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Linear Algebra
 
Michael E. Taylor University of North Carolina, Chapel Hill, Chapel Hill, NC
Linear Algebra
Softcover ISBN:  978-1-4704-5670-2
Product Code:  AMSTEXT/45
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
eBook ISBN:  978-1-4704-5918-5
Product Code:  AMSTEXT/45.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-5670-2
eBook: ISBN:  978-1-4704-5918-5
Product Code:  AMSTEXT/45.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
Michael E. Taylor University of North Carolina, Chapel Hill, Chapel Hill, NC
Softcover ISBN:  978-1-4704-5670-2
Product Code:  AMSTEXT/45
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
eBook ISBN:  978-1-4704-5918-5
Product Code:  AMSTEXT/45.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-5670-2
eBook ISBN:  978-1-4704-5918-5
Product Code:  AMSTEXT/45.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: 452020; 306 pp
    MSC: Primary 15;

    This text develops linear algebra with the view that it is an important gateway connecting elementary mathematics to more advanced subjects, such as advanced calculus, systems of differential equations, differential geometry, and group representations. The purpose of this book is to provide a treatment of this subject in sufficient depth to prepare the reader to tackle such further material.

    The text starts with vector spaces, over the sets of real and complex numbers, and linear transformations between such vector spaces. Later on, this setting is extended to general fields. The reader will be in a position to appreciate the early material on this more general level with minimal effort.

    Notable features of the text include a treatment of determinants, which is cleaner than one often sees, and a high degree of contact with geometry and analysis, particularly in the chapter on linear algebra on inner product spaces. In addition to studying linear algebra over general fields, the text has a chapter on linear algebra over rings. There is also a chapter on special structures, such as quaternions, Clifford algebras, and octonions.

    Readership

    Undergraduate and graduate students interested in linear algebra.

  • Table of Contents
     
     
    • Cover
    • Title Page
    • Preface
    • Some basic notation
    • Chapter 1. Vector spaces, linear transformations, and matrices
    • 1.1. Vector spaces
    • 1.2. Linear transformations and matrices
    • 1.3. Basis and dimension
    • 1.4. Matrix representation of a linear transformation
    • 1.5. Determinants and invertibility
    • 1.6. Applications of row reduction and column reduction
    • Chapter 2. Eigenvalues, eigenvectors, and generalized eigenvectors
    • 2.1. Eigenvalues and eigenvectors
    • 2.2. Generalized eigenvectors and the minimal polynomial
    • 2.3. Triangular matrices and upper triangularization
    • 2.4. The Jordan canonical form
    • Chapter 3. Linear algebra on inner product spaces
    • 3.1. Inner products and norms
    • 3.2. Norm, trace, and adjoint of a linear transformation
    • 3.3. Self-adjoint and skew-adjoint transformations
    • 3.4. Unitary and orthogonal transformations
    • 3.5. Schur’s upper triangular representation
    • 3.6. Polar decomposition and singular value decomposition
    • 3.7. The matrix exponential
    • 3.8. The discrete Fourier transform
    • Chapter 4. Further basic concepts: duality, convexity, quotients, positivity
    • 4.1. Dual spaces
    • 4.2. Convex sets
    • 4.3. Quotient spaces
    • 4.4. Positive matrices and stochastic matrices
    • Chapter 5. Multilinear algebra
    • 5.1. Multilinear mappings
    • 5.2. Tensor products
    • 5.3. Exterior algebra
    • 5.4. Isomorphism 𝑆𝑘𝑒𝑤kern .5𝑝𝑡(𝑉)≈Λ²𝑉 and the Pfaffian
    • Chapter 6. Linear algebra over more general fields
    • 6.1. Vector spaces over more general fields
    • 6.2. Rational matrices and algebraic numbers
    • Chapter 7. Rings and modules
    • 7.1. Rings and modules
    • 7.2. Modules over principal ideal domains
    • 7.3. The Jordan canonical form revisited
    • 7.4. Integer matrices and algebraic integers
    • 7.5. Noetherian rings and Noetherian modules
    • 7.6. Polynomial rings over UFDs
    • Chapter 8. Special structures in linear algebra
    • 8.1. Quaternions and matrices of quaternions
    • 8.2. Algebras
    • 8.3. Clifford algebras
    • 8.4. Octonions
    • Appendix A. Complementary results
    • A.1. The fundamental theorem of algebra
    • A.2. Averaging rotations
    • A.3. Groups
    • A.4. Finite fields and other algebraic field extensions
    • Bibliography
    • Index
    • Selected Published Titles in This Series
    • 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: 452020; 306 pp
MSC: Primary 15;

