Linear Algebra
Besides the very obvious change from German to English, the second edition of this book contains many additions as weil as a great many other changes. It might even be called a new book altogether were it not for the fact that the essential character of t
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MATHEMATISCHEN WI S SEN SCHAFTE N IN EINZELDARSTELLUNGEN MIT BESONDERER BERUCKSICHTIGUNG DER ANWENDUNGSGEBIETE HERAUSGEGEBEN VON
J.L.DOOB·R.GRAMMEL· E.HEINZ F.HIRZEBRUCH· E.HOPF· H.HOPF. W.MAAK W. MAGNUS . F. K. SCHMIDT . K. STEIN GESCHAFTSFUHRENDE HERAUSGEBER
B. ECKMANN UND B. L. VAN DER WAERDEN ZURICH
BAND 97
Springer-Verlag Berlin Heidelberg GmbH
LINEAR ALGEBRA BY
WERNER H. GREUB MATHEMATICS DEPARTMENT UNIVERSITY OF TORONTO
SECOND EDITION
WITH 6 FIGURES
1963
Springer-Verlag Berlin Heidelberg GmbH
Library oj Congress Cata/og Card Number 63-17509 All rigbts reserved No pari oj Ibis book mqy be reproduced in any jorm, by microft/m or any otber means, witbou written permission jrom tbe publisbers
ISBN 978-3-662-01547-6 ISBN 978-3-662-01545-2 (eBook) DOI 10.1007/978-3-662-01545-2
© BY SPRINGER-VERLAGBERLlNHEIDELBERG 1963 Originally published by Springer-Verlag OHG. Berlin . Göttingen . Heidelberg in 1963.
Softcover reprint of the hardcover 2nd edition 1963
Managing Editors:
Prof. Dr. Prof. Dr.
B. L.
B.
Eckmann.
Eidgenössische Technische Hochschule Zürich
van der Waerden.
Mathematisches Institut der Universität Zarich
TO
ROLF NEVANLINNA
Preface to the second edition Besides the very obvious change from German to English, the second edition of this book contains many additions as weil as a great many other changes. It might even be called a new book altogether were it not for the fact that the essential character of the book has remained the same; in other words, the entire presentation continues to be based on an axiomatic treatment of linear spaces. In this second edition, the thorough-going restriction to linear spaces of finite dimension has been removed. Another complete change is the restriction to linear spaces with real or complex coefficients, thereby removing a number of relatively involved discussions which did not really contribute substantially to the subject. On p.6 there is a list of those chapters in which the presentation can be transferred directly to spaces over an arbitrary coefficient field. Chapter I deals with the general properties of a linear space. Those concepts which are only valid for finitely many dimensions are discussed in a special paragraph. Chapter 11 now covers only linear transformations while the treatment of matrices has been delegated to a new chapter, chapter 111. The discussion of dual spaces has been changed; dual spaces are now introduced abstractly and the connection with the space of linear functions is not established untillater. Chapters IV and V, dealing with determinants and orientation respectively, do not contain substantial changes. Brief reference should be made here to the new paragraph in chapter IV on the trace of an endomorphism - a concept which is used quite consistently throughout the book from that time on. Special emphasize is given to tensors. The original chapter on Multilinear Algebra is now spread over four chapters: Multilinear Mappings (Ch. VI), Tensor Algebra (Ch. VII), Exterior Algebra (Ch. VIII) and Duality in Exterior
