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A Course in Linear Algebra with Applications Solutions to the Exercises This page is intentionally left blank B O U R SE IN LINEAR ALGEBRA WITH APPLICATIONS SOLUTIONS J> TO THE EXERCISES Derek J S Robinson Department of Mathematics University of Illinois at Urbana-Champaign USA World Scientific Singapore • New Jersey • London • Hong Kong Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. First published 1991 Reprinted 1995, 1999 A COURSE IN LINEAR ALGEBRA WITH APPLICATIONS SOLUTIONS TO THE EXERCISES Copyright © 1991 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-02-1048-5 (pbk) Printed in Singapore. V FOREWORD This solution manual contains complete solutions to all the problems in the book "A Course in Linear Algebra with Applications". It is hoped that the manual will prove useful to lecturer and student alike. In particular it should be an effective tool for self-study when used properly, as a check on the correctness of the reader's solution, or to supply a hint in a case where, after several attempts, no solution has been found. All page, theorem, example, and exercise numbers refer to the book. Derek Robinson, Urbana, Illinois, U.S.A. July, 1992. July, 1992. This page is intentionally left blank Vll CONTENTS Foreword v Chapter One: Matrix Algebra 1 Exercises 1.1 1 Exercises 1.2 3 Exercises 1.3 14 Chapter Two: Systems of Linear Equations 17 Exercises 2.1 17 Exercises 2.2 22 Exercises 2.3 25 Chapter Three: Determinants 33 Exercises 3.1 33 Exercises 3.2 40 Exercises 3.3 46 Chapter Four: Introduction to Vector Spaces 51 Exercises 4.1 51 Exercises 4.2 52 Exercises 4.3 57 Chapter Five: Basis and Dimension 63 Exercises 5.1 63 Exercises 5.2 69 Exercises 5.3 74 Chapter Six: Linear Transformations 83 Exercises 6.1 83 Exercises 6.2 86 Exercises 6.3 94 Chapter Seven: Orthogonality in Vector Spaces 101 Exercises 7.1 101 Exercises 7.2 111 Exercises 7.3 117 Exercises 7.4 127 viii Contents Chapter Eight: Eigenvectors and Eigenvalues 135 Exercises 8.1 135 Exercises 8.2 143 Exercises 8.3 154 Chapter Nine: Some More Advanced Topics 167 Exercises 9.1 167 Exercises 9.2 173 Exercises 9.3 179 Exercises 9.4 186 Appendix: Mathematical Induction 193 Exercises 193 CHAPTER ONE MATRIX ALGEBRA EXERCISES 1.1. 1. Write out in extended form the matrix [(-l)2 ^ (i + j )] 2,4' Solution. Use the given formula to calculate the (i ,j ) entry of the matrix for i = 1, 2 and j = 1, 2, 3, 4. Then the extended form of the matrix is [ 2 -3 •33 4 4444 -5-- --55551 111 i, —O ** — --33 444 --55 --55 66 6 2. Find a formula for the (i ,j ) entry of each of the following matrices: rf ---11l 111 i ----111i l 1 11 f 112 22 3 33 4 44 11 (a) 11i ---i11 -1 i11 1 j ,, , (b) 55 666 777 888 I [[ ---11i 111i ---"11iI JJjJ 99 1100 1111 1122 [ 13 14 15 16 J Solution. (a) By inspection we see that the (i J ) entry is +1 if i + j is odd and - 1 if i + j is even. So the (i ,j ) entry is (-1) ^ ~ • 1 2 Chapter One: Matrix Algebra (b) Notice the pattern: the ; th entry in the first row is ; , the ; tn entry of the second row is j + 4. Similarly the j th entries of the third and fourth rows are ,; + 8 and ; 4- 12 respectively. Thus the j th entry of the t th row, that is, the (t J ) entry, is ; + 4(t - 1) = 4i + j-4. 3. Using the fact that matrices have a rectangular shape, say how many different zero matrices can be formed using a total of 12 zeros. Solution. The possible sizes of zero matrices are m % n where mn = 12. Therefore there are six zero matrices, namely, 0 p Og 2' ^4 V ^3 4* ^2 6* ^1 12 * 12 4. For every integer n > 1 there are always at least two zero matrices that can be formed using a total of n zeros. For which n are there exactly two such zero matrices? Solution. The possible zero matrices are 0 where rs = n . Two cases that T,S always occur are r = 1, s = n and r = n , s = 1, that is, 0 and t 1,71 0 ^ . There will be no other zero matrices precisely when n is a prime n number. 5. Which matrices are both upper and lower triangular?