Linear Algebra with Applications, 9/e

Contents:

Preface

Part 1: Linear Equations, Vectors, and Matrices

Chapter 1: Linear Equations and Vectors • Matrices and Systems of Linear Equations • Gauss-Jordan Elimination • The Vector Space Rⁿ • Subspaces of Rⁿ • Basis and Dimension • Dot Product, Norm, Angle, and Distance (Option: This section can be deferred to just before Section 4.6.) • Curve Fitting, Electrical Networks, and Traffic Flow • Chapter 1 Review Exercises

Chapter 2: Matrices and Linear Transformations • Addition, Scalar Multiplication, and Multiplication of Matrices • Properties of Matrix Operations • Symmetric Matrices and Seriation in Archaeology • The Inverse of a Matrix and Cryptography • Matrix Transformations, Rotations, and Dilations • Linear Transformations, Graphics, and Fractals • The Leontief Input-Output Model in Economics • Markov Chains, Population Movements, and Genetics • A Communication Model and Group Relationships in Sociology • Chapter 2 Review Exercises

Chapter 3: Determinants and Eigenvectors • Introduction to Determinants • Properties of Determinants • Determinants, Matrix Inverses, and Systems of Linear Equations • Eigenvalues and Eigenvectors (Option: Diagonalization of Matrices, Section 5.3, may be discussed at this time.) • Google, Demography, Weather Prediction, and Leslie Matrix Models • Chapter 3 Review Exercises

Part 2: Vector Spaces

Chapter 4: General Vector Spaces • General Vector Spaces and Subspaces • Linear Combinations of Vectors • Linear Independence of Vectors • Properties of Bases • Rank • Projections, Gram-Schmidt Process, and QR Factorization • Orthogonal Complement • Kernel, Range, and the Rank/Nullity Theorem • One-to-One Transformations and Inverse Transformations • Transformations and Systems of Linear Equations • Chapter 4 Review Exercises

Chapter 5: Coordinate Representations • Coordinate Vectors • Matrix Representations of Linear Transformations • Diagonalization of Matrices • Quadratic Forms, Difference Equations, and Normal Modes • Linear Differential Equations (Calculus Prerequisite) • Chapter 5 Review Exercises

Chapter 6: Inner Product Spaces • Inner Product Spaces • Non-Euclidean Geometry and Special Relativity • Approximation of Functions and Coding Theory • Least Squares Solutions • Chapter 6 Review Exercises

Part 3: Numerical Linear Algebra

Chapter 7: Numerical Methods • Gaussian Elimination • The Method of LU Decomposition • Practical Difficulties in Solving Systems of Equations • Iterative Methods for Solving Systems of Linear Equations • Eigenvalues by Iteration and Connectivity of Networks • The Singular Value Decomposition • Chapter 7 Review Exercises

Chapter 8: Linear Programming • A Geometrical Introduction to Linear Programming • The Simplex Method • Geometrical Explanation of the Simplex Method • Chapter 8 Review Exercises

Appendices • Cross Product • Equations of Planes and Lines in Three-Space • Graphing Calculator Manual • Reduced Echelon Form of a Matrix • Matrix Operations • Powers of a Matrix • Transpose of a Matrix • Inverse of a Matrix • Determinant of a Matrix • Summary of Formats for Row Operations • MATLAB Manual • Entering and Displaying a Matrix (Section 1.1) • Solving Systems of Linear Equations (Sections 1.1, 1.2, 1.7) • Dot Product, Norm, Angle, Distance (Section 1.6) • Matrix Operations (Sections 2.1-2.3) • Computational Considerations (Section 2.2) • Inverse of a Matrix (Section 2.4) • Solving Systems of Equations Using Matrix Inverse (Section 2.4) • Cryptography (Section 2.4) • Transformations Defined by Matrices (Sections 2.5, 2.6) • Fractals (Section 2.6) • Leontief I/O Model (Section 2.7) • Markov Chains (Sections 2.8, 3.5) • Digraphs (Section 2.9) • Determinants (Sections 3.1-3.3) • Cramer's Rule (Section 3.3) • Eigenvalues and Eigenvectors (Sections 3.4, 3.5) • Linear Combinations, Dependence, Basis, Rank (Sections 1.3, 4.2-4.5) • Projection, Gram-Schmidt Orthogonalization (Section 4.6) • QR Factorization (Section 4.6) • Kernel and Range (Section 4.8) • Inner Product, Non-Euclidean Geometry (Sections 6.1, 6 2) • Space-Time Travel (Section 6.2) • Pseudoinverse and Least Squares Curves (Section 6.4) • LU Decomposition (Section 7.2) • Condition Number of a Matrix (Section 7.3) • Jacobi and Gauss-Seidel Iterative Methods (Section 7.4) • Singular Value Decomposition (Section 7.6) • The Simplex Method in Linear Programming (Section 8.2) • Cross Product (Appendix A) • MATLAB Commands, Functions, and M-Files • The Linear Algebra with Applications Toolbox M-Files

Answers to Selected Exercises

Index

About the Author:

Gareth Williams - Stetson University

Gareth Williams earned a B.S. and a Ph.D. in applied mathematics from the University of Wales, and did graduate work at Kings College University of London. He has taught mathematics at the University of Florida, the University of Denver, and Stetson University. He has been an exchange professor at the Paedagogische Hochschule, in Freiburg, Germany and has spent sabbaticals at the University of Wales. His mathematical interests include linear algebra, relativity, mathematical modeling, and the computer in mathematics education.

His publications include “Fine Topologies for Minkowski Space”, “Motions in Relativistic Space”, “Mathematics in Archaeology”, and books on Finite Mathematics, Calculus, and College Algebra. He is the co-developer (with Lisa Coulter) of the linear algebra software package, “The Linear Algebra Toolbox” for MATLAB.

Dr. Williams is a member of The Mathematics Association of America.

Gareth Williams - Stetson University

Gareth Williams earned a B.S. and a Ph.D. in applied mathematics from the University of Wales, and did graduate work at Kings College University of London. He has taught mathematics at the University of Florida, the University of Denver, and Stetson University. He has been an exchange professor at the Paedagogische Hochschule, in Freiburg, Germany and has spent sabbaticals at the University of Wales. His mathematical interests include linear algebra, relativity, mathematical modeling, and the computer in mathematics education.

His publications include “Fine Topologies for Minkowski Space”, “Motions in Relativistic Space”, “Mathematics in Archaeology”, and books on Finite Mathematics, Calculus, and College Algebra. He is the co-developer (with Lisa Coulter) of the linear algebra software package, “The Linear Algebra Toolbox” for MATLAB.
Dr. Williams is a member of The Mathematics Association of America.