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Undergraduate Texts in Mathematics
Serge Lang
Introduction to
Linear Algebra
Second Edition

Springer
Undergraduate Texts In Mathematics
Editors
s.
Axler
F. W. Gehring
K. A. Ribet
Springer
New York
Berlin
Heidelberg
Hong Kong
London
Milan
Paris
Tokyo
Springer Books on Elementary Mathematics by Serge Lang
MATH! Encounters with High School Students
1985, ISBN 96129-1
The Beauty of Doing Mathematics
1985, ISBN 96149-6
Geometry: A High School Course (with G. Murrow), Second Edition
1988, ISBN 96654-4
Basic Mathematics
1988, ISBN 96787-7
A First Course in Calculus, Fifth Edition
1986, ISBN 96201-8
Calculus of Several Variables, Third Edition
1987, ISBN 96405-3
Introduction to Linear Algebra, Second Edition
1986, ISBN 96205-0
Linear Algebra, Third Edition
1987, ISBN 96412-6
Undergraduate Algebra, Second Edition
1990, ISBN 97279-X
Undergraduate Analysis, Second Edition
1997, ISBN 94841-4
Complex Analysis, Fourth Edition
1999, ISBN 98592-1
Real and Functional Analysis, Third Edition
1993, ISBN 94001-4
Serge Lang
Introduction
to Linear Algebra
Second Edition
With 66 Illustrations
Springer
Serge Lang
Department of Mathematics
Yale University
New Haven, CT 06520
U.S.A.
Editorial Board
S. Axler
Department of Mathematics
Michigan State University
East Lansing, MI 48824
U.S.A.
F. W. Gehring
Department of Mathematics
University of Michigan
Ann Arbor. MI 48019
U.S.A.
K.A. Ribet
Department of Mathelnatics
University of California
at Berkeley
Berkeley, CA 94720-3840
U.S.A.
Mathematics Subjects Classifications (2000): 15-01
Library of Congress Cataloging in Publication Data
Lang, Serge, 1927Introduction to linear algebra.
(Undergraduate texts in mathematics)
Includes index.
1. Algebras, Linear. I. Title. II. Series.
QA184.L37 1986
512'.5
85-14758
Printed on acid-free paper.
The first edition of this book was published by Addison-Wesley Publishing Company, Inc., in 1970.
© 1970, 1986 by Springer-Verlag New York Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A.),
except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any
form of information storage and retrieval, electronic adaptation, computer software, or by similar or
dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the
former are not especially identified, is not to be taken as a sign that such names, as understood by
the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
Printed in the United States of America
987
Springer-Verlag
(ASC/EB)
SPIN 10977149