CHAPTER 1 Linear Equations in Linear Algebra

  Introductory Example: Linear Models in Economics and Engineering

  1.1 Systems of Linear Equations

  1.2 Row Reduction and Echelon Forms

  1.3 Vector Equations

  1.4 The Matrix Equation Ax = b

  1.5 Solution Sets of Linear Systems

  1.6 Applications of Linear Systems

  1.7 Linear Independence

  1.8 Introduction to Linear Transformations

  1.9 The Matrix of a Linear Transformations

  1.1 0Linear Models in Business, Science, and Engineering

  Supplementary Exercises

CHAPTER 2 Matrix Algebra

  Introductory Example: Computer Models in Aircraft Design

  2.1 Matrix Operations

  2.2 The Inverse of a Matrix

  2.3 Characterizations of Invertible Matrices

  2.4 Partioned Matrices

  2.5 Matrix Factorizations

  2.6 The Leontief Input-Output Modes

  2.7 Applications to Computer Graphics

  2.8 Subspaces of Rn

  2.9 Dimension and Rank

  Supplementary Exercises

CHAPTER 3 Determinants

  Introductory Example: Determinants in Analytic Geometry

  3.1 Introduction to Determinants

  3.2 Properties of Determinants

  3.3 Cramer’s Rule, Volume, and Linear Transformations

  Supplementary Exercises

CHAPTER 4 Vector Spaces

  Introductory Example: Space Flight and Control Systems

  4.1 Vector Spaces and Subspaces

  4.2 Null Space, Column Spaces, and Linear Transformations

  4.3 Linearly Independent Sets: Bases

  4.4 Coordinate Systems

  4.5 The Dimension of a Vector Space

  4.6 Rank

  4.7 Change of Basis

  4.8 Applications to Difference Equations

  4.9 Applications to Markov Chains

  Supplementary Exercises

CHAPTER 5 Eigenvalues and Eigenvectors

  Introductory Example: Dynamical Systems and Spotted Owls

  5.1 Eigenvectors and Eignevalues

  5.2 The Characteristic Equation

  5.3 Diagonalization

  5.4 Eigenvectors and Linear Transformations

  5.5 Complex Eigenvalues

  5.6 Discrete Dynamical Systems

  5.7 Applications to Differential Equations

  5.8 Iterative Estimates for Eigenvalues

  Supplementary Exercises

CHAPTER 6 Orthogonality and Least Squares

  Introductory Example: Readjusting the North American Datum

  6.1 Inner Product, Length, and Orthogonality

  6.2 Orthogonal Sets

  6.3 Orthogonal Projections

  6.4 The Gram-Schmidt Process

  6.5 Least-Squares Problems

  6.6 Applications to Linear Models

  6.7 Inner Product Spaces

  6.8 Applications of Inner Product Spaces

  Supplementary Exercises

CHAPTER 7 Symmetric Matrices and Quadratic Forms

  Introductory Example: Multichannel Image Processing

  7.1 Diagonalization of Symmetric Matices

  7.2 Quadratic Forms

  7.3 Constrained Optimization

  7.4 The Singular Value Deposition

  7.5 Applications to Image Processing and Statistics

  Supplementary Exercises

Appendixes

  A Uniqueness of the Reduced Echelon Form

  B Complex Numbers

Glossary

Answers to Odd-Numbered Exercises

Index