Essential Student Algebra: Volume Four: Linear Algebra

132 Pages·1986·4.155 MB·English

by  T. S. Blyth, E. F. Robertson

Essential VOLUME FOUR
Student
Algebra
Linear
Algebra
T.S. BLYTH & E.F. ROBERTSON

Essential VOLUME FOUR
Student
Algebra Bs
—— Linear
Algebra
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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
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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.
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