elements of abstract and linear algebra

elements of abstract and linear algebra

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时间:2018-07-28

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1、ElementsofAbstractandLinearAlgebraE.H.ConnelliiE.H.ConnellDepartmentofMathematicsUniversityofMiamiP.O.Box249085CoralGables,Florida33124USAec@math.miami.eduMathematicalSubjectClassi cations(1991):12-01,13-01,15-01,16-01,20-01c1999E.H.ConnellMarch30,2001[http://www.math.miami.edu/ec/book/]iiiIn

2、troductionIn1965I rsttaughtanundergraduatecourseinabstractalgebra.Itwasfuntoteachbecausethematerialwasinterestingandtheclasswasoutstanding.FiveofthosestudentslaterearnedaPh.D.inmathematics.SincethenIhavetaughtthecourseaboutadozentimesfromvarioustexts.OvertheyearsIdevelopedasetoflecturenotesand

3、in1985Ihadthemtypedsotheycouldbeusedasatext.Theynowappear(inmodi edform)asthe rst vechaptersofthisbook.Hereweresomeofmymotivesatthetime.1)Tohavesomethingasshortandinexpensiveaspossible.Inmyexperience,studentslikeshortbooks.2)Toavoidallinnovation.Toorganizethematerialinthemostsimple-mindedstrai

4、ghtforwardmanner.3)Toorderthemateriallinearly.Totheextentpossible,eachsectionshouldusetheprevioussectionsandbeusedinthefollowingsections.4)Toomitasmanytopicsaspossible.Thisisafoundationalcourse,notatopicscourse.Ifatopicisnotusedlater,itshouldnotbeincluded.Therearethreegoodreasonsforthis.First,

5、linearalgebrahastoppriority.Itisbettertogoforwardanddomorelinearalgebrathantostopanddomoregroupandringtheory.Second,itismoreimportantthatstudentslearntoorganizeandwriteproofsthemselvesthantocovermoresubjectmatter.Algebraisaperfectplacetogetstartedbecausetherearemanyeasy"theoremstoprove.Therea

6、remanyroutinetheoremsstatedherewithoutproofs,andtheymaybeconsideredasexercisesforthestudents.Third,thematerialshouldbesofundamentalthatitbeappropriateforstudentsinthephysicalsciencesandincomputerscience.Zillionsofstudentstakecalculusandcookbooklinearalgebra,butfewtakeabstractalgebracourses.Som

7、ethingiswronghere,andonethingwrongisthatthecoursestrytodotoomuchgroupandringtheoryandnotenoughmatrixtheoryandlinearalgebra.5)Too eranalternativeforcomputersciencemajorstothestandarddiscretemathematicscourses.Mostofthematerialinthe rstfo

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