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" Linear Algebra "
by Larry Smith.
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BL
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573576
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b402795
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| Main Entry
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Smith, Larry.
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| Title & Author
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Linear Algebra\ by Larry Smith.
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| Edition Statement
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Third Edition.
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| Publication Statement
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New York, NY :: Springer New York :: Imprint: Springer,, 1998.
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| Series Statement
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Undergraduate Texts in Mathematics,
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| ISBN
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9781461216704
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: 9781461272380
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| Contents
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1. Vectors in the Plane and in Space -- 1.1 First Steps -- 1.2 Exercises -- 2. Vector Spaces -- 2.1 Axioms for Vector Spaces -- 2.2 Cartesian (or Euclidean) Spaces -- 2.3 Some Rules for Vector Algebra -- 2.4 Exercises -- 3. Examples of Vector Spaces -- 3.1 Three Basic Examples -- 3.2 Further Examples of Vector Spaces -- 3.3 Exercises -- 4. Subspaces -- 4.1 Basic Properties of Vector Subspaces -- 4.2 Examples of Subspaces -- 4.3 Exercises -- 5. Linear Independence and Dependence -- 5.1 Basic Definitions and Examples -- 5.2 Properties of Independent and Dependent Sets -- 5.3 Exercises -- 6. Finite-Dimensional Vector Spaces and Bases -- 6.1 Finite-Dimensional Vector Spaces -- 6.2 Properties of Bases -- 6.3 Using Bases -- 6.4 Exercises -- 7. The Elements of Vector Spaces: A Summing Up -- 7.1 Numerical Examples -- 7.2 Exercises -- 8. Linear Transformations -- 8.1 Definition of Linear Transformations -- 8.2 Examples of Linear Transformations -- 8.3 Properties of Linear Transformations -- 8.4 Images and Kernels of Linear Transformations -- 8.5 Some Fundamental Constructions -- 8.6 Isomorphism of Vector Spaces -- 8.7 Exercises -- 9. Linear Transformations: Examples and Applications -- 9.1 Numerical Examples -- 9.2 Some Applications -- 9.3 Exercises -- 10. Linear Transformations and Matrices -- 10.1 Linear Transformations and Matrices in IR3 -- 10.2 Some Numerical Examples -- 10.3 Matrices and Their Algebra -- 10.4 Special Types of Matrices -- 10.5 Exercises -- 11. Representing Linear Transformations by Matrices -- 11.1 Representing a Linear Transformation by a Matrix -- 11.2 Basic Theorems -- 11.3 Change of Bases -- 11.4 Exercises -- 12. More on Representing Linear Transformations by Matrices -- 12.1 Projections -- 12.2 Nilpotent Transformations -- 12.3 Cyclic Transformations -- 12.4 Exercises -- 13. Systems of Linear Equations -- 13.1 Existence Theorems -- 13.2 Reduction to Echelon Form -- 13.3 The Simplex Method -- 13.4 Exercises -- 14. The Elements of Eigenvalue and Eigenvector Theory -- 14.1 The Rank of an Endomorphism -- 14.2 Eigenvalues and Eigenvectors -- 14.3 Determinants -- 14.4 The Characteristic Polynomial -- 14.5 Diagonalization Theorems -- 14.6 Exercises -- 15. Inner Product Spaces -- 15.1 Scalar Products -- 15.2 Inner Product Spaces -- 15.3 Isometries -- 15.4 The Riesz Representation Theorem -- 15.5 Legendre Polynomials -- 15.6 Exercises -- 16. The Spectral Theorem and Quadratic Forms -- 16.1 Self-Adjoint Transformations -- 16.2 The Spectral Theorem -- 16.3 The Principal Axis Theorem for Quadratic Forms -- 16.4 A Proof of the Spectral Theorem in the General Case -- 16.5 Exercises -- 17. Jordan Canonical Form -- 17.1 Invariant Subspaces -- 17.2 Nilpotent Transformations -- 17.3 The Jordan Normal Form -- 17.4 Square Roots -- 17.5 The Hamilton-Cayley Theorem -- 17.6 Inverses -- 17.7 Exercises -- 18. Application to Differential Equations -- 18.1 Linear Differential Systems: Basic Definitions -- 18.2 Diagonalizable Systems -- 18.3 Application of Jordan Form -- 18.4 Exercises -- 19. The Similarity Problem -- 19.1 The Fundamental Problem of Linear Algebra -- 19.2 A Bit of Invariant Theory -- 19.3 Exercises -- A. Multilinear Algebra and Determinants -- A.1 Multilinear Forms -- A.2 Determinants -- A.3 Exercises -- B. Complex Numbers -- B.1 The Complex Numbers -- B.2 Exercises -- Font Usage -- Notations.
