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Linear Algebra for the Young Mathematician
 
Steven H. Weintraub :  Lehigh University, Bethlehem, PA
Linear Algebra for the Young Mathematician
Hardcover ISBN:  978-1-4704-5084-7
Product Code:  AMSTEXT/42
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-5378-7
Product Code:  AMSTEXT/42.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-1-4704-5084-7
eBook: ISBN:  978-1-4704-5378-7
Product Code:  AMSTEXT/42.B
List Price: $174.00 $131.50
MAA Member Price: $156.60 $118.35
AMS Member Price: $139.20 $105.20
Add to cart
Linear Algebra for the Young Mathematician
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Linear Algebra for the Young Mathematician
Steven H. Weintraub :  Lehigh University, Bethlehem, PA
Hardcover ISBN:  978-1-4704-5084-7
Product Code:  AMSTEXT/42
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-5378-7
Product Code:  AMSTEXT/42.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-1-4704-5084-7
eBook ISBN:  978-1-4704-5378-7
Product Code:  AMSTEXT/42.B
List Price: $174.00 $131.50
MAA Member Price: $156.60 $118.35
AMS Member Price: $139.20 $105.20
Add to cart
  • Book Details
     
     
    Pure and Applied Undergraduate Texts
    Volume: 42; 2019; 389 pp
    MSC: Primary 15

    Linear Algebra for the Young Mathematician is a careful, thorough, and rigorous introduction to linear algebra. It adopts a conceptual point of view, focusing on the notions of vector spaces and linear transformations, and it takes pains to provide proofs that bring out the essential ideas of the subject. It begins at the beginning, assuming no prior knowledge of the subject, but goes quite far, and it includes many topics not usually treated in introductory linear algebra texts, such as Jordan canonical form and the spectral theorem. While it concentrates on the finite-dimensional case, it treats the infinite-dimensional case as well. The book illustrates the centrality of linear algebra by providing numerous examples of its application within mathematics. It contains a wide variety of both conceptual and computational exercises at all levels, from the relatively straightforward to the quite challenging.

    Readers of this book will not only come away with the knowledge that the results of linear algebra are true, but also with a deep understanding of why they are true.
    Readership

    Undergraduate students interested in learning and teaching linear algebra.
  • Table of Contents
     
     
    • Cover
    • Title page
    • Preface
    • Part I . Vector spaces
    • Chapter 1. The basics
    • 1.1. The vector space Fⁿ
    • 1.2. Linear combinations
    • 1.3. Matrices and the equation 𝐴𝑥=𝑏
    • 1.4. The basic counting theorem
    • 1.5. Matrices and linear transformations
    • 1.6. Exercises
    • Chapter 2. Systems of linear equations
    • 2.1. The geometry of linear systems
    • 2.2. Solving systems of equations—setting up
    • 2.3. Solving linear systems—echelon forms
    • 2.4. Solving systems of equations—the reduction process
    • 2.5. Drawing some consequences
    • 2.6. Exercises
    • Chapter 3. Vector spaces
    • 3.1. The notion of a vector space
    • 3.2. Linear combinations
    • 3.3. Bases and dimension
    • 3.4. Subspaces
    • 3.5. Affine subspaces and quotient vector spaces
    • 3.6. Exercises
    • Chapter 4. Linear transformations
    • 4.1. Linear transformations I
    • 4.2. Matrix algebra
    • 4.3. Linear transformations II
    • 4.4. Matrix inversion
    • 4.5. Looking back at calculus
    • 4.6. Exercises
    • Chapter 5. More on vector spaces and linear transformations
    • 5.1. Subspaces and linear transformations
    • 5.2. Dimension counting and applications
    • 5.3. Bases and coordinates: vectors
    • 5.4. Bases and matrices: linear transformations
    • 5.5. The dual of a vector space
    • 5.6. The dual of a linear transformation
    • 5.7. Exercises
    • Chapter 6. The determinant
    • 6.1. Volume functions
    • 6.2. Existence, uniqueness, and properties of the determinant
    • 6.3. A formula for the determinant
    • 6.4. Practical evaluation of determinants
    • 6.5. The classical adjoint and Cramer’s rule
    • 6.6. Jacobians
    • 6.7. Exercises
    • Chapter 7. The structure of a linear transformation
    • 7.1. Eigenvalues, eigenvectors, and generalized eigenvectors
    • 7.2. Polynomials in cT
    • 7.3. Application to differential equations
    • 7.4. Diagonalizable linear transformations
    • 7.5. Structural results
    • 7.6. Exercises
    • Chapter 8. Jordan canonical form
    • 8.1. Chains, Jordan blocks, and the (labelled) eigenstructure picture of cT
    • 8.2. Proof that cT has a Jordan canonical form
    • 8.3. An algorithm for Jordan canonical form and a Jordan basis
    • 8.4. Application to systems of first-order differential equations
    • 8.5. Further results
    • 8.6. Exercises
    • Part II . Vector spaces with additional structure
    • Chapter 9. Forms on vector spaces
    • 9.1. Forms in general
    • 9.2. Usual types of forms
    • 9.3. Classifying forms I
    • 9.4. Classifying forms II
    • 9.5. The adjoint of a linear transformation
    • 9.6. Applications to algebra and calculus
    • 9.7. Exercises
    • Chapter 10. Inner product spaces
    • 10.1. Definition, examples, and basic properties
    • 10.2. Subspaces, complements, and bases
    • 10.3. Two applications: symmetric and Hermitian forms, and the singular value decomposition
    • 10.4. Adjoints, normal linear transformations, and the spectral theorem
    • 10.5. Exercises
    • Appendix A. Fields
    • A.1. The notion of a field
    • A.2. Fields as vector spaces
    • Appendix B. Polynomials
    • B.1. Statement of results
    • B.2. Proof of results
    • Appendix C. Normed vector spaces and questions of analysis
    • C.1. Spaces of sequences
    • C.2. Spaces of functions
    • Appendix D. A guide to further reading
    • Index
    • Back Cover
  • Additional Material
     
