Linear Algebra and Its Applications

About the Authors

Preface

A Note to Students

Chapter 1 Linear Equations in LinearAlgebra

• 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 Transformation
• 1.10 Linear Models in Business,Science, and Engineering
• Projects
• 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 Partitioned Matrices
• 2.5 Matrix Factorizations
• 2.6 The Leontief Input—Output Model
• 2.7 Applications to Computer Graphics
• 2.8 Subspaces of ?ⁿ
• 2.9 Dimension and Rank
• Projects
• Supplementary Exercises

Chapter 3 Determinants

• Introductory Example: Random Paths and Distortion
• 3.1 Introduction to Determinants
• 3.2 Properties of Determinants
• 3.3 Cramer's Rule, Volume, and Linear Transformations
• Projects
• Supplementary Exercises

Chapter 4 Vector Spaces

• Introductory Example: Space Flightand Control Systems
• 4.1 Vector Spaces and Subspaces
• 4.2 Null Spaces, 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 Change of Basis
• 4.7 Digital Signal Processing
• 4.8 Applications to Difference Equations
• Projects
• Supplementary Exercises

Chapter 5 Eigenvalues and Eigenvectors

• Introductory Example: Dynamical Systems and Spotted Owls
• 5.1 Eigenvectors and Eigenvalues
• 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
• 5.9 Markov Chains
• Projects
• Supplementary Exercises

Chapter 6 Orthogonality and Least Squares

• Introductory Example: Artificial Intelligence and Machine Learning
• 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 Machine Learning and LinearModels
• 6.7 Inner Product Spaces
• 6.8 Applications of Inner Product Spaces
• Projects
• Supplementary Exercises

Chapter 7 Symmetric Matrices and Quadratic Forms

• Introductory Example: Multichannel Image Processing
• 7.1 Diagonalization of Symmetric Matrices
• 7.2 Quadratic Forms
• 7.3 Constrained Optimization
• 7.4 The Singular Value Decomposition
• 7.5 Applications to ImageProcessing and Statistics
• Projects
• Supplementary Exercises

Chapter 8 The Geometry of Vector Spaces

• Introductory Example: The Platonic Solids
• 8.1 Affine Combinations
• 8.2 Affine Independence
• 8.3 Convex Combinations
• 8.4 Hyperplanes
• 8.5 Polytopes
• 8.6 Curves and Surfaces
• Projects
• Supplementary Exercises

Chapter 9 Optimization

• Introductory Example: The Berlin Airlift
• 9.1 Matrix Games
• 9.2 Linear Programming-Geometric Method
• 9.3 Linear Programming-Simplex Method
• 9.4 Duality
• Projects
• Supplementary Exercises

Chapter 10 Finite-State Markov Chains(Online Only)

• Introductory Example: Googling Markov Chains
• 10.1 Introduction and Examples
• 10.2 The Steady-State Vector andGoogle's PageRank
• 10.3 Communication Classes
• 10.4 Classification of States andPeriodicity
• 10.5 The Fundamental Matrix
• 10.6 Markov Chains and BaseballStatistics

Appendixes

1. Uniqueness of the Reduced Echelon Form
2. Complex Numbers

Credits

Glossary

Answers to Odd-Numbered Exercises

Index