Linear Algebra PDF
Preview Linear Algebra
Linear Algebra Sterling K. Berberian The University of Texas at Austin Oxford New York Tokyo OXFORD UNIVERSITY PRESS 1992 Oxford University Press, Walton Street. Oxford OX2 6DP Oxford New York Toronto Delhi Bombay Calculla Madras Karachi Kuala Lumpur Singapore Hong Kong Tokyo Nairobi Dares Salaam Cape Town Melbourne Auckland and associated companies in Berlin lbadan Oxford is a trade mark of Oxford University Press Published in the United States by Oxford University Press, New York © S. K. Berberian 1992 All rights reserved. No port 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 the prior permission of Oxford University Press This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold. 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 A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data Berberian, Sterling K., 1926- Linear algebra/Sterling K. Berberian. Includes indexes. ISBN 0-19-853436-1 ISBN 0-19-853435-3 (PBKJ I. Algebras. Linear. I. Title. QA/84.B47 /992 512'.5-dc20 90-28672 Typeset by Keytec Typeselling Ltd, Bridpon, Dorset Printed in the U.S.A. For Jim and Susan Preface This book grew out of my experiences in teaching two one-semester courses in linear algebra, the first at the immediate post-calculus level, the second at the upper-undergraduate level. The latter course takes up general determin ants and standard forms for matrices, and thus requires some familiarity with permutation groups and the factorization of polynomials into irreducible polynomials; this material is normally covered in a one-semester abstract algebra course taken between the two linear algebra courses. Part 1 of the book (Chapters 1-6) mirrors the first of the above-mentioned linear algebra courses, Part 2 the second (Chapters 7-13); the information on factorization needed from the transitional abstract algebra course is thoroughly reviewed in an Appendix. The underlying plan of Part 1 is simple: first things first. For the benefit of the reader with little or no experience in formal mathematics (axioms, theorem-proving), the proofs in Part 1 are especially detailed, with frequent comments on the logical strategy of the proof. The more experienced reader can simply skip over superfluous explanations, but the theorems themselves are cast in the form needed for the more advanced chapters of the book; apart from Chapter 6 (an elementary treatment of 2 x 2 and 3 x 3 deter minants}, nothing in Part 1 has to be redone for Part 2. Part 2 goes deeper, addressing topics that are more demanding, even difficult: general determinant theory, similarity and canonical forms for matrices, spectral theory in real and complex inner product spaces, tensor products. {Not to worry: in mathematics, 'difficult' means only that it takes more time to make it easy.} Courses with different emphases can be based on various combinations of chapters: (A) An introductory course that serves also as an initiation into formal mathematics: Chapter 1-6. (B) An advanced course, for students who have met matrices and linear mappings before (on an informal, relatively proofless level}, and have had a theorem-proving type course in elementary abstract algebra (groups and rings): a rapid tour of Chapters 1-5, followed by a selection of chapters from Part 2 tailored to the needs of the class. The selection can be tilted towards algebra or towards analysis/geometry, as indicated in the flow chart following this preface. It is generally agreed that a course in linear algebra should begin with a discussion of examples; I concur wholeheartedly (Chapter 1, §§1 and 2). Now comes the hard decision: which to take up first, (i) linear equations and matrices, or (ii) vector spaces and linear mappings? I have chosen the latter course, partly on grounds of efficiency, partly because matrices are addictive and linear mappings are not. My experience in introductory courses is that viii PREFACE once a class has tasted the joy of matrix computation, it is hard to get anyone to focus on something so austere as a linear mapping on a vector space. Eventually (from Chapter 4 onward} the reader will, I believe, perceive the true relation between linear mappings and matrices to be symbiotic: each is indispensable for an understanding of the other. It is equally a joy to see computational aspects (matrices, determinants) fall out almost effortlessly when the proper conceptual foundation has been laid (linear mappings). A word about the role of Appendix A ('Foundations'). The book starts right off with vectors (I usually cover Section 1 of Chapter 1 on Day 1), but before taking up Section 2 it is a good idea to go over briefly Appendix A.2 on set notations. Similarly, before undertaking Chapter 2 (linear mappings}, a brief discussion of Appendix A.3 on functions is advisable. Appendix A.1 (an informal discussion of the logical organization of proofs) is cited in the text wherever it can heighten our understanding of what's going on in a proof. In short, Appendix A is mainly for reference; what it contains is important and needs to be talked about, often and in little bits, but not for a whole hour at a stretch. The wide appeal of linear algebra lies in its importance for other branches of mathematics and its adaptability to concrete problems in mathematical sciences. Explicit applications are not abundant in the book but they are not entirely neglected, and applicability is an ever-present consideration in the choice of topics. For example, Hilbert space operator theory and the representation of groups by matrices (the applications with which I am most familiar) are not taken up explicitly in the text, but the chapters on inner product spaces consciously prepare the way for Hilbert space and the reader who studies group representations will find the chapters on similarity and tensor products helpful. Systems of linear equations and the reduction of matrices to standard forms are applications that belong to everyone; they are treated thoroughly. For the class that has the leisure to take them up, the applications to analytic geometry in Chapter 6 are instructive and rewarding. In brief, linear algebra is a feast for all tastes. Bon appetit! Sterling Berberian Austin, Texas August 1990 IX Flow chart of chapters 1-4 I I 7 5 5 I I 8 6 7 I I First course 10 8 I I II 9 I I 13 12 Algebraic Analytic, Geometric
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