     
  • Reviews
     
     

    • I enjoyed this book. It contains as clear an exposition of the JCF as I've seen anywhere, many applications which illustrate how transparent certain facts from calculus and differential equations are when viewed in the context of linear algebra and is very well-written.
      Benjamin Linowitz, Oberlin College, MAA Reviews
  • 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
Pure and Applied Undergraduate Texts
Volume: 42; 2019; 389 pp
MSC: Primary 15

Linear Algebra for the Young Mathematician is a careful, thorough, and rigorous introduction to linear algebra. It adopts a conceptual point of view, focusing on the notions of vector spaces and linear transformations, and it takes pains to provide proofs that bring out the essential ideas of the subject. It begins at the beginning, assuming no prior knowledge of the subject, but goes quite far, and it includes many topics not usually treated in introductory linear algebra texts, such as Jordan canonical form and the spectral theorem. While it concentrates on the finite-dimensional case, it treats the infinite-dimensional case as well. The book illustrates the centrality of linear algebra by providing numerous examples of its application within mathematics. It contains a wide variety of both conceptual and computational exercises at all levels, from the relatively straightforward to the quite challenging.

Readers of this book will not only come away with the knowledge that the results of linear algebra are true, but also with a deep understanding of why they are true.
Readership

Undergraduate students interested in learning and teaching linear algebra.
  • Cover
  • Title page
  • Preface
  • Part I . Vector spaces
  • Chapter 1. The basics
  • 1.1. The vector space Fⁿ
  • 1.2. Linear combinations
  • 1.3. Matrices and the equation 𝐴𝑥=𝑏
  • 1.4. The basic counting theorem
  • 1.5. Matrices and linear transformations
  • 1.6. Exercises
  • Chapter 2. Systems of linear equations
  • 2.1. The geometry of linear systems
  • 2.2. Solving systems of equations—setting up
  • 2.3. Solving linear systems—echelon forms
  • 2.4. Solving systems of equations—the reduction process
  • 2.5. Drawing some consequences
  • 2.6. Exercises
  • Chapter 3. Vector spaces
  • 3.1. The notion of a vector space
  • 3.2. Linear combinations
  • 3.3. Bases and dimension
  • 3.4. Subspaces
  • 3.5. Affine subspaces and quotient vector spaces
  • 3.6. Exercises
  • Chapter 4. Linear transformations
  • 4.1. Linear transformations I
  • 4.2. Matrix algebra
  • 4.3. Linear transformations II
  • 4.4. Matrix inversion
  • 4.5. Looking back at calculus
  • 4.6. Exercises
  • Chapter 5. More on vector spaces and linear transformations
  • 5.1. Subspaces and linear transformations
  • 5.2. Dimension counting and applications
  • 5.3. Bases and coordinates: vectors
  • 5.4. Bases and matrices: linear transformations
  • 5.5. The dual of a vector space
  • 5.6. The dual of a linear transformation
  • 5.7. Exercises
  • Chapter 6. The determinant
  • 6.1. Volume functions
  • 6.2. Existence, uniqueness, and properties of the determinant
  • 6.3. A formula for the determinant
  • 6.4. Practical evaluation of determinants
  • 6.5. The classical adjoint and Cramer’s rule
  • 6.6. Jacobians
  • 6.7. Exercises
  • Chapter 7. The structure of a linear transformation
  • 7.1. Eigenvalues, eigenvectors, and generalized eigenvectors
  • 7.2. Polynomials in cT
  • 7.3. Application to differential equations
  • 7.4. Diagonalizable linear transformations
  • 7.5. Structural results
  • 7.6. Exercises
  • Chapter 8. Jordan canonical form
  • 8.1. Chains, Jordan blocks, and the (labelled) eigenstructure picture of cT
  • 8.2. Proof that cT has a Jordan canonical form
  • 8.3. An algorithm for Jordan canonical form and a Jordan basis
  • 8.4. Application to systems of first-order differential equations
  • 8.5. Further results
  • 8.6. Exercises
  • Part II . Vector spaces with additional structure
  • Chapter 9. Forms on vector spaces
  • 9.1. Forms in general
  • 9.2. Usual types of forms
  • 9.3. Classifying forms I
  • 9.4. Classifying forms II
  • 9.5. The adjoint of a linear transformation
  • 9.6. Applications to algebra and calculus
  • 9.7. Exercises
  • Chapter 10. Inner product spaces
  • 10.1. Definition, examples, and basic properties
  • 10.2. Subspaces, complements, and bases
  • 10.3. Two applications: symmetric and Hermitian forms, and the singular value decomposition
  • 10.4. Adjoints, normal linear transformations, and the spectral theorem
  • 10.5. Exercises
  • Appendix A. Fields
  • A.1. The notion of a field
  • A.2. Fields as vector spaces
  • Appendix B. Polynomials
  • B.1. Statement of results
  • B.2. Proof of results
  • Appendix C. Normed vector spaces and questions of analysis
  • C.1. Spaces of sequences
  • C.2. Spaces of functions
  • Appendix D. A guide to further reading
  • Index
  • Back Cover

  • I enjoyed this book. It contains as clear an exposition of the JCF as I've seen anywhere, many applications which illustrate how transparent certain facts from calculus and differential equations are when viewed in the context of linear algebra and is very well-written.
    Benjamin Linowitz, Oberlin College, MAA Reviews
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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