Linear Algebra PDF
Preview Linear Algebra
Linear Algebra Linear Algebra is intended primarily as an undergraduate textbook but is written in such a way that it can also be a valuable resource for independent learning. The narrative of the book takes a matrix approach: the exposition is intertwined with matrices either as the main subject or as tools to explore the theory. Each chapter contains a description of its aims, a summary at the end of the chapter, exercises, and solutions. The reader is carefully guided through the theory and techniques presented which are outlined throughout in ‘How to…’ text boxes. Common mis- takes and pitfalls are also pointed out as one goes along. Features • Written to be essentially self-contained • Ideal as a primary textbook for an undergraduate course in linear algebra • Applications of the general theory which are of interest to disciplines outside of mathematics, such as engineering Linear Algebra Lina Oliveira First edition published 2022 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2022 Taylor & Francis, LLC CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and pub- lisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. 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Library of Congress Cataloging-in-Publication Data Names: Oliveira, Lina, author. Title: Linear algebra / Lina Oliveira. Description: First edition. | Boca Raton : Chapman & Hall/CRC Press, 2022. | Includes bibliographical references and index. Identifiers: LCCN 2021061942 (print) | LCCN 2021061943 (ebook) | ISBN 9781032287812 (hardback) | ISBN 9780815373315 (paperback) | ISBN 9781351243452 (ebook) Subjects: LCSH: Algebras, Linear. Classification: LCC QA184.2 .O43 2022 (print) | LCC QA184.2 (ebook) | DDC 512/.5--dc23/eng20220415 LC record available at https://lccn.loc.gov/2021061942 LC ebook record available at https://lccn.loc.gov/2021061943 ISBN: 9781032287812 (hbk) ISBN: 9780815373315 (pbk) ISBN: 9781351243452 (ebk) DOI: 10.1201/9781351243452 Typeset in CMR10 font by KnowledgeWorks Global Ltd. Publisher’s note: This book has been prepared from camera-ready copy provided by the authors. To my daughters. Contents Preface xi Symbol Description xiii Biography xv 1 Matrices 1 1.1 Real and Complex Matrices . . . . . . . . . . . . . . . . . . . 1 1.2 Matrix Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.3 Matrix Inverses . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.4 Elementary Matrices . . . . . . . . . . . . . . . . . . . . . . 37 1.4.1 LU and LDU factorisations . . . . . . . . . . . . . . . 43 1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.6 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2 Determinant 55 2.1 Axiomatic Definition . . . . . . . . . . . . . . . . . . . . . . 55 2.2 Leibniz’s Formula . . . . . . . . . . . . . . . . . . . . . . . . 66 2.3 Laplace’s Formula . . . . . . . . . . . . . . . . . . . . . . . . 70 2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.5 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3 Vector Spaces 81 3.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.2 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . 87 3.3 Bases and Dimension . . . . . . . . . . . . . . . . . . . . . . 93 3.3.1 Matrix spaces and spaces of polynomials . . . . . . . . 99 3.3.2 Existence and construction of bases . . . . . . . . . . 101 3.4 Null Space, Row Space, and Column Space . . . . . . . . . . 107 3.4.1 Ax=b . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.5 Sum and Intersection of Subspaces . . . . . . . . . . . . . . . 116 3.6 Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.8 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 vii viii Contents 4 Eigenvalues and Eigenvectors 131 4.1 Spectrum of a Matrix . . . . . . . . . . . . . . . . . . . . . . 131 4.2 Spectral Properties . . . . . . . . . . . . . . . . . . . . . . . 134 4.3 Similarity and Diagonalisation . . . . . . . . . . . . . . . . . 139 4.4 Jordan Canonical Form . . . . . . . . . . . . . . . . . . . . . 149 4.4.1 Nilpotent matrices . . . . . . . . . . . . . . . . . . . . 149 4.4.2 Generalised eigenvectors . . . . . . . . . . . . . . . . . 156 4.4.3 Jordan canonical form . . . . . . . . . . . . . . . . . . 160 4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.6 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5 Linear Transformations 175 5.1 Linear Transformations . . . . . . . . . . . . . . . . . . . . . 176 5.2 Matrix Representations . . . . . . . . . . . . . . . . . . . . . 179 5.3 Null Space and Image . . . . . . . . . . . . . . . . . . . . . . 185 5.3.1 Linear transformations T :Kn →Kk . . . . . . . . . . 185 5.3.2 Linear transformations T :U →V . . . . . . . . . . . 187 5.4 Isomorphisms and Rank-nullity Theorem . . . . . . . . . . . 189 5.5 Composition and Invertibility . . . . . . . . . . . . . . . . . 191 5.6 Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.7 Spectrum and Diagonalisation . . . . . . . . . . . . . . . . . 198 5.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5.9 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 6 Inner Product Spaces 205 6.1 Real Inner Product Spaces . . . . . . . . . . . . . . . . . . . 205 6.2 Complex Inner Product Spaces . . . . . . . . . . . . . . . . . 214 6.3 Orthogonal Sets . . . . . . . . . . . . . . . . . . . . . . . . . 218 6.3.1 Orthogonal complement . . . . . . . . . . . . . . . . . 220 6.3.2 Orthogonal projections. . . . . . . . . . . . . . . . . . 228 6.3.3 Gram–Schmidt process. . . . . . . . . . . . . . . . . . 235 6.4 Orthogonal and Unitary Diagonalisation . . . . . . . . . . . 238 6.5 Singular Value Decomposition . . . . . . . . . . . . . . . . . 245 6.6 Affine Subspaces of Rn . . . . . . . . . . . . . . . . . . . . . 249 6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 6.8 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 7 Special Matrices by Example 257 7.1 Least Squares Solutions . . . . . . . . . . . . . . . . . . . . . 257 7.2 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . 260 7.2.1 Google matrix and PageRank . . . . . . . . . . . . . . 265 7.3 Population Dynamics . . . . . . . . . . . . . . . . . . . . . . 266 7.4 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 7.5 Differential Equations . . . . . . . . . . . . . . . . . . . . . . 275 7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Contents ix 7.7 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 8 Appendix 285 8.1 Uniqueness of Reduced Row Echelon Form . . . . . . . . . . 285 8.2 Uniqueness of Determinant . . . . . . . . . . . . . . . . . . . 286 8.3 Direct Sum of Subspaces . . . . . . . . . . . . . . . . . . . . 287 9 Solutions 289 9.1 Solutions to Chapter 1 . . . . . . . . . . . . . . . . . . . . . 289 9.2 Solutions to Chapter 2 . . . . . . . . . . . . . . . . . . . . . 294 9.3 Solutions to Chapter 3 . . . . . . . . . . . . . . . . . . . . . 294 9.4 Solutions to Chapter 4 . . . . . . . . . . . . . . . . . . . . . 299 9.5 Solutions to Chapter 5 . . . . . . . . . . . . . . . . . . . . . 300 9.6 Solutions to Chapter 6 . . . . . . . . . . . . . . . . . . . . . 301 9.7 Solutions to Chapter 7 . . . . . . . . . . . . . . . . . . . . . 303 Bibliography 307 Index 309
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