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Book Description

For courses in Advanced Linear Algebra.

Illustrates the power of linear algebra through practical applications

This acclaimed theorem-proof text presents a careful treatment of the principal topics of linear algebra.

It emphasizes the symbiotic relationship between linear transformations and matrices, but states theorems in the more general infinite-dimensional case where appropriate. Applications to such areas as differential equations, economics, geometry, and physics appear throughout, and can be included at the instructor's discretion.

Table of Contents

  1. Linear Algebra
  2. Contents
  3. Preface
    1. Suggested Course Outlines
    2. Overview of Contents
    3. Differences Between the Fourth and Fifth Editions
  4. To the Student
  5. 1 Vector Spaces
    1. 1.1 Introduction
      1. Exercises
    2. 1.2 Vector Spaces
      1. Exercises
    3. 1.3 Subspaces
      1. Exercises
        1. Definition.
        2. Definition.
    4. 1.4 Linear Combinations and Systems of Linear Equations
      1. Exercises
    5. 1.5 Linear Dependence and Linear Independence
      1. Exercises
    6. 1.6 Bases and Dimension
      1. An Overview of Dimension and Its Consequences
      2. The Dimension of Subspaces
        1. Corollary.
      3. The Lagrange Interpolation Formula
      4. Exercises
    7. 1.7* Maximal Linearly Independent Subsets
      1. Exercises
    8. Index of Definitions for Chapter 1
  6. 2 Linear Transformations and Matrices
    1. 2.1 Linear Transformations, Null Spaces, and Ranges
      1. Exercises
        1. Definitions.
        2. Definitions.
    2. 2.2 The Matrix Representation of a Linear Transformation
      1. Exercises
    3. 2.3 Composition of Linear Transformations and Matrix Multiplication
      1. Applications*
      2. Exercises
    4. 2.4 Invertibility and Isomorphisms
      1. Exercises
    5. 2.5 The Change of Coordinate Matrix
      1. Exercises
    6. 2.6* Dual Spaces
      1. Exercises
    7. 2.7* Homogeneous Linear Differential Equations with Constant Coefficients
      1. Exercises
    8. Index of Definitions for Chapter 2
  7. 3 Elementary Matrix Operations and Systems of Linear Equations
    1. 3.1 Elementary Matrix Operations and Elementary Matrices
      1. Exercises
    2. 3.2 The Rank of a Matrix and Matrix Inverses
      1. The Inverse of a Matrix
        1. Definition.
      2. Exercises
    3. 3.3 Systems of Linear Equations—Theoretical Aspects Systems of Linear Equations—Theoretical Aspects
      1. An Application
      2. Exercises
    4. 3.4 Systems of Linear Equations—Computational Aspects Systems of Linear Equations—Computational Aspects
      1. Exercises
    5. Index of Definitions for Chapter 3
  8. 4 Determinants
    1. 4.1 Determinants of Order 2
      1. Exercises
    2. 4.2 Determinants of Order n
      1. Exercises
    3. 4.3 Properties of Determinants
      1. Exercises
        1. Definition.
    4. 4.4 Summary—Important Facts about Determinants Summary—Important Facts about Determinants
      1. Properties of the Determinant
      2. Exercises
    5. 4.5* A Characterization of the Determinant
      1. Exercises
    6. Index of Definitions for Chapter 4
  9. 5 Diagonalization
    1. 5.1 Eigenvalues and Eigenvectors
      1. Exercises
    2. 5.2 Diagonalizability
      1. Test for Diagonalizability
      2. Systems of Differential Equations
      3. Direct Sums*
        1. Definition.
        2. Definition.
        3. Proof.
        4. Proof.
      4. Exercises
        1. Definitions.
    3. 5.3* Matrix Limits and Markov Chains
      1. Applications*
      2. Exercises
        1. Definition.
    4. 5.4 Invariant Subspaces and the Cayley-Hamilton Theorem
      1. The Cayley-Hamilton Theorem
        1. Proof.
        2. Corollary (Cayley-Hamilton Theorem for Matrices).
        3. Proof.
      2. Invariant Subspaces and Direct Sums3
        1. Proof.
        2. Definition.
        3. Proof.
      3. Exercises
        1. Definition.
    5. Index of Definitions for Chapter 5
  10. 6 Inner Product Spaces
    1. 6.1 Inner Products and Norms
      1. Exercises
        1. Definition.
    2. 6.2 The Gram-Schmidt Orthogonalization Process and Orthogonal Complements
      1. Exercises
    3. 6.3 The Adjoint of a Linear Operator
      1. Proof.
      2. Proof.
      3. Proof.
        1. Corollary.
      4. Proof.
      5. Proof.
        1. Corollary.
      6. Proof.
      7. Least Squares Approximation
        1. Lemma 1.
        2. Proof.
        3. Lemma 2.
        4. Proof.
          1. Corollary.
        5. Minimal Solutions to Systems of Linear Equations
        6. Proof.
        7. Exercises
          1. Definition.
