Table Of ContentLinear Algebra
Sterling K. Berberian
The University of Texas at Austin
Oxford New York Tokyo
OXFORD UNIVERSITY PRESS
1992
Oxford University Press, Walton Street. Oxford OX2 6DP
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© S. K. Berberian 1992
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A catalogue record for this book is available from the British Library
Library of Congress Cataloging in Publication Data
Berberian, Sterling K., 1926-
Linear algebra/Sterling K. Berberian.
Includes indexes.
ISBN 0-19-853436-1
ISBN 0-19-853435-3 (PBKJ
I. Algebras. Linear. I. Title.
QA/84.B47 /992 512'.5-dc20 90-28672
Typeset by Keytec Typeselling Ltd, Bridpon, Dorset
Printed in the U.S.A.
For Jim and Susan
Preface
This book grew out of my experiences in teaching two one-semester courses
in linear algebra, the first at the immediate post-calculus level, the second at
the upper-undergraduate level. The latter course takes up general determin
ants and standard forms for matrices, and thus requires some familiarity with
permutation groups and the factorization of polynomials into irreducible
polynomials; this material is normally covered in a one-semester abstract
algebra course taken between the two linear algebra courses.
Part 1 of the book (Chapters 1-6) mirrors the first of the above-mentioned
linear algebra courses, Part 2 the second (Chapters 7-13); the information on
factorization needed from the transitional abstract algebra course is
thoroughly reviewed in an Appendix.
The underlying plan of Part 1 is simple: first things first. For the benefit of
the reader with little or no experience in formal mathematics (axioms,
theorem-proving), the proofs in Part 1 are especially detailed, with frequent
comments on the logical strategy of the proof. The more experienced reader
can simply skip over superfluous explanations, but the theorems themselves
are cast in the form needed for the more advanced chapters of the book;
apart from Chapter 6 (an elementary treatment of 2 x 2 and 3 x 3 deter
minants}, nothing in Part 1 has to be redone for Part 2.
Part 2 goes deeper, addressing topics that are more demanding, even
difficult: general determinant theory, similarity and canonical forms for
matrices, spectral theory in real and complex inner product spaces, tensor
products. {Not to worry: in mathematics, 'difficult' means only that it takes
more time to make it easy.}
Courses with different emphases can be based on various combinations of
chapters:
(A) An introductory course that serves also as an initiation into formal
mathematics: Chapter 1-6.
(B) An advanced course, for students who have met matrices and linear
mappings before (on an informal, relatively proofless level}, and have had a
theorem-proving type course in elementary abstract algebra (groups and
rings): a rapid tour of Chapters 1-5, followed by a selection of chapters from
Part 2 tailored to the needs of the class. The selection can be tilted towards
algebra or towards analysis/geometry, as indicated in the flow chart following
this preface.
It is generally agreed that a course in linear algebra should begin with a
discussion of examples; I concur wholeheartedly (Chapter 1, §§1 and 2). Now
comes the hard decision: which to take up first, (i) linear equations and
matrices, or (ii) vector spaces and linear mappings? I have chosen the latter
course, partly on grounds of efficiency, partly because matrices are addictive
and linear mappings are not. My experience in introductory courses is that
viii PREFACE
once a class has tasted the joy of matrix computation, it is hard to get anyone
to focus on something so austere as a linear mapping on a vector space.
Eventually (from Chapter 4 onward} the reader will, I believe, perceive the
true relation between linear mappings and matrices to be symbiotic: each is
indispensable for an understanding of the other. It is equally a joy to see
computational aspects (matrices, determinants) fall out almost effortlessly
when the proper conceptual foundation has been laid (linear mappings).
A word about the role of Appendix A ('Foundations'). The book starts
right off with vectors (I usually cover Section 1 of Chapter 1 on Day 1), but
before taking up Section 2 it is a good idea to go over briefly Appendix A.2
on set notations. Similarly, before undertaking Chapter 2 (linear mappings}, a
brief discussion of Appendix A.3 on functions is advisable. Appendix A.1 (an
informal discussion of the logical organization of proofs) is cited in the text
wherever it can heighten our understanding of what's going on in a proof. In
short, Appendix A is mainly for reference; what it contains is important and
needs to be talked about, often and in little bits, but not for a whole hour at
a stretch.
The wide appeal of linear algebra lies in its importance for other branches
of mathematics and its adaptability to concrete problems in mathematical
sciences. Explicit applications are not abundant in the book but they are not
entirely neglected, and applicability is an ever-present consideration in the
choice of topics. For example, Hilbert space operator theory and the
representation of groups by matrices (the applications with which I am most
familiar) are not taken up explicitly in the text, but the chapters on inner
product spaces consciously prepare the way for Hilbert space and the reader
who studies group representations will find the chapters on similarity and
tensor products helpful. Systems of linear equations and the reduction of
matrices to standard forms are applications that belong to everyone; they are
treated thoroughly. For the class that has the leisure to take them up, the
applications to analytic geometry in Chapter 6 are instructive and rewarding.
In brief, linear algebra is a feast for all tastes. Bon appetit!
Sterling Berberian
Austin, Texas
August 1990
IX
Flow chart of chapters
1-4
I I
7 5 5
I I
8 6 7
I I
First
course
10 8
I I
II 9
I I
13 12
Algebraic Analytic,
Geometric