This text develops linear algebra with the view that it is an important gateway connecting elementary mathematics to more advanced subjects, such as advanced calculus, systems of differential equations, differential geometry, and group representations. The purpose of this book is to provide a treatment of this subject in sufficient depth to prepare the reader to tackle such further material.

The text starts with vector spaces, over the sets of real and complex numbers, and linear transformations between such vector spaces. Later on, this setting is extended to general fields. The reader will be in a position to appreciate the early material on this more general level with minimal effort.

Notable features of the text include a treatment of determinants, which is cleaner than one often sees, and a high degree of contact with geometry and analysis, particularly in the chapter on linear algebra on inner product spaces. In addition to studying linear algebra over general fields, the text has a chapter on linear algebra over rings. There is also a chapter on special structures, such as quaternions, Clifford algebras, and octonions.

Readership

Undergraduate and graduate students interested in linear algebra.

  • Cover
  • Title Page
  • Preface
  • Some basic notation
  • Chapter 1. Vector spaces, linear transformations, and matrices
  • 1.1. Vector spaces
  • 1.2. Linear transformations and matrices
  • 1.3. Basis and dimension
  • 1.4. Matrix representation of a linear transformation
  • 1.5. Determinants and invertibility
  • 1.6. Applications of row reduction and column reduction
  • Chapter 2. Eigenvalues, eigenvectors, and generalized eigenvectors
  • 2.1. Eigenvalues and eigenvectors
  • 2.2. Generalized eigenvectors and the minimal polynomial
  • 2.3. Triangular matrices and upper triangularization
  • 2.4. The Jordan canonical form
  • Chapter 3. Linear algebra on inner product spaces
  • 3.1. Inner products and norms
  • 3.2. Norm, trace, and adjoint of a linear transformation
  • 3.3. Self-adjoint and skew-adjoint transformations
  • 3.4. Unitary and orthogonal transformations
  • 3.5. Schur’s upper triangular representation
  • 3.6. Polar decomposition and singular value decomposition
  • 3.7. The matrix exponential
  • 3.8. The discrete Fourier transform
  • Chapter 4. Further basic concepts: duality, convexity, quotients, positivity
  • 4.1. Dual spaces
  • 4.2. Convex sets
  • 4.3. Quotient spaces
  • 4.4. Positive matrices and stochastic matrices
  • Chapter 5. Multilinear algebra
  • 5.1. Multilinear mappings
  • 5.2. Tensor products
  • 5.3. Exterior algebra
  • 5.4. Isomorphism 𝑆𝑘𝑒𝑤kern .5𝑝𝑡(𝑉)≈Λ²𝑉 and the Pfaffian
  • Chapter 6. Linear algebra over more general fields
  • 6.1. Vector spaces over more general fields
  • 6.2. Rational matrices and algebraic numbers
  • Chapter 7. Rings and modules
  • 7.1. Rings and modules
  • 7.2. Modules over principal ideal domains
  • 7.3. The Jordan canonical form revisited
  • 7.4. Integer matrices and algebraic integers
  • 7.5. Noetherian rings and Noetherian modules
  • 7.6. Polynomial rings over UFDs
  • Chapter 8. Special structures in linear algebra
  • 8.1. Quaternions and matrices of quaternions
  • 8.2. Algebras
  • 8.3. Clifford algebras
  • 8.4. Octonions
  • Appendix A. Complementary results
  • A.1. The fundamental theorem of algebra
  • A.2. Averaging rotations
  • A.3. Groups
  • A.4. Finite fields and other algebraic field extensions
  • Bibliography
  • Index
  • Selected Published Titles in This Series
  • 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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