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Elementary Linear Algebra A Matrix Approach L. Spence A. Insel S. Friedberg Second Edition ISBN 10: 1-292-02503-4 ISBN 13: 978-1-292-02503-2 Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk © Pearson Education Limited 2014 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affi liation with or endorsement of this book by such owners. ISBN 10: 1-292-02503-4 ISBN 13: 978-1-292-02503-2 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Printed in the United States of America 122345556699295257822175193191133 P E A R S O N C U S T O M L I B R AR Y Table of Contents Chapter 1. Matrices, Vectors, and Systems of Linear Equations Lawrence E. Spence/Arnold J. Insel/Stephen H. Friedberg 1 Chapter 2. Matrices and Linear Transformations Lawrence E. Spence/Arnold J. Insel/Stephen H. Friedberg 93 Chapter 3. Determinants Lawrence E. Spence/Arnold J. Insel/Stephen H. Friedberg 197 Chapter 4. Subspaces and Their Properties Lawrence E. Spence/Arnold J. Insel/Stephen H. Friedberg 225 Chapter 5. Eigenvalues, Eigenvectors, and Diagonalization Lawrence E. Spence/Arnold J. Insel/Stephen H. Friedberg 291 Chapter 7. Vector Spaces Lawrence E. Spence/Arnold J. Insel/Stephen H. Friedberg 359 Chapter 6. Orthogonality Lawrence E. Spence/Arnold J. Insel/Stephen H. Friedberg 423 Appendices Lawrence E. Spence/Arnold J. Insel/Stephen H. Friedberg 551 Bibliography Lawrence E. Spence/Arnold J. Insel/Stephen H. Friedberg 579 Answers to Selected Exercises Lawrence E. Spence/Arnold J. Insel/Stephen H. Friedberg 581 List of Frequently Used Symbols Lawrence E. Spence/Arnold J. Insel/Stephen H. Friedberg 621 Index 623 I This page intentionally left blank 1 INTRODUCTION For computers to process digital images, whether satellite photos or x-rays, there is a need to recognize the edges of objects. Image edges, which are rapid changes or discontinuities in image intensity, reflect a boundary between dissimilar regions in an image and thus are important basic characteristicsofanimage.They oftenindi- cate the physical extent of objects in the Ideal Edge Real Edge image or a boundary between light and shadowonasinglesurfaceorotherregions ofinterest. The lowermost two figures at the left indicate the changes in image intensity of the ideal and real edges above, when moving from right to left. Weseethatrealintensitiescanchangerapidly,but not instantaneously. In principle, the edge may be foundbylookingfor verylargechangesoversmall distances. However,adigitalimageisdiscreteratherthan continuous: it is a matrix of nonnegative entries thatprovidenumericaldescriptionsoftheshadesofgrayforthepixelsinthe image,wheretheentriesvaryfrom0forawhitepixelto1forablackpixel.An analysismustbedoneusingthediscreteanalogofthederivativetomeasure therateofchangeofimageintensityintwodirections. From Chapter 1 of Elementary Linear Algebra, Second Edition. Lawrence E. Spence, Arnold J. Insel, Stephen H. Friedberg. Copyright © 2008 by Pearson Education, Inc. All rights reserved. 1 2 1 Introduction   −1 0 1 the direction and magnitude of the intensity change TheSobelmatrices,S = −2 0 2 andS = at the pixel. This vector may be thought of as the dis- 1 2   −1 0 1 creteanalogofthegradientvectorofafunctionoftwo 1 2 1 variablesstudiedincalculus.  0 0 0  provide a method for measuring Replace each of the original pixel values by the −1 −2 −1 lengths of these vectors, and choose an appropriate these intensity changes. Apply the Sobel matrices S thresholdvalue.Thefinalimage,calledthethresholded 1 and S in turn to the 3x3 subimage centered on each image,isobtainedbychangingtoblackeverypixelfor 2 pixelintheoriginalimage.Theresultsarethechangesof whichthelengthofthevectorisgreaterthanthethresh- intensitynearthepixelinthehorizontalandthevertical old value, and changing to white all the other pixels. directions, respectively. The ordered pair of numbers (Seetheimagesbelow.) thatareobtainedisavectorintheplanethatprovides Original Image Thresholded Image Notice how the edges are emphasized in the image.Likewise,arapidchangeinimageintensity,which thresholdedimage.Inregionswhereimageintensityis occursatanedgeofanobject,resultsinarelativelydark constant,thesevectorshavelengthzero,andhencethe coloredboundaryinthethresholdedimage. correspondingregionsappearwhiteinthethresholded 2 CHAPTER MATRICES, VECTORS, 1 AND SYSTEMS OF LINEAR EQUATIONS T he most common use of linear algebra is to solve systemsof linear equations, which arise in applications to such diverse disciplines as physics, biology, economics, engineering, and sociology. In this chapter, we describe the most efficientalgorithmforsolvingsystemsoflinearequations,Gaussianelimination.This algorithm, or some variation of it, is used by most mathematics software (such as MATLAB). Wecanwritesystemsoflinearequationscompactly,usingarrayscalledmatrices and vectors. More importantly, the arithmetic properties of these arrays enable us to compute solutions of such systems or to determine if no solutions exist. This chapter begins by developing the basic properties of matrices and vectors. In Sections 1.3 and 1.4, we begin our study of systems of linear equations. In Sections 1.6 and 1.7, we introduce two other important concepts of vectors, namely, generating sets and linear independence, which provide information about the existence and uniqueness of solutions of a system of linear equations. 