    4. 6.4 Normal and Self-Adjoint Operators
      1. Exercises
        1. Definitions.
    5. 6.5 Unitary and Orthogonal Operators and Their Matrices
      1. Rigid Motions*
        1. Definition.
        2. Proof.
      2. Orthogonal Operators on R2
        1. Proof.
          1. Corollary.
      3. Conic Sections
      4. Exercises
        1. Definition.
    6. 6.6 Orthogonal Projections and the Spectral Theorem
      1. Exercises
    7. 6.7* The Singular Value Decomposition and the Pseudoinverse
      1. The Polar Decomposition of a Square Matrix
        1. Proof.
      2. The Pseudoinverse
        1. Definition.
      3. The Pseudoinverse of a Matrix
      4. The Pseudoinverse and Systems of Linear Equations
        1. Proof.
        2. Proof.
      5. Exercises
    8. 6.8* Bilinear and Quadratic Forms
      1. Bilinear Forms
        1. Definition.
        2. Definitions.
        3. Proof.
        4. Definition.
        5. Proof.
          1. Corollary 1.
        6. Proof.
          1. Corollary 2.
            1. Corollary 3.
        7. Definition.
        8. Proof.
          1. Corollary.
        9. Proof.
      2. Symmetric Bilinear Forms
        1. Definition.
        2. Proof.
        3. Definition.
          1. Corollary.
        4. Proof.
        5. Proof.
        6. Proof.
          1. Corollary.
        7. Proof.
      3. Diagonalization of Symmetric Matrices
      4. Quadratic Forms
        1. Definition.
      5. Quadratic Forms on a Real Inner Product Space
        1. Proof.
          1. Corollary.
        2. Proof.
      6. The Second Derivative Test for Functions of Several Variables
        1. Proof.
      7. Sylvester’s Law of Inertia
        1. Proof.
        2. Definitions.
        3. Corollary 1 (Sylvester’s Law of Inertia for Matrices).
        4. Definitions.
          1. Corollary 2.
        5. Proof.
          1. Corollary 3.
      8. Exercises
    9. 6.9* Einstein’s Special Theory of Relativity Einstein’s Special Theory of Relativity
      1. Time Contraction
      2. Exercises
    10. 6.10* Conditioning and the Rayleigh Quotient
      1. Exercises
    11. 6.11* The Geometry of Orthogonal Operators
      1. Exercises
    12. Index of Definitions for Chapter 6
  11. 7 Canonical Forms
    1. 7.1 The Jordan Canonical Form I
      1. Exercises
    2. 7.2 The Jordan Canonical Form II
      1. Exercises
        1. Definitions.
        2. Definition.
    3. 7.3 The Minimal Polynomial
      1. Exercises
        1. Definition.
    4. 7.4* The Rational Canonical Form
      1. Uniqueness of the Rational Canonical Form
        1. Lemma 1.
        2. Lemma 2.
        3. Corollary.
        4. Definitions.
        5. Direct Sums*
        6. Proof.
        7. Proof.
      2. Exercises
    5. Index of Definitions for Chapter 7
  12. Appendices
    1. Appendix A Sets
    2. Appendix B Functions
    3. Appendix C Fields
      1. Definitions.
      2. Proof.
      3. Corollary.
      4. Proof.
      5. Proof.
      6. Corollary.
    4. Appendix D Complex Numbers
      1. Definitions.
      2. Proof.
      3. Definition.
      4. Proof.
      5. Definition.
      6. Proof.
      7. Proof.
      8. Corollary.
      9. Proof.
    5. Appendix E Polynomials
      1. Definition.
      2. Proof.
      3. Corollary 1.
      4. Proof.
      5. Corollary 2.
      6. Proof.
      7. Definition.
      8. Lemma.
      9. Proof.
      10. Proof.
      11. Definitions.
      12. Proof.
      13. Proof.
      14. Proof.
      15. Definitions.
      16. Proof.
      17. Proof.
      18. Proof.
      19. Corollary.
      20. Proof.
      21. Proof.
      22. Proof.
  13. Answers to Selected Exercises
    1. Chapter 1
      1. Section 1.1
      2. Section 1.2
      3. Section 1.3
      4. Section 1.4
      5. Section 1.5
      6. Section 1.6
      7. Section 1.7
    2. Chapter 2
      1. Section 2.1
      2. Section 2.2
      3. Section 2.3
      4. Section 2.4
      5. Section 2.5
      6. Section 2.6
      7. Section 2.7
    3. Chapter 3
      1. Section 3.1
      2. Section 3.2
      3. Section 3.3
      4. Section 3.4
    4. Chapter 4
      1. Section 4.1
      2. Section 4.2
      3. Section 4.3
      4. Section 4.4
      5. Section 4.5
    5. Chapter 5
      1. Section 5.1
      2. Section 5.2
      3. Section 5.3
      4. Section 5.4
    6. Chapter 6
      1. Section 6.1
      2. Section 6.2
      3. Section 6.3
      4. Section 6.4
      5. Section 6.5
      6. Section 6.6
      7. Section 6.7
      8. Section 6.8
      9. Section 6.9
      10. Section 6.10
      11. Section 6.11
    7. Chapter 7
      1. Section 7.1
      2. Section 7.2
      3. Section 7.3
      4. Section 7.4
  14. Index
  15. List of Symbols
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