# Gareth Williams

A sophomore level introduction to linear algebra which provides a solid foundation in the mathematics and at the same time teaches the student how to use linear algebra. Real applications are introduced. These include an analysis of traffic flow in Jacksonville, predicting weather patterns in Tel Aviv, determining the role of Moscow on the trade routes of Russia, and understanding the behavior of time on a space voyage to Alpha Centauri. Other applications come from such fields as archaeology, demography, sociology, electrical engineering, physics and fractal geometry.

Companion Software:
The Linear Algebra with Applications Toolbox by Gareth Williams and Lisa Coulter - a set of MATLAB M-files.
Mathematica Workbook

This text is available from Jones and Bartlett Publishers , Sudbury, Massachusetts. Tel: 978-579-8218.
The Spanish version is available from McGraw-Hill Publishers

CONTENTS

Preface

1 Systems of Linear Equations

1.1 Matrices and Systems of Linear Equations.

1.2 Gauss-Jordan Elimination.

1.3 Curve Fitting, Electrical Networks and Traffic Flow.

2 Matrices

2.1 Addition, Scalar Multiplication, and Multiplication of Matrices

2.2 Properties of Matrix Operations.

2.3 Symmetric Matrices and Seriation in Archaeology.

2.4 The Inverse of a Matrix and Cryptography.

2.5 The Leontief Input-Output Model in Economics.

2.6 Markov Chains, Population Movements, and Genetics.

2.7 A Communication Model and Group Relationships in Sociology.

3 Determinants

3.1 Introduction to Determinants.

3.2 Properties of Determinants.

3.3 Numerical Evaluation of a Determinant.

3.4 Determinants, Matrix Inverses, and Systems of Linear Equations.

4 The Vector Space Rn

4.1 Introduction to Vectors.

4.2 Dot Product, Norm, Angle, and Distance.

4.3 Introduction to Linear Transformations.

4.4 Matrix Transformations, Computer Graphics, and Fractals.

5 General Vector Spaces

5.1 Vector Spaces.

5.2 Subspaces.

5.3 Linear Combinations of Vectors.

5.4 Linear Dependence and Independence.

5.5 Bases and Dimension.

5.6 Rank of a Matrix.

5.7 Orthonormal Vectors and Projections in Rn

6 Eigenvalues and Eigenvectors

6.1 Eigenvalues and Eigenvectors.

6.2 Demography and Weather Prediction.

6.3 Diagonalization of Matrices.

6.4 Quadratic Forms, Difference Equations, and Normal Modes.

7 Linear Transformations

7.1 Linear Transformations, Kernel, and Range.

7.2 Transformations and Systems of Linear Equations.

7.3 Coordinate Vectors.

7.4 Matrix Representations of Linear Transformations.

8 Inner Product Spaces

8.1 Inner Product Spaces.

8.2 Non-Euclidean Geometry and Special Relativity.

8.3 Approximation of Functions and Coding Theory.

8.4 Least-Squares Curves.

9 Numerical Techniques

9.1 Gaussian Elimination.

9.2 The Method of LU Decomposition.

9.3 Practical Difficulties in Solving Systems of Equations.

9.4 Iterative Methods for Solving Systems of Linear Equations.

9.5 Eigenvalues by Iteration. Connectivity of Networks.

10 Linear Programming

10.1 A Geometrical Introduction to Linear Programming.

10.2 The Simplex Method.

10.3 Geometrical Explanation of the Simplex Method.