroupandringtheoryandnotenoughmatrixtheoryandlinearalgebra.5)Too®eranalter
8、nativeforcomputersciencemajorstothestandarddiscretemathematicscourses.Mostofthematerialinthe¯rst
