Linear Geometry
This is essentially a book on linear algebra. But the approach is somewhat unusual in that we emphasise throughout the geometric aspect of the subject. The material is suitable for a course on linear algebra for mathe matics majors at North American Univ
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49 Editorial Board
F. W. Gehring P. R. Halmos Managing Editor
c. C. Moore
K. W. Gruenberg A.J. Weir
Linear Geometry 2nd Edition
Springer Science+Business Media, LLC
K. W. Gruenberg
A. J. Weir
Department of Pure Mathematics Queen Mary College University of London England
School of Mathematical and Physical Sciences University of Sussex England
Editorial Board
P. R. Halmos
F. W. Gehring
c. C. Moore
Managing Editor Department of Mathematics Department of Mathematics Department of Mathematics University of Michigan University of California at Berkeley University of California Ann Arbor, Michigan 48104 Berkeley, California 94720 Santa Barbara, California 93106
AMS Subject Classification: 50D4O
Library of Congress Cataloging in Publication Data Gruenberg, Karl W Linear geometry. (Graduate texts in mathematics ; 49) I. Geometry, Algebraic. 2. Algebras, Linear. I. Weir, Alan J., joint author. II. Title. 1977 516'.35 76-27693 QA564.G72
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. © 1967 by K. W. Gruenberg and A. J. Weir. © 1977 by Springer Science+Business Media New York Originally published by Springer-Verlag New York in 1977
First edition published 1967 by D. Van Nostrand Company. 9 876 5 432 1 ISBN 978-1-4419-2806-1 ISBN 978-1-4757-4101-8 (eBook) DOI 10.1007/978-1-4757-4101-8
Preface
This is essentially a book on linear algebra. But the approach is somewhat unusual in that we emphasise throughout the geometric aspect of the subject. The material is suitable for a course on linear algebra for mathematics majors at North American Universities in their junior or senior year and at British Universities in their second or third year. However, in view of the structure of undergraduate courses in the United States, it is very possible that, at many institutions, the text may be found more suitable at the beginning graduate level. The book has two aims: to provide a basic course in linear algebra up to, and including, modules over a principal ideal domain; and to explain in rigorous language the intuitively familiar concepts of euclidean, affine, and projective geometry and the relations between them. It is increasingly recognised that linear algebra should be approached from a geometric point of VIew. This applies not only to mathematics majors but also to mathematically-oriented natural scientists and engineers. The material in this book has been taught for many years at Queen Mary College in the University of London and one of us has used portions of it at the University of Michigan and at Cornell University. It can be covered adequately in a full one-year course. But suitable parts can also be used for one-semester courses with either a geometric or a purely algebraic flavor. We shall give below explicit and detailed suggestions on how this can be done (in the "Guide to the Reader"). The first chapter contains in fairly concise form the definition and most elementary properties of a vector space. Chapter 2 then de
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