Elementary Linear Algebra
11th Edition

Elementary Linear Algebra
Howard Anton

© 2014

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Topics covered

CHAPTER 1 Systems of Linear Equations and Matrices

  1. Introduction to Systems of Linear Equations
  2. Gaussian Elimination
  3. Matrices and Matrix Operations
  4. Inverses; Algebraic Properties of Matrices
  5. Elementary Matrices and a Method for Finding −1
  6. More on Linear Systems and Invertible Matrices
  7. Diagonal, Triangular, and Symmetric Matrices
  8. Matrix Transformations
  9. Applications of Linear Systems
  10. Network Analysis (Traffic Flow)
  11. Electrical Circuits
  12. Balancing Chemical Equations
  13. Polynomial Interpolation
  14. Application: Leontief Input-Output Models

CHAPTER 2 Determinants

  1. Determinants by Cofactor Expansion
  2. Evaluating Determinants by Row Reduction
  3. Properties of Determinants; Cramer’s Rule

CHAPTER 3 Euclidean Vector Spaces

  1. Vectors in 2-Space, 3-Space, and -Space
  2. Norm, Dot Product, and Distance in
  3. Orthogonality
  4. The Geometry of Linear Systems
  5. Cross Product

CHAPTER 4 General Vector Spaces

  1. Real Vector Spaces
  2. Subspaces
  3. Linear Independence
  4. Coordinates and Basis
  5. Dimension
  6. Change of Basis
  7. Row Space, Column Space, and Null Space
  8. Rank, Nullity, and the Fundamental Matrix Spaces
  9. Basic Matrix Transformations in 2 and 3
  10. Properties of Matrix Transformations
  11. Application: Geometry of Matrix Operators on 2

CHAPTER 5 Eigenvalues and Eigenvectors

  1. Eigenvalues and Eigenvectors
  2. Diagonalisation
  3. Complex Vector Spaces
  4. Application: Differential Equations
  5. Application: Dynamical Systems and Markov Chains

CHAPTER 6 Inner Product Spaces

  1. Inner Products
  2. Angle and Orthogonality in Inner Product Spaces
  3. Gram–Schmidt Process; -Decomposition
  4. Best Approximation; Least Squares
  5. Application: Mathematical Modeling Using Least Squares
  6. Application: Function Approximation; Fourier Series

CHAPTER 7 Diagonalisation and Quadratic Forms

  1. Orthogonal Matrices
  2. Orthogonal Diagonalisation
  3. Quadratic Forms
  4. Optimisation Using Quadratic Forms
  5. Hermitian, Unitary, and Normal Matrices

CHAPTER 8 General Linear Transformations

  1. General Linear Transformation
  2. Compositions and Inverse Transformations
  3. Isomorphism
  4. Matrices for General Linear Transformations
  5. Similarity

CHAPTER 9 Numerical Methods

  1. -Decompositions
  2. The Power Method
  3. Comparison of Procedures for Solving Linear Systems
  4. Singular Value Decomposition
  5. Application: Data Compression Using Singular Value Decomposition

APPENDIX A Working with Proofs

APPENDIX B Complex Numbers

Answers to Exercises