Linear algebra is relatively easy for students during the early stages of the course, when the material is presented in a familiar, concrete setting. But when abstract concepts are introduced, students often hit a brick wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations), are not easily understood, and require time to assimilate. Since they are fundamental to the study of linear algebra, students' understanding of these concepts is vital to their mastery of the subject. Lay introduces these concepts early in a familiar, concrete Rn setting, develops them gradually, and returns to them again and again throughout the text so that when discussed in the abstract, these concepts are more accessible.'

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