Linear Algebra and Its Applications with CD-ROM, Update

David C. Lay holds a B.A. from Aurora University (Illinois), and an M.A. and Ph.D. from the University of California at Los Angeles. Lay has been an educator and research mathematician since 1966, mostly at the University of Maryland, College Park. He has also served as a visiting professor at the University of Amsterdam, the Free University in Amsterdam, and the University of Kaiserslautern, Germany. He has over 30 research articles published in functional analysis and linear algebra. As a founding member of the NSF-sponsored Linear Algebra Curriculum Study Group, Lay has been a leader in the current movement to modernize the linear algebra curriculum. Lay is also co-author of several mathematics texts, including Introduction to Functional Analysis, with Angus E. Taylor, Calculus and Its Applications, with L.J. Goldstein and D.I. Schneider, and Linear Algebra Gems-Assets for Undergraduate Mathematics, with D. Carlson, C.R. Johnson, and A.D. Porter. A top-notch educator, Professor Lay has received four university awards for teaching excellence, including, in 1996, the title of Distinguished Scholar-Teacher of the University of Maryland. In 1994, he was given one of the Mathematical Association of America's Awards for Distinguished College or Unviersity Teaching of Mathematics. He has been elected by the university students to membership in Alpha Lambda Delta National Scholastic Honor Society and Golden Key National Honor Society. In 1989, Aurora University conferred on him the Outstanding Alumnus award. Lay is a member of the American Mathematical Society, the Canadian Mathematical Society, the International Linear Algebra Society, the Mathematical Association of America, Sigma Xi, and the Society for Industrial and Applied Mathematics. Since 1992, he has served several terms on the national board of the Association of Christians in the Mathematical Sciences.

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