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Elementary Linear Algebra: Applications Version

Book Description

Anton's Elementary Linear Algebra continues to provide a strong recourse for readers due to his sound mathematics and clear exposition. This classic treatment of linear algebra presents the fundamentals in the clearest possible way, examining basic ideas by means of computational examples and geometrical interpretation. It proceeds from familiar concepts to the unfamiliar, from the concrete to the abstract. Readers consistently praise this outstanding text for its expository style and clarity of presentation.

Table of Contents

  1. Cover Page
  2. Title Page
  3. Copyright
  4. Dedication
  5. About the Author
  6. PREFACE
  7. Contents
  8. CHAPTER 1: Systems of Linear Equations and Matrices
    1. 1.1 Introduction to Systems of Linear Equations
    2. 1.2 Gaussian Elimination
    3. 1.3 Matrices and Matrix Operations
    4. 1.4 Inverses; Algebraic Properties of Matrices
    5. 1.5 Elementary Matrices and a Method for Finding A −1
    6. 1.6 More on Linear Systems and Invertible Matrices
    7. 1.7 Diagonal, Triangular, and Symmetric Matrices
    8. 1.8 Matrix Transformations
    9. 1.9 Applications of Linear Systems
    10. 1.10 Leontief Input-Output Models
  9. CHAPTER 2: Determinants
    1. 2.1 Determinants by Cofactor Expansion
    2. 2.2 Evaluating Determinants by Row Reduction
    3. 2.3 Properties of Determinants; Cramer's Rule
  10. CHAPTER 3: Euclidean Vector Spaces
    1. 3.1 Vectors in 2-Space, 3-Space, and n -Space
    2. 3.2 Norm, Dot Product, and Distance in R n
    3. 3.3 Orthogonality
    4. 3.4 The Geometry of Linear Systems
    5. 3.5 Cross Product
  11. CHAPTER 4: General Vector Spaces
    1. 4.1 Real Vector Spaces
    2. 4.2 Subspaces
    3. 4.3 Linear Independence
    4. 4.4 Coordinates and Basis
    5. 4.5 Dimension
    6. 4.6 Change of Basis
    7. 4.7 Row Space, Column Space, and Null Space
    8. 4.8 Rank, Nullity, and the Fundamental Matrix Spaces
    9. 4.9 Basic Matrix Transformations in R 2 and R 3
    10. 4.10 Properties of Matrix Transformations
    11. 4.11 Geometry of Matrix Operators on R 2
  12. CHAPTER 5: Eigenvalues and Eigenvectors
    1. 5.1 Eigenvalues and Eigenvectors
    2. 5.2 Diagonalization
    3. 5.3 Complex Vector Spaces
    4. 5.4 Differential Equations
    5. 5.5 Dynamical Systems and Markov Chains
  13. CHAPTER 6: Inner Product Spaces
    1. 6.1 Inner Products
    2. 6.2 Angle and Orthogonality in Inner Product Spaces
    3. 6.3 Gram–Schmidt Process; QR -Decomposition
    4. 6.4 Best Approximation; Least Squares
    5. 6.5 Mathematical Modeling Using Least Squares
    6. 6.6 Function Approximation; Fourier Series
  14. CHAPTER 7: Diagonalization and Quadratic Forms
    1. 7.1 Orthogonal Matrices
    2. 7.2 Orthogonal Diagonalization
    3. 7.3 Quadratic Forms
    4. 7.4 Optimization Using Quadratic Forms
    5. 7.5 Hermitian, Unitary, and Normal Matrices
  15. CHAPTER 8: General Linear Transformations
    1. 8.1 General Linear Transformations
    2. 8.2 Compositions and Inverse Transformations
    3. 8.3 Isomorphism
    4. 8.4 Matrices for General Linear Transformations
    5. 8.5 Similarity
  16. CHAPTER 9: Numerical Methods
    1. 9.1 LU -Decompositions
    2. 9.2 The Power Method
    3. 9.3 Comparison of Procedures for Solving Linear Systems
    4. 9.4 Singular Value Decomposition
    5. 9.5 Data Compression Using Singular Value Decomposition
  17. APPENDIX A WORKING WITH PROOFS
  18. APPENDIX B COMPLEX NUMBERS
  19. ANSWERS TO EXERCISES
  20. INDEX