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Linear Algebra: Vector Spaces and Linear Transformations
 
Meighan I. Dillon Kennesaw State University, Marietta, GA
Linear Algebra
Softcover ISBN:  978-1-4704-6986-3
Product Code:  AMSTEXT/57
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
eBook ISBN:  978-1-4704-7200-9
Product Code:  AMSTEXT/57.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-6986-3
eBook: ISBN:  978-1-4704-7200-9
Product Code:  AMSTEXT/57.B
List Price: $170.00 $127.50
MAA Member Price: $153.00 $114.75
AMS Member Price: $136.00 $102.00
Linear Algebra
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Linear Algebra: Vector Spaces and Linear Transformations
Meighan I. Dillon Kennesaw State University, Marietta, GA
Softcover ISBN:  978-1-4704-6986-3
Product Code:  AMSTEXT/57
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
eBook ISBN:  978-1-4704-7200-9
Product Code:  AMSTEXT/57.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-6986-3
eBook ISBN:  978-1-4704-7200-9
Product Code:  AMSTEXT/57.B
List Price: $170.00 $127.50
MAA Member Price: $153.00 $114.75
AMS Member Price: $136.00 $102.00
  • Book Details
     
     
    Pure and Applied Undergraduate Texts
    Volume: 572023; 367 pp
    MSC: Primary 15

    This textbook is directed towards students who are familiar with matrices and their use in solving systems of linear equations. The emphasis is on the algebra supporting the ideas that make linear algebra so important, both in theoretical and practical applications. The narrative is written to bring along students who may be new to the level of abstraction essential to a working understanding of linear algebra. The determinant is used throughout, placed in some historical perspective, and defined several different ways, including in the context of exterior algebras. The text details proof of the existence of a basis for an arbitrary vector space and addresses vector spaces over arbitrary fields. It develops LU-factorization, Jordan canonical form, and real and complex inner product spaces. It includes examples of inner product spaces of continuous complex functions on a real interval, as well as the background material that students may need in order to follow those discussions. Special classes of matrices make an entrance early in the text and subsequently appear throughout. The last chapter of the book introduces the classical groups.

    Readership

    Undergraduate and graduate students interested in linear algebra.

  • Table of Contents
     
     
    • Cover
    • Title page
    • Contents
    • List of Figures
    • Preface
    • How To Use This Book
    • Notation and Terminology
    • To the Student
    • Introduction
    • Chapter 1. Vector Spaces
    • 1.1. Fields
    • 1.2. Vector Spaces
    • 1.3. Spanning and Linear Independence
    • 1.4. Bases
    • 1.5. Polynomials
    • 1.6. ℝ and ℂ in Linear Algebra
    • Chapter 2. Linear Transformations and Subspaces
    • 2.1. Linear Transformations
    • 2.2. Cosets and Quotient Spaces
    • 2.3. Affine Sets and Mappings
    • 2.4. Isomorphism and the Rank Theorem
    • 2.5. Sums, Products, and Projections
    • Chapter 3. Matrices and Coordinates
    • 3.1. Matrices
    • 3.2. Coordinate Vectors
    • 3.3. Change of Basis
    • 3.4. Vector Spaces of Linear Transformations
    • 3.5. Equivalences
    • Chapter 4. Systems of Linear Equations
    • Introduction
    • 4.1. The Solution Set
    • 4.2. Elementary Matrices
    • 4.3. Reduced Row Echelon Form
    • 4.4. Row Equivalence
    • 4.5. An Early Use of the Determinant
    • 4.6. LU-Factorization
    • Chapter 5. Introductions
    • 5.1. Dual Spaces
    • 5.2. Transposition and Duality
    • 5.3. Bilinear Forms, Their Matrices, and Duality
    • 5.4. Linear Operators and Direct Sums
    • 5.5. Groups of Matrices
    • 5.6. Self-Adjoint and Unitary Matrices
    • Chapter 6. The Determinant Is a Multilinear Mapping
    • 6.1. Multilinear Mappings
    • 6.2. Alternating Multilinear Mappings
    • 6.3. Permutations, Part I
    • 6.4. Permutations, Part II
    • 6.5. The Determinant
    • 6.6. Properties of the Determinant
    • Chapter 7. Inner Product Spaces
    • 7.1. The Dot Product: Under the Hood
    • 7.2. Inner Products
    • 7.3. Length and Angle
    • 7.4. Orthonormal Sets
    • 7.5. Orthogonal Complements
    • 7.6. Inner Product Spaces of Functions
    • 7.7. Unitary Transformations
    • 7.8. The Adjoint of an Operator
    • 7.9. A Fundamental Theorem
    • Chapter 8. The Life of a Linear Operator
    • 8.1. Factoring Polynomials
    • 8.2. The Minimal Polynomial
    • 8.3. Eigenvalues
    • 8.4. The Characteristic Polynomial
    • 8.5. Diagonalizability
    • 8.6. Self-Adjoint Matrices Are Diagonalizable
    • 8.7. Rotations and Translations
    • Chapter 9. Similarity
    • 9.1. Triangularization
    • 9.2. The Primary Decomposition
    • 9.3. Nilpotent Operators, Part I
    • 9.4. Nilpotent Operators, Part II
    • 9.5. Jordan Canonical Form
    • Chapter 10. 𝐺𝐿_{𝑛}(𝔽) and Friends
    • 10.1. More about Groups
    • 10.2. Homomorphisms and Normal Subgroups
    • 10.3. The Quaternions
    • 10.4. The Special Linear Group
    • 10.5. The Projective Group
    • 10.6. The Orthogonal Group
    • 10.7. The Unitary Group
    • 10.8. The Symplectic Group
    • Appendix A. Background Review
    • A.1. Logic and Proof
    • A.2. Sets
    • A.3. Well-Definedness
    • A.4. Counting
    • A.5. Equivalence Relations
    • A.6. Mappings
    • A.7. Binary Operations
    • Appendix B. ℝ² and ℝ³
    • B.1. Vectors
    • B.2. The Real Plane
    • B.3. The Complex Numbers and ℝ²
    • B.4. Real 3-Space
    • B.5. The Dot Product
    • B.6. The Cross-Product
    • Appendix C. More Set Theory
    • C.1. Partially Ordered Sets
    • C.2. Zorn’s Lemma
    • Appendix D. Infinite Dimension
    • Bibliography
    • Index
    • Back Cover
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 572023; 367 pp
MSC: Primary 15

