Linear Algebra Through Geometry

315 Pages·1992·8.59 MB·English

by  Thomas Banchoff

Undergraduate Texts in Mathematics
Editors
s. Axler
F.W. Gehring
P.R. Halmos
Springer
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Heidelberg
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Undergraduate Texts in Mathematics
Anglin: Mathematics: A Concise History Elaydi: Introduction to Difference
and Philosophy. Equations.
Readings in Mathematics. Fischer: Intermediate Real Analysis.
AnglinlLambek: The Heritage of Thales. Flanigan/Kazdan: Calculus Two: Linear
Readings in Mathematics. and Nonlinear Functions. Second
Apostol: Introduction to Analytic edition.
Number Theory. Second edition. Fleming: Functions of Several Variables.
Armstrong: Basic Topology. Second edition.
Armstrong: Groups and Symmetry. Foulds: Combinatorial Optimization for
BaklNewman: Complex Analysis. Undergraduates.
BanchofflWermer: Linear Algebra Foulds: Optimization Techniques: An
Through Geometry. Second edition. Introduction.
Berberian: A First Course in Real Franklin: Methods of Mathematical
Analysis. Economics.
Bremaud: An Introduction to HairerIWanner: Analysis by Its History.
Probabilistic Modeling. Readings in Mathematics.
Bressoud: Factorization and Primality Halmos: Finite-Dimensional Vector
Testing. Spaces. Second edition.
Bressoud: Second Year Calculus. Halmos: Naive Set Theory.
Readings in Mathematics. HiimmerlinIHoffmann: Numerical
Brickman: Mathematical Introduction Mathematics.
to Linear Programming and Game Readings in Mathematics.
Theory. Iooss/Joseph: Elementary Stability and
Browder: Mathematical Analysis: Bifurcation Theory. Second edition.
An Introduction. Isaac: The Pleasures of Probability.
Cederberg: A Course in Modem Readings in Mathematics.
Geometries. James: Topological and Uniform Spaces.
Childs: A Concrete Introduction to Jiinich: Linear Algebra.
Higher Algebra. Second edition. Jiinich: Topology.
Chung: Elementary Probability Theory Kemeny/Snell: Finite Markov Chains.
with Stochastic Processes. Third Kinsey: Topology of Surfaces.
edition. Klambauer: Aspects of Calculus.
CoxILittleJO'Shea: Ideals, Varieties, Lang: A First Course in Calculus. Fifth
and Algorithms. edition.
Croom: Basic Concepts of Algebraic Lang: Calculus of Several Variables.
Topology. Third edition.
Curtis: Linear Algebra: An Introductory Lang: Introduction to Linear Algebra.
Approach. Fourth edition. Second edition.
Devlin: The Joy of Sets: Fundamentals Lang: Linear Algebra. Third edition.
of Contemporary Set Theory. Second Lang: Undergraduate Algebra. Second
edition. edition.
Dixmier: General Topology. Lang: Undergraduate Analysis.
Driver: Why Math? LaxIBursteinlLax: Calculus with
Ebbinghaus/FIumffhomas: Applications and Computing.
Mathematical Logic. Second edition. Volume l.
Edgar: Measure, Topology, and Fractal LeCuyer: College Mathematics with
Geometry. 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 F.W. Gehring P.R. Halmos
Department of Department of Department of
Mathematics Mathematics Mathematics
Michigan State University University of Michigan Santa Clara University
East Lansing, MI 48824 Ann Arbor, MI 48109 Santa Clara, CA 95053
USA USA USA
Library of Congress Cataloging-in-Publication Data
Banchoff, Thomas.
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 91-18083
Printed on acid-free paper.
© 1992 Springer-Verlag New York, Inc.
Softcover reprint of the hardcover 2nd edition 1992
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
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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
To our wives Lynore and Kerstin
Preface to the Second Edition
In this book we lead the student to an understanding of elementary linear
algebra by emphasizing the geometric significance of the subject.
Our experience in teaching undergraduates over the years has convinced
us that students learn the new ideas of linear algebra best when these
ideas are grounded in the familiar geometry of two and three dimensions.