1.1 MATRICESAND VECTORS Many types of numerical data are best displayed in two-dimensional arrays, such as tables. For example, suppose that a company owns two bookstores, each of which sells newspapers, magazines, and books. Assume that the sales (in hundreds of dollars) of thetwobookstoresforthemonthsofJulyandAugustarerepresentedbythefollowing tables: July August Store 1 2 Store 1 2 Newspapers 6 8 and Newspapers 7 9 Magazines 15 20 Magazines 18 31 Books 45 64 Books 52 68 The first column of the July table shows that store 1 sold $1500 worth of magazines and$4500worthofbooksduringJuly.WecanrepresenttheinformationonJulysales more simply as   6 8   15 20 . 45 64 3 4 CHAPTER1 Matrices,Vectors,andSystemsofLinearEquations Sucharectangulararrayofrealnumbersiscalledamatrix.1 Itiscustomarytoreferto realnumbersasscalars(originallyfromthewordscale)whenworkingwithamatrix. We denote the set of real numbers by R. Definitions Amatrix(plural,matrices)isarectangulararrayofscalars.Ifthematrix has m rows and n columns, we say that the size of the matrix is m by n, written m×n. The matrix is square if m =n. The scalar in the ith row and jth column is called the (i, j)-entry of the matrix. If A is a matrix, we denote its (i,j)-entry by a . We say that two matrices A and ij B are equal if they have the same size and have equal corresponding entries; that is, a =b for all i and j. Symbolically, we write A=B. ij ij Inourbookstoreexample,theJulyandAugustsalesarecontainedinthematrices     6 8 7 9 B =15 20 and C =18 31. 45 64 52 68 Notethatb =8andc =9,soB (cid:1)=C.BothB andC are3×2matrices.Because 12 12 of the context in which these matrices arise, they are called inventorymatrices. Other examples of matrices are   (cid:5) (cid:6) 23 −4 0 , 38, and (cid:7)−2 0 1 1(cid:8). π 1 6 4 Thefirstmatrixhassize2×3,thesecondhassize3×1,andthethirdhassize1×4. (cid:5) (cid:6) 4 2 PracticeProblem1 (cid:1) Let A= . 1 3 (a) What is the (1,2)-entry of A? (b) What is a ? (cid:1) 22 Sometimes we are interested in only a part of the information contained in a matrix. For example, suppose that we are interested in only magazine and book sales in July. Then the relevant information is contained in the last two rows of B; that is, in the matrix E defined by (cid:5) (cid:6) 15 20 E = . 45 64 E is called a submatrix of B. In general, a submatrix of a matrix M is obtained by deleting from M entire rows, entire columns, or both. It is permissible, when forming a submatrix of M, to delete none of the rows or none of the columns of M. Asanotherexample,ifwedeletethefirstrowandthesecondcolumnofB,weobtain the submatrix (cid:5) (cid:6) 15 . 45 1JamesJosephSylvester(1814–1897)coinedthetermmatrixinthe1850s. 4 1.1 MatricesandVectors 5 MATRIXSUMSANDSCALARMULTIPLICATION Matrices are more than convenient devices for storing information. Their usefulness liesintheirarithmetic.Asanexample,supposethatwewanttoknowthetotalnumbers of newspapers, magazines, and books sold by both stores during July and August. It is natural to form one matrix whose entries are the sum of the corresponding entries of the matrices B and C, namely, Store  1 2  Newspapers 13 17  . Magazines 33 51 Books 97 132 If A and B are m×n matrices, the sum of A and B, denoted by A+B, is the m×n matrixobtainedbyaddingthecorrespondingentriesofAandB;thatis,A+B is the m×n matrix whose (i,j)-entry is a +b . Notice that the matrices A and B ij ij must have the same size for their sum to be defined. Supposethatinourbookstoreexample,Julysalesweretodoubleinallcategories. Then the new matrix of July sales would be   12 16   30 40 . 90 128 We denote this matrix by 2B. Let A be an m×n matrix and c be a scalar. The scalar multiple cA is the m×n matrix whose entries are c times the corresponding entries of A; that is, cA is the m×n matrix whose (i,j)-entry is ca . Note that 1A=A. We denote the matrix ij (−1)A by −A and the matrix 0A by O. We call the m×n matrix O in which each entry is 0 the m×n zero matrix. Example1 Compute the matrices A+B, 3A, −A, and 3A+4B, where (cid:5) (cid:6) (cid:5) (cid:6) 3 4 2 −4 1 0 A= and B = . 2 −3 0 5 −6 1 Solution We have (cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6) −1 5 2 9 12 6 −3 −4 −2 A+B = , 3A= , −A= , 7 −9 1 6 −9 0 −2 3 0 and (cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6) 9 12 6 −16 4 0 −7 16 6 3A+4B = + = . 6 −9 0 20 −24 4 26 −33 4 Justaswehavedefinedadditionofmatrices,wecanalsodefinesubtraction.For anymatricesAandB ofthesamesize,wedefineA−B tobethematrix obtainedby subtracting each entry of B from the corresponding entry of A. Thus the (i,j)-entry of A−B is a −b . Notice that A−A=O for all matrices A. ij ij 5