Essential Student Algebra Volume Two: Matrices and Vector Spaces
H, 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 re
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VOLUME TWO
Matrices and Vector Spaces
Essential Student Algebra
VOLUME TWO
Matrices and Vector Spaces T. S. BLYTH & E. F. ROBERTSON University of St Andrews
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
© 1986 T. S. Blyth and E. F. Robertson Originally published by Chapman and Han in 1986 ISBN 978-0-412-27870-9 ISBN 978-94-017-2213-1 (eBook) DOI 10.1007/978-94-017-2213-1 This paperback edition is sold subject to the condition that it shal! not, by way oftrade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent in any form ofbinding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. An rights reserved. No part ofthis 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. VoI 2: Matrices and vector spaces 1. Algebra 1. Title II. Robertson, E. F. 512 QA155 ISBN 978-0-412-27870-9
Contents
Preface Chapter One : The algebra of matrices
1
Chapter Two : Some applications of matrices
13
Chapter Three : Systems of linear equations
21
Chapter Four : Invertible matrices
42
Chapter Five : Vector spaces
48
Chapter Six : Linear mappings
64
Chapter Seven: The matrix connection
74
Chapter Eight : Determinants
86
Chapter Nine : Eigenvalues and eigenvectors
100
Index
119
Preface
H, 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 mathematical 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 selection of illustrative examples, we have not included any exercises. 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.
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