This textbook is directed towards students who are familiar with matrices and their use in solving systems of linear equations. The emphasis is on the algebra supporting the ideas that make linear algebra so important, both in theoretical and practical applications. The narrative is written to bring along students who may be new to the level of abstraction essential to a working understanding of linear algebra. The determinant is used throughout, placed in some historical perspective, and defined several different ways, including in the context of exterior algebras. The text details proof of the existence of a basis for an arbitrary vector space and addresses vector spaces over arbitrary fields. It develops LU-factorization, Jordan canonical form, and real and complex inner product spaces. It includes examples of inner product spaces of continuous complex functions on a real interval, as well as the background material that students may need in order to follow those discussions. Special classes of matrices make an entrance early in the text and subsequently appear throughout. The last chapter of the book introduces the classical groups.

Readership

Undergraduate and graduate students interested in linear algebra.

  • Cover
  • Title page
  • Contents
  • List of Figures
  • Preface
  • How To Use This Book
  • Notation and Terminology
  • To the Student
  • Introduction
  • Chapter 1. Vector Spaces
  • 1.1. Fields
  • 1.2. Vector Spaces
  • 1.3. Spanning and Linear Independence
  • 1.4. Bases
  • 1.5. Polynomials
  • 1.6. ℝ and ℂ in Linear Algebra
  • Chapter 2. Linear Transformations and Subspaces
  • 2.1. Linear Transformations
  • 2.2. Cosets and Quotient Spaces
  • 2.3. Affine Sets and Mappings
  • 2.4. Isomorphism and the Rank Theorem
  • 2.5. Sums, Products, and Projections
  • Chapter 3. Matrices and Coordinates
  • 3.1. Matrices
  • 3.2. Coordinate Vectors
  • 3.3. Change of Basis
  • 3.4. Vector Spaces of Linear Transformations
  • 3.5. Equivalences
  • Chapter 4. Systems of Linear Equations
  • Introduction
  • 4.1. The Solution Set
  • 4.2. Elementary Matrices
  • 4.3. Reduced Row Echelon Form
  • 4.4. Row Equivalence
  • 4.5. An Early Use of the Determinant
  • 4.6. LU-Factorization
  • Chapter 5. Introductions
  • 5.1. Dual Spaces
  • 5.2. Transposition and Duality
  • 5.3. Bilinear Forms, Their Matrices, and Duality
  • 5.4. Linear Operators and Direct Sums
  • 5.5. Groups of Matrices
  • 5.6. Self-Adjoint and Unitary Matrices
  • Chapter 6. The Determinant Is a Multilinear Mapping
  • 6.1. Multilinear Mappings
  • 6.2. Alternating Multilinear Mappings
  • 6.3. Permutations, Part I
  • 6.4. Permutations, Part II
  • 6.5. The Determinant
  • 6.6. Properties of the Determinant
  • Chapter 7. Inner Product Spaces
  • 7.1. The Dot Product: Under the Hood
  • 7.2. Inner Products
  • 7.3. Length and Angle
  • 7.4. Orthonormal Sets
  • 7.5. Orthogonal Complements
  • 7.6. Inner Product Spaces of Functions
  • 7.7. Unitary Transformations
  • 7.8. The Adjoint of an Operator
  • 7.9. A Fundamental Theorem
  • Chapter 8. The Life of a Linear Operator
  • 8.1. Factoring Polynomials
  • 8.2. The Minimal Polynomial
  • 8.3. Eigenvalues
  • 8.4. The Characteristic Polynomial
  • 8.5. Diagonalizability
  • 8.6. Self-Adjoint Matrices Are Diagonalizable
  • 8.7. Rotations and Translations
  • Chapter 9. Similarity
  • 9.1. Triangularization
  • 9.2. The Primary Decomposition
  • 9.3. Nilpotent Operators, Part I
  • 9.4. Nilpotent Operators, Part II
  • 9.5. Jordan Canonical Form
  • Chapter 10. 𝐺𝐿_{𝑛}(𝔽) and Friends
  • 10.1. More about Groups
  • 10.2. Homomorphisms and Normal Subgroups
  • 10.3. The Quaternions
  • 10.4. The Special Linear Group
  • 10.5. The Projective Group
  • 10.6. The Orthogonal Group
  • 10.7. The Unitary Group
  • 10.8. The Symplectic Group
  • Appendix A. Background Review
  • A.1. Logic and Proof
  • A.2. Sets
  • A.3. Well-Definedness
  • A.4. Counting
  • A.5. Equivalence Relations
  • A.6. Mappings
  • A.7. Binary Operations
  • Appendix B. ℝ² and ℝ³
  • B.1. Vectors
  • B.2. The Real Plane
  • B.3. The Complex Numbers and ℝ²
  • B.4. Real 3-Space
  • B.5. The Dot Product
  • B.6. The Cross-Product
  • Appendix C. More Set Theory
  • C.1. Partially Ordered Sets
  • C.2. Zorn’s Lemma
  • Appendix D. Infinite Dimension
  • Bibliography
  • Index
  • Back Cover
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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