K. W. Gruenberg, A. J. Weir - Linear Geometry (1977) [978-1-4757-4101-8].pdf

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Graduate Texts in Mathematics
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
Department of Pure Mathematics
Queen Mary College
University of London
England
A.
J.
Weir
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 mathe-
matics 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 defines affine and
projective geometries in terms of vector spaces and establishes explicitly the
connexion between these two types of geometry. In Chapter 3, the idea of
isomorphism is carried over from vector spaces to affine and projective
geometries. In particular, we include a simple proof of the basic theorem of
projective geometry, in §3.5. This chapter is also the one in which systems
of linear equations make their first appearance (§3.3). They reappear in
increasingly sophisticated forms in §§4.5 and 4.6.
Linear algebra proper is continued in Chapter 4 with the usual topics
centred on linear. mappings. In this chapter the important concept of
duality in vector spaces is linked to the idea of dual geometries. In our
treatment of bilinear forms in Chapter 5 we take the theory up to, and
including, the classification of symmetric forms over the complex and real
fields. The geometric significance of bilinear forms in terms of quadrics is
v

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