Linear algebra : pure and applied

This is a matrix-oriented approach to linear algebra that covers the traditional material of the courses generally known as “Linear Algebra I” and “Linear Algebra II” throughout North America, but it also includes more advanced topics such as the pseudoinverse and the singular value decomposition that make it appropriate for a more advanced course as well. As is becoming increasingly the norm, the book begins with the geometry of Euclidean 3-space so that important concepts like linear combination, linear independence and span can be introduced early and in a “real” context. The book reflects the author's background as a pure mathematician — all the major definitions and theorems of basic linear algebra are covered rigorously — but the restriction of vector spaces to Euclidean n-space and linear transformations to matrices, for the most part, and the continual emphasis on the system Ax=b, make the book less abstract and more attractive to the students of today than some others. As the subtitle suggests, however, applications play an important role too. Coding theory and least squares are recurring themes. Other applications include electric circuits, Markov chains, quadratic forms and conic sections, facial recognition and computer graphics. Readership: Undergraduates in mathematics Cover 1 S Title 2 LINEAR ALGEBRA, Pure & Applied 4 Copyright@ 2014 by World Scientific Publishing 5 ISBN 978-981-4508-36-0 5 QA184.2.G67 2014 512'.5--dc23 5 LCCN 2013022750 5 Contents 6 Preface 10 Organization, Philosophy, Style 11 A Course Outline 12 Acknowledgements 13 To My Students 14 Suggested Lecture Schedule 16 Chapter 1 Euclidean n-Space 18 1.1 Vectors and Arrows 18 1.2 Length and Direction 43 1.3 lines, Planes and the Cross Product 60 1.4 Projections 83 1.5 linear Dependence and Independence 94 Review Exercises for Chapter 1 101 Chapter 2 Matrices and Linear Equations 110 2.1 The Algebra of Matrices 110 2.2 Application: Generating Codes with Matrices 138 2.3 The Inverse and Transpose of a Matrix 144 2.4 Systems of linear Equations 158 2.5 Application: Electric Circuits 186 2.6 Homogeneous Systems; More on linear Independence 194 2.7 Elementary Matrices and LU Factorization 202 2.8 IDU Factorization 224 2.9 More on the Inverse of a Matrix 233 Review Exercises for Chapter 2 245 Chapter 3 Determinants, Eigenvalues, Eigenvectors 254 3.1. The Determinant of a Matrix 254 3.2 Properties of Determinants 265 3.3 Application: Graphs 286 3.4 Eigenvalues and Eigenvectors 296 3.5 Similarity and Diagonalization 309 3.6 Application: Llnear Recurrence Relations 323 3.7 Application: Markov Chains 328 Review Exercises for Chapter 3 342 Chapter 4 Vector Spaces 348 4.1 The Theory of linear Equations 348 4.2 Subspaces 364 4.3 Basis and Dimension 378 4.4 Finite Dimensional Vector Spaces 387 4.5 One-sided Inverses 405 Review Exercises for Chapter 4 413 Chapter 5 Linear Transformations 418 5.1 Fundamentals 418 5.2 Matrix Multiplication Revisited 432 5.3 Application: Computer Graphics 440 5.4 The Matrices of a linear Transformation 445 5.5 Changing Coordinates 458 Review Exercises for Chapter 5 475 Chapter 6 Orthogonality 478 6.1 Projection Matrices 478 6.2 Application: Data Fitting 497 6.3 The Gram-Schmidt Algorithm and QR Factorization 503 6.4 Orthogonal Subspaces and Complements 520 6.5 The Pseudoinverse of a Matrix 541 Review Exercises for Chapter 6 552 Chapter 7 The Spectral Theorem 558 7.1 Complex Vectors and Matrices 558 7.2 Unitary Diagonalization 571 7.3 Real Symmetric Matrices 587 7.4 Application: Quadratic Forms, Conic Sections 593 7.5 The Singular Value Decomposition 598 Review Exercises for Chapter 7 606 Appendix A: Complex Numbers 610 Appendix B: Show and Prove 624 Appendix C: Things I Must Remember 632 Answers to True/False and BB Exercises 638 Glossary 710 Index 728 Back Cover 734 Euclidean N-space -- Matrices And Linear Equations -- Determinants, Eigenvalues, Eigenvectors -- Vector Spaces -- Linear Transformations -- Orthogonality -- The Spectral Theorem. By Edgar G Goodaire, Memorial University, Canada. Includes Bibliographical References And Index.