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Essential VOLUME FOUR Student Algebra Linear Algebra T.S. BLYTH & E.F. ROBERTSON Essential VOLUME FOUR Student Algebra Bs —— Linear Algebra : — < 7 aa= a i. s a at s or aa’ e Essential VOLUME FOUR Student Algebra Linear Algebra T. S. BLYTH & E. F. ROBERTSON University of StA ndrews London New York CHAPMAN AND HALL First published in 1986 by Chapman and Hall Ltd 11 New Fetter Lane, London EC4P 4EE Published in the USA by Chapman and Hall 29 West 35th Street, New York NY 10001 © 1986 T.S. Blyth and E. F. Robertson Printed in Great Britain by J. W. Arrowsmith Ltd., Bristol ISBN 0 412 27850 2 This paperback edition is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher’s prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. All rights reserved. No part of this book may be reprinted or reproduced, or utilized in any form or by any electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or in any information storage and retrieval system, without permission in writing from the publisher. British Library Cataloguing in Publication Data Blyth, T.S. Essential student algebra. Vol 4: Linear algebra 1. Algebra I. Title II. Robertson, E. F. 512 QA155 ISBN 0-412-27850-2 Contents Preface Chapter One: The minimum polynomial Chapter Two : Direct sums of subspaces Chapter Three : Reduction to triangular form 22 Chapter Four : Reduction to Jordan form 31 Chapter Five: The rational and classical forms 44 Chapter Six : Dual spaces 58 Chapter Seven : Inner product spaces 69 Chapter Eight : Orthogonal direct sums 87 Chapter Nine : Bilinear and quadratic forms 99 Chapter Ten : Real normality 108 Index 119 P ahi het,e te TH i: _ b a: 27| 9s 7%F s Bs a1 eri r. eS J 4 Wyhreat) aghast ist, a > cor atin on saii? wi ~ evant MMh oqeane x ess s= e amen ne a wat est) + ramj aune ee ae Preface If, as it is often said, mathematics is the queen of science then algebra is surely the jewel in her crown. In the course of its vast development over the last half-century, algebra has emerged as the subject in which one can observe pure mathe- matical reasoning at its best. Its elegance is matched only by the ever-increasing number of its applications to an extraordinarily wide range of topics in areas other than ‘pure’ mathematics. Here our objective is to present, in the form of a series of five concise volumes, the fundamentals of the subject. Broadly speaking, we have covered in all the now traditional syllabus that is found in first and second year university courses, as well as some third year material. Further study would be at the level of ‘honours options’. The reasoning that lies behind this modular presentation is simple, namely to allow the student (be he a mathematician or not) to read the subject in a way that is more appropriate to the length, content, and extent, of the various courses he has to take. Although we have taken great pains to include a wide selec- tion of illustrative examples, we have not included any exer- cises. For a suitable companion collection of worked examples, we would refer the reader to our series Algebra through practice (Cambridge University Press), the first five books of which are appropriate to the material covered here. T.S.B., E.F.R. Wnts ae et ee ee ae oPmamie =e a ae ‘Selle he a ; <_ “a . 7 xe e 7 : 7 > - 7 _ \ aa one rs _ : a ae egitii.se te we 2 up wh a saiddereveds bias pete = # re aarignes Woe eb _ even ced ae [nen oe weet on — } and ahr ‘mle yralere ee el wel Seve. Saosaqetee rn a t 7 ae et eceq sttade veo gam Ee wt Susuew a¢ d = a ake 73 haw. balsaeas <i sorte get's aii et roe da 3y/ie0 i gitsaiGroarnsa) aa. oF Bort: oe 'p od > weiri g: : Qolacmebions ‘ewig wasit tedie see18 2; SegoT seen @ Ef ent od) ae torr ot wvitonide 9 = : pcorstt Soap +eo s te swells vee : " « gputabiven by seibart won. odd ie ai Beiseco" Jd - flee es: teehee ywariee tear hee Ia tb 5 ee hisew eowre onitioll Sekmtea: saref . ea Bx mia t nell aad) guabareusiee at Ssumandega 2j0 ot pel sedate ecle of cisarw cabguniaa escsas heist vere of train ory ceer ot (tes = ts as ate. , noes hee tentang otigstsl od oe ih = aU that onead Ad 2 S : -jele whaet s sites nina ong meineoael ss ~tena ure babatned dem penal ter amen sth ’ aod 2 barre @ aorta artnnn. aos 8 ak weg Beyagact se deaht ashes xo a) 3 bit to ahaned vit spt od2 {oe wha Syeweey