Linear Algebra and Its Applications with CD-ROM, Update, 3rd Edition

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 Transformation

1.10                 Linear 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                   Partitioned Matrices

2.5                   Matrix Factorizations

2.6                   The Leontief Input=Output Model

2.7                   Applications to Computer Graphics

2.8                   Subspaces of R^n

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 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                   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 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

        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 Matrices

7.2                   Quadratic Forms

7.3                   Constrained Optimization

7.4                   The Singular Value Decomposition

7.5                   Applications to Image Processing and Statistics

        Supplementary Exercises

 

ONLINE ONLY 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

        Supplementary Exercises

 

ONLINE ONLY 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

        Supplementary Exercises

 

 

 

Appendices

 

A                    Uniqueness of the Reduced Echelon Form

B                    Complex Numbers

 

Glossary

 

Answers to Odd-Numbered Exercises

 

Index