Linear Algebra with Applications, 9/e

1 Linear Algebra with ApplicationsLeonNINTH edITIoNGLobAL edITIoNGLobAL edITIoNThis is a special edition of an established title widely used by colleges and universities throughout the world. Pearson published this exclusive edition for the benefit of students outside the United States and Canada. If you purchased this book within the United States or Canada you should be aware that it has been imported without the approval of the Publisher or Global EditionGLobAL edITIoNFor these Global editions, the editorial team at Pearson has collaborated with educators across the world to address a wide range of subjects and requirements, equipping students with the best possible learning tools. This Global edition preserves the cutting-edge approach and pedagogy of the original, but also features alterations, customization, and adaptation from the North American Algebra with Applications NINTH edITIoN Steven J.

2 124/12/14 1:09 PM7th January 2015A01_LEON2218_09_SE_FMpage 1 Linear Algebra with ApplicationsNinth EditionGlobal EditionSteven J. LeonUniversity of Massachusetts, DartmouthBoston Columbus Indianapolis New York San FranciscoAmsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal TorontoDelhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo7th January 2015A01_LEON2218_09_SE_FMpage 2 Editor in Chief: Deirdre LynchSenior Production Manufacturing Controller,Head of Learning Asset Acquisition,Global Editions: Trudy KimberGlobal Editions: Laura DentProcurement Manager: Carol MelvilleAcquisitions Editor: William HoffmanMedia Production Manager, Global Editions:Acquisitions Editor, Global Editions:Vikram KumarMurchana BorthakurSenior Author Support/Technology Specialist:Editorial Assistant.

3 Salena CashaJoe VetereProject Editor, Global Editions:Cover Designer: Lumina NeelakantanImage Manager: Diahanne LucasExecutive Marketing Manager: Jeff WeidenaarCover Image: MY_NEW_IMAGES\shutterstockMarketing Assistant: Brooke SmithProduction Coordination, Composition,Project Management Team Lead: Christina LepreIllustrations: IntegraProduction Project Manager: Mary SangerPearson Education LimitedEdinburgh GateHarlowEssex CM20 2 JEEnglandand Associated Companies throughout the Pearson Education Limited 2015 The right of Steven J. Leon to be identified as the author of this work has been asserted by him inaccordance with the Copyright, Designs and Patents Act adaptation from the United States edition, entitled Linear Algebra with Applications, 9thedition, ISBN 978-0-321-96221-8, by Steven J.

4 Leon, published by Pearson Education rights reserved. No part of this publication may be reproduced, stored in a retrieval system, ortransmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise,without either the prior written permission of the publisher or a license permitting restricted copying inthe United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6 10 Kirby Street,London EC1N trademarks used herein are the property of their respective owners. The use of any trademark in thistext does not vest in the author or publisher any trademark ownership rights in such trademarks, nor doesthe use of such trademarks imply any affiliation with or endorsement of this book by such 10:1-292-07059-5 ISBN 13: 978-1-292-07059-9 British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library10987654321 Typeset by Integra Software Services, Pvt.

5 Ltd. in Times and bound by Courier Westford in The United States of January 2015A01_LEON2218_09_SE_FMpage 3To the memories ofFlorence and Rudolph Leon,devoted and loving parentsand to the memories ofGene Golub, Germund Dahlquist, and Jim Wilkinson,friends, mentors, and role models7th January 2015A01_LEON2218_09_SE_FMpage 47th January 2015A01_LEON2218_09_SE_FMpage 5 ContentsPreface91 Matrices and Systems of Systems of Linear Row Echelon Matrix Matrix Elementary Partitioned Matrices86 MATLAB Exercises96 Chapter Test A True or False100 Chapter Test The Determinant of a Properties of Additional Topics and Applications117 MATLAB Exercises125 Chapter Test A True or False127 Chapter Test B1273 Vector Definition and Linear Basis and Change of Row Space and Column Space173 MATLAB Exercises181 Chapter Test A True or False182 Chapter Test B18357th January 2015A01_LEON2218_09_SE_FMpage 66 Contents4 Linear Definition and Matrix Representations of Linear Transformations