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| Abstract
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Thistextwasoriginallywrittenforaonesemestercourseinlinearal gebraatthe(U. S. )sophomoreundergraduatelevel,preferablydirectly followingaonevariablecalculuscourse,sothatlinearalgebracould beusedinacourseonmultidimensionalcalculusand/ordifferential equations. Studentsatthislevelgenerallyhavehadlittlecontactwith complexnumbersorabstractmathematics,sothebookdealsalmostex clusivelywithrealfinite-dimensionalvectorspaces,butinasettingand formulationthatpermitseasygeneralizationtoabstractvectorspaces. Theparallelcomplextheoryisdevelopedinpartintheexercises. Thegoalofthefirsttwoeditionswastheprincipalaxistheoremfor realsymmetriclineartransformations. Twentyyearsofteachingin Germany,wherelinearalgebraisaoneyearcoursetakeninthefirst yearofstudyattheuniversity,hasmodifiedthatgoal. Theprincipal axistheorembecomesthefirstoftwogoals,andtobeachievedas originallyplannedinonesemester,amoreorlessdirectpathisfollowed toitsproof. Asaconsequencetherearemanysubjectsthatarenot developed,andthisisintentional:thisisonlyanintroductiontolinear algebra. Ascompensation,awideselectionofexamplesofvectorspaces andlineartransformationsispresented,toserveasatestingground forthetheory. Studentswithaneedtolearnmorelinearalgebra candosoinacourseinabstractalgebra,whichistheappropriate setting. Throughthisbooktheywillbetakenonanexcursiontothe algebraic/analyticzoo,andintroducedtosomeoftheanimalsforthe firsttime. Furtherexcursionscanteachthemmoreaboutthecurious habitsofsomeoftheseremarkablecreatures. InthesecondeditionofthebookIadded,amongotherthings,asafari intothewildernessofcanonicalforms,wherethehardystudentcould vii viii Preface pursuetheJordanform,whichhasbecomethesecondgoalofthisbook, withthetoolsdevelopedintheprecedingchapters. Inthisedition Ihaveaddedthetipoftheicebergofinvarianttheorytoshowthat linearalgebraaloneisnotcapableofsolvingthesecanonicalforms problems,eveninthesimplestcaseof2x2complexmatrices. Gottingen,Germany,February1998 LarrySmitfi Contents Preface vii 1. Vectors in the Plane and in Space 1 1. 1FirstSteps 1 1. 2Exercises 12 2. Vector Spaces 15 2. 1AxiomsforVectorSpaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2. 2Cartesian(orEuclidean)Spaces. . . . . . . . . . . . . . . . . . . . . . . 18 2. 3SomeRulesforVectorAlgebra 21 2. 4Exercises 22 3. ExamplesofVector Spaces 25 3. 1ThreeBasicExamples 25 3. 2FurtherExamplesofVectorSpaces. . . . . . . . . . . . . . . . . . . 27 3. 3Exercises . . . . . . . . . . 30 4. Subspaces 35 4. 1BasicPropertiesofVectorSubspaces 35 4. 2ExamplesofSubspaces 41 4. 3Exercises 42 5. Linear Independence and Dependence 47 5. 1BasicDefinitionsandExamples. . . . . . . . . . . . . . . . . . . . . . . 47 5. 2PropertiesofIndependentandDependentSets 50 5. 3Exercises 53 ix x Contents 6. Finite-Dimensional Vector Spaces and Bases 57 6. 1Finite-DimensionalVectorSpaces. . . . . . . . . . . . . . . . . . . . . 57 6. 2PropertiesofBases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6. 3UsingBases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6. 4Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 7. The Elements ofVector Spaces: A SummingUp 75 7. 1NumericalExamples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7. 2Exercises 82 8. Linear Transformations 85 8. 1DefinitionofLinearTransformations 85 8. 2ExamplesofLinearTransformations. . . . . . . . . . . . . . . . . . 89 8. 3PropertiesofLinearTransformations 91 8. 4ImagesandKernelsofLinearTransformations 94 8. 5SomeFundamentalConstructions 98 8. 6IsomorphismofVectorSpaces 102 8. 7Exercises 109 9. LinearTransformations: Examples andApplications 113 9. 1NumericalExamples 113 9. 2SomeApplications 123 9. 3Exercises 124 10. LinearTransformations and Matrices 129 3 10. 1LinearTransformationsandMatricesinm. 129 10. 2SomeNumericalExamples 134 10. 3MatricesandTheirAlgebra 136 10. 4SpecialTypesofMatrices 141 10. 5Exercises 151 11. Representing LinearTransformations by Matrices 159 11. 1RepresentingaLinearTransformationbyaMatrix. . 159 11. 2BasicTheorems 165 11. 3ChangeofBases 174 11. 4Exercises 178 12.
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Mathematics.
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Matrix theory.
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SpringerLink (Online service)
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