Many important notions of linear algebra already occur in these dimen
sions in a non-trivial way, and a student with a confident grasp of the
ideas will encounter little difficulty in extending them to higher dimensions
and to more abstract algebraic systems. Moreover, we feel that this
geometric approach provides a solid basis for the linear algebra needed in
engineering, physics, biology, and chemistry, as well as in economics and
statistics.
The great advantage of beginning with a thorough study of the linear
algebra of the plane is that students are introduced quickly to the most
important new concepts while they are still on the familiar ground of
two-dimensional geometry. In short order, the student sees and uses the
notions of dot product, linear transformations, determinants, eigenvalues,
and quadratic forms. This is done in Chapters 2.0-2.7.
Then, the very same outline is used in Chapters 3.0-3.7 to present the
linear algebra of three-dimensional space, so that the former ideas are
reinforced while new concepts are being introduced.
In Chapters 4.0-4.2, we deal with geometry in IR" for n ;;:::: 4. We intro
duce linear transformations and matrices in 1R4, and we point out that the
step from 1R4 to IR" with n > 4 is now almost immediate. In Chapters 4.3
and 4.4, we treat systems of linear equations in n variables.
In the present edition, we have added Chapter 5 on vector spaces,
Chapter 6 on inner products on a vector space, and Chapter 7 on
viii Linear Algebra Through Geometry
symmetric n x n matrices and quadratic forms in n variables. Finally, in
Chapter 8 we deal with three applications:
(1) differential systems, that is, systems of linear first-order differential
equations;
(2) least-squares method in data analysis; and
(3) curvature of surfaces in 1ij3, which are given as graphs of functions of
two variables.
Except for Chapter 8, the student need only know basic high-school
algebra and geometry and introductory trigonometry in order to read this
book. In fact, we believe that high-school seniors who are interested in
mathematics could read much of this book on their own. To read Chapter
8, students should have a knowledge of elementary calculus.
Acknowledgments
We would like to thank the many students in our classes, whose interest
and suggestions have helped in the development of this book. We particu
larly thank Curtis Hendrickson and Davide Cervone, who produced the
computer-generated illustrations in this new edition. Our special thanks
go to Dale Cavanaugh, Natalie Johnson, and Carol Oliveira of the Brown
University Mathematics Department for their assistance.
Contents
Preface to the Second Edition vii
Acknowledgments ix
1.0 Vectors in the Line
2.0 The Geometry of Vectors in the Plane 3
2.1 Transformations of the Plane 23
2.2 Linear Transformations and Matrices 29
2.3 Sums and Products of Linear Transformations 39
2.4 Inverses and Systems of Equations 50
2.5 Determinants 61
2.6 Eigenvalues 75
2.7 Classification of Conic Sections 85
3.0 Vector Geometry in 3-Space 98
3.1 Transformations of 3-Space 113
3.2 Linear Transformations and Matrices 117
3.3 Sums and Products of Linear Transformations 122
3.4 Inverses and Systems of Equations 133
3.5 Determinants 151
3.6 Eigenvalues 163
3.7 Symmetric Matrices 178
3.8 Classification of Quadric Surfaces 190
XII Linear Algebra Through Geometry
4.0 Vector Geometry in n-Space, n ~ 4 197
4.1 Transformations of n-Space, n ~ 4 205
4.2 Linear Transformations and Matrices 213
4.3 Homogeneous Systems of Equations in n-Space 218
4.4 Inhomogeneous Systems of Equations in n-Space 226
5.0 Vector Spaces 235
5.1 Bases and Dimensions 238
5.2 Existence and Uniqueness of Solutions 245
5.3 The Matrix Relative to a Given Basis 247
6.0 Vector Spaces with an Inner Product 253
6.1 Orthonormal Bases 255
6.2 Orthogonal Decomposition of a Vector Space 260
7.0 Symmetric Matrices in n Dimensions 263
7.1 Quadratic Forms in n Variables 269
8.0 Differential Systems 274
8.1 Least Squares Approximation 291
8.2 Curvature of Function Graphs 296
Index 303