Updated and revised to increase clarity and further improve student learning, the Eighth Edition of Gareth Williams classic text is designed for the introductory course in linear algebra. It provides a flexible blend of theory and engaging applications for students within engineering, science, mathematics, business management, and physics. It is organized into three parts that contain core and optional sections. There is then ample time for the instructor to select the material that gives the course the desired flavor. Part 1 introduces the basics, presenting systems of linear equations, vectors and subspaces of Rn, matrices, linear transformations, determinants, and eigenvectors. Part 2 builds on the material presented in Part1 and goes on to introduce the concepts of general vector spaces, discussing properties of bases, developing the rank/nullity theorem, and introducing spaces of matrices and functions. Part 3 completes the course with important ideas and methods of numerical linear algebra, such as ill-conditioning, pivoting, and LU decomposition. Throughout the text the author takes care to fully and clearly develop the mathematical concepts and provide modern applications to reinforce those concepts. The applications range from theoretical applications within differential equations and least square analysis, to practical applications in fields such as archeology, demography, electrical engineering and more. New exercises can be found throughout that tie back to the modern examples in the text. Key Features of the Eighth Edition: • Updated and revised throughout with new section material and exercises. • Each section begins with a motivating introduction, which ties material to the previously learned topics. • Carefully explained examples illustrate key concepts throughout the text. • Includes such new topics such as QR Factorization and Singular Value Decomposition. • Includes new applications such as a Leslie Matrix model that is used to predict birth and death patterns of animals. • Includes discussions of the role of linear algebra in many areas, such as the operation of the search engine Google and the global structure of the worldwide air transportation network. • A MATLAB manual that ties into the regular course material is included as an appendix. These ideas can be implemented on any matrix algebra software package. This manual consists of 28 sections that tie into the regular course material. • Graphing Calculator Manual included as an appendix. • A Student Solutions Manual that contains solutions to selected exercises is available as a supplement. An Instructors Complete Solutions Manual, test bank, and PowerPoint Lecture Outlines are also available. • Available with WebAssign Online Homework & Assessment
Author(s): Gareth Williams
Edition: 8
Publisher: Jones & Bartlett Learning
Year: 2012
Language: English
Pages: 599
Cover......Page 1
Preface......Page 5
Part 1. Linear Equations, Vectors, and Matrices......Page 13
1.1 Matrices and Systems of Linear Equations......Page 15
1.2 Gauss-Jordan Elimination......Page 27
1.3 The Vector Space Rn......Page 36
1.4 Subspaces of Rn......Page 43
1.5 Basis and Dimension in Rn......Page 49
1.6 Dot Product, Norm, Angle, and Distance......Page 55
1.7 Curve Fitting, Electrical Networks, and Traffic Flow......Page 67
2.1 Addition, Scalar Multiplication, and Multiplication of Matrices......Page 79
2.2 Properties of Matrix Operations......Page 92
2.3 Symmetric Matrices and Seriation in Archaeology......Page 104
2.4 The Inverse of a Matrix and Cryptography......Page 114
2.5 Matrix Transformations, Rotations, and Dilations......Page 127
2.6 Linear Transformations, Graphics, and Fractals......Page 139
2.7 The Leontief Input-Output Model in Economics......Page 149
2.8 Markov Chains, Population Movements, and Genetics......Page 155
2.9 A Communication Model and Group Relationships in Sociology......Page 161
3.1 Introduction to Determinants......Page 175
3.2 Properties of Determinants......Page 183
3.3 Determinants, Matrix Inverses, and Systems of Linear Equations......Page 192
3.4 Eigenvalues and Eigenvectors......Page 200
3.5 Google, Demography, Weather Prediction, and Leslie Matrix Models......Page 207
Part 2. Vector Spaces......Page 223
4.1 General Vector Spaces and Subspaces......Page 225
4.2 Linear Combinations of Vectors......Page 234
4.3 Linear Independence of Vectors......Page 241
4.4 Properties of Bases......Page 247
4.5 Rank......Page 255
4.6 Projections, Gram-Schmidt Process, and QR Factorization......Page 264
4.7 Orthogonal Complement......Page 276
4.8 Kernel, Range, and the Rank/Nullity Theorem......Page 282
4.9 One-to-One Transformations and Inverse Transformations......Page 294
4.10 Transformations and Systems of Linear Equations......Page 299
5.1 Coordinate Vectors......Page 309
5.2 Matrix Representations of Linear Transformations......Page 317
5.3 Diagonalization of Matrices......Page 326
5.4 Quadratic Forms, Difference Equations, and Normal Modes......Page 339
6.1 Inner Product Spaces......Page 353
6.2 Non-Euclidean Geometry and Special Relativity......Page 362
6.3 Approximation of Functions and Coding Theory......Page 367
6.4 Least Squares Solutions......Page 374
Appendices......Page 389
7.1 Gaussian Elimination......Page 391
7.2 The Method of LU Decomposition......Page 397
7.3 Practical Difficulties in Solving Systems of Equations......Page 404
7.4 Iterative Methods for Solving Systems of Linear Equations......Page 413
7.5 Eigenvalues by Iteration and Connectivity of Networks......Page 417
7.6 The Singular Value Decomposition......Page 427
8.1 A Geometrical Introduction to Linear Programming......Page 441
8.2 The Simplex Method......Page 449
8.3 Geometrical Explanation of the Simplex Method......Page 455
Appendix A. Cross Product......Page 463
Appendix B. Equations of Planes and Lines in Three-Space......Page 473
Appendix C. Graphing Calculator Manual......Page 481
Appendix D. MATLAB Manual......Page 485
Answers to Selected Exercises......Page 543
Index......Page 593
Photo Credits......Page 599