6 Similarity208 MATLAB Exercises214 Chapter Test A True or False215 Chapter Test The Scalar Product Orthogonal Least Squares Inner Product Orthonormal The Gram Schmidt Orthogonalization Orthogonal Polynomials291 MATLAB Exercises299 Chapter Test A True or False301 Chapter Test Eigenvalues and Systems of Linear Differential Hermitian The Singular Value quadratic Positive Definite Nonnegative Matrices393 MATLAB Exercises403 Chapter Test A True or False409 Chapter Test B4097 Numerical Linear Floating-Point Gaussian Elimination4207th January 2015A01_LEON2218_09_SE_FMpage Pivoting Matrix Norms and Condition Orthogonal The Eigenvalue Least Squares Problems467 MATLAB Exercises479 Chapter Test A True or False484 Chapter Test B4848 Iterative MethodsOnline Basic Iterative Methods9 Canonical FormsOnline Nilpotent The Jordan Canonical FormAppendix: MATLAB487 Bibliography499 Answers to Selected Exercises502 Index515 Online: The supplemental Chapters 8 and 9 can be downloaded from the Internet.

7 See the section of thePreface on supplementary January 2015A01_LEON2218_09_SE_FMpage 87th January 2015A01_LEON2218_09_SE_FMpage 9 PrefaceI am pleased to see the text reach its ninth edition. The continued support and enthusi-asm of the many users has been most gratifying. Linear Algebra is more exciting nowthan at almost any time in the past. Its applications continue to spread to more and morefields. Largely due to the computer revolution of the last 75 years, Linear Algebra hasrisen to a role of prominence in the mathematical curriculum rivaling that of software has also made it possible to dramatically improve the way the courseis first edition of this book was published in 1980. Many significant changeswere made for the second edition (1986), most notably the exercise sets were greatlyexpanded and the Linear transformations chapter of the book was completely of the following editions has seen significant modifications including the additionof comprehensive sets of MATLAB computer exercises, a dramatic increase in thenumber of applications, and many revisions in the various sections of the book.

8 I havebeen fortunate to have had outstanding reviewers and their suggestions have led tomany important improvements in the book. For the ninth edition we have given specialattention to Chapter 7 as it is the only chapter that has not seen major revisions in anyof the previous editions. The following is an outline of the most significant revisionsthat were made for the ninth s New in the Ninth Edition? Subsection Added to Chapter 3 Section 2 of Chapter 3 deals with the topic of subspaces. One important exampleof a subspace occurs when we find all solutions to a homogeneous system of linearequations. This type of subspace is referred to as anull space. A new subsection hasbeen added to show how the null space is also useful in finding the solution set to anonhomogeneous Linear system.

9 The subsection contains a new theorem and a newfigure that provides a geometric illustration of the theorem. Three related problemshave been added to the exercises at the end of Section Applications Added to Chapters 1, 5, 6, and 7In Chapter 1, we introduce an important application to the field of ManagementScience. Management decisions often involve making choices between a number ofalternatives. We assume that the choices are to be made with a fixed goal in mindand should be based on a set of evaluation criteria. These decisions often involvea number of human judgments that may not always be completely consistent. Theanalytic hierarchy process is a technique for rating the various alternatives basedon a chart consisting of weighted criteria and ratings that measure how well eachalternative satisfies each of the January 2015A01_LEON2218_09_SE_FMpage 1010 PrefaceIn Chapter 1, we see how to set up such a chart or decision tree for the weights and ratings have been assigned to each entry in the chart, an overallranking of the alternatives is calculated using simple matrix-vector operations.

10 InChapters 5 and 6, we revisit the application and discuss how to use advanced matrixtechniques to determine appropriate weights and ratings for the decision in Chapter 7, we present a numerical algorithm for computing the weightvectors used in the decision 1 of Chapter 7 Revised and Two Subsections AddedSection has been revised and modernized. A new subsection on IEEE floating-point representation of numbers and a second subsection on accuracy and stabilityof numerical algorithms have been added. New examples and additional exerciseson these topics are also 5 of Chapter 7 RevisedThe discussion of Householder transformations has been revised and expanded. Anew subsection has been added, which discusses the practicalities of using QR fac-torizations for solving Linear systems.