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Springer
New
York
Berlin
Heidelberg
Barcelona
Budapest
Hong
Kong
London
Milan
Paris
Santa
Clara
Singapore
Tokyo
Undergraduate Texts in Mathematics
Editors
s.
Axler
F.W. Gehring
P.R. Halmos

Undergraduate Texts in Mathematics
Anglin: Mathematics: A Concise History
and Philosophy.
Readings in Mathematics.
AnglinlLambek: The Heritage
of
Thales.
Readings in Mathematics.
Apostol: Introduction
to
Analytic
Number Theory. Second edition.
Armstrong: Basic Topology.
Armstrong: Groups and Symmetry.
BaklNewman: Complex Analysis.
BanchofflWermer: Linear Algebra
Through Geometry. Second edition.
Berberian: A First Course in Real
Analysis.
Bremaud:
An Introduction
to
Probabilistic Modeling.
Bressoud: Factorization and Primality
Testing.
Bressoud: Second Year Calculus.
Readings in Mathematics.
Brickman: Mathematical Introduction
to
Linear Programming and Game
Theory.
Browder: Mathematical Analysis:
An Introduction.
Cederberg: A Course
in
Modem
Geometries.
Childs: A Concrete Introduction
to
Higher Algebra. Second edition.
Chung: Elementary Probability Theory
with Stochastic Processes. Third
edition.
CoxILittleJO'Shea: Ideals, Varieties,
and Algorithms.
Croom: Basic Concepts of Algebraic
Topology.
Curtis: Linear Algebra:
An
Introductory
Approach. Fourth edition.
Devlin: The Joy
of
Sets: Fundamentals
of
Contemporary Set Theory. Second
edition.
Dixmier: General Topology.
Driver: Why Math?
Ebbinghaus/FIumffhomas:
Mathematical Logic. Second edition.
Edgar:
Measure, Topology, and Fractal
Geometry.
Elaydi: Introduction
to
Difference
Equations.
Fischer: Intermediate Real Analysis.
Flanigan/Kazdan: Calculus Two: Linear
and Nonlinear Functions. Second
edition.
Fleming: Functions
of
Several Variables.
Second edition.
Foulds: Combinatorial Optimization for
Undergraduates.
Foulds: Optimization Techniques: An
Introduction.
Franklin: Methods
of
Mathematical
Economics.
HairerIWanner: Analysis by Its History.
Readings in Mathematics.
Halmos: Finite-Dimensional Vector
Spaces. Second edition.
Halmos: Naive Set Theory.
HiimmerlinIHoffmann: Numerical
Mathematics.
Readings in Mathematics.
Iooss/Joseph: Elementary Stability and
Bifurcation Theory. Second edition.
Isaac: The Pleasures of Probability.
Readings in Mathematics.
James: Topological
and
Uniform Spaces.
Jiinich: Linear Algebra.
Jiinich: Topology.
Kemeny/Snell: Finite Markov Chains.
Kinsey: Topology of Surfaces.
Klambauer: Aspects
of
Calculus.
Lang: A First Course in Calculus. Fifth
edition.
Lang: Calculus
of
Several Variables.
Third edition.
Lang: Introduction
to
Linear Algebra.
Second edition.
Lang: Linear Algebra. Third edition.
Lang: Undergraduate Algebra. Second
edition.
Lang: Undergraduate Analysis.
LaxIBursteinlLax: Calculus with
Applications and Computing.
Volume
l.
LeCuyer: College Mathematics with
APL.
(continued after index)

Thomas Banchoff
John Wermer
Linear
Algebra
Through Geometry
Second Edition
With
92
Illustrations
Springer

Thomas
BanchofT
John
Wermer
Department
of
Mathematics
Brown
University
Providence,
RI
02912
USA
Editorial Board
S.
Axler
Department
of
Mathematics
Michigan State University
East Lansing,
MI
48824
USA
F.W. Gehring
Department
of
Mathematics
University
of
Michigan
Ann Arbor, MI
48109
USA
Library of Congress Cataloging-in-Publication Data
Banchoff, Thomas.
P.R.
Halmos
Department
of
Mathematics
Santa
Clara
University
Santa
Clara,
CA
95053
USA
Linear algebra through geometry, 2nd ed./Thomas Banchoff, John Wermer.
p.
cm.-(Undergraduate texts
in
mathematics)
Includes bibliographical references and index.
ISBN-13: 978-1-4612-8752-0 e-ISBN-13: 978-1-4612-4390-8
DOl 10.1007/978-1-4612-4390-8
1.
Algebras, Linear. I. Wermer, John.
II.
Title. III.
Series.
QA184.B36
1991
512'.5-dc20
Printed on acid-free paper.
©
1992
Springer-Verlag
New
York, Inc.
Softcover reprint
of
the hardcover 2nd edition 1992
91-18083
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
New
York Inc.,
175
Fifth Avenue,
New
York, NY
10010,
USA),
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.
Typeset
by
Asco
Trade Typesetting Ltd., Hong Kong.
9 8 7
654
3

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