Linear Algebra - K.Hoffman, R.Kunze, Prentice-Hall, 1971.pdf - MAT Algebra - kf.mtsw

LINEAR ALGEBRA

Second Edition

KENNETH HOFFMAN

Professor of Mathematics

Massachusetts Institute of Technology

RAY KUNZE

Professor of Mathematics

University of California, Irvine

PRENTICE-HALL,

INC., Englewood Cliffs, New Jersey

@ 1971, 1961 by

Prentice-Hall, Inc.

Englewood Cliffs, New Jersey

reproduced in any form or by any means without

permission in writing from the publisher.

PRENTICE-HALL~NTERNATXONAL,INC.,

London

PRENTICE-HALLOFAUSTRALIA,PTY.

LTD., Sydney

PRENTICE-HALL

OF CANADA,

LTD., Toronto

PRENTICE-HALLOFINDIA

PRIVATE

LIMITED,

New Delhi

PRENTICE-HALL

OFJAPAN,INC.,

Tokyo

Current printing (last digit) :

10 9 8 7 6

Library of Congress Catalog Card No. 75142120

Printed in the United States of America

re ace

Our original purpose in writing this book was to provide a text for the under-

graduate linear algebra course at the Massachusetts Institute of Technology. This

course was designed for mathematics majors at the junior level, although three-

fourths of the students were drawn from other scientific and technological disciplines

and ranged from freshmen through graduate students. This description of the

M.I.T. audience for the text remains generally accurate today. The ten years since

the first edition have seen the proliferation of linear algebra courses throughout

the country and have afforded one of the authors the opportunity

to teach the

basic material to a variety of groups at Brandeis University,

Washington Univer-

sity (St. Louis), and the University of California (Irvine).

Our principal aim in revising Linear Algebra has been to increase the variety

of courses which can easily be taught from it. On one hand, we have structured the

chapters, especially the more difficult ones, so that there are several natural stop-

ping points along the way, allowing the instructor in a one-quarter or one-semester

course to exercise a considerable amount of choice in the subject matter. On the

other hand, we have increased the amount of material in the text, so that it can be

used for a rather comprehensive

one-year course in linear algebra and even as a

reference book for mathematicians.

The major changes have been in our treatments of canonical forms and inner

product spaces. In Chapter 6 we no longer begin with the general spatial theory

which underlies the theory of canonical forms. We first handle characteristic values

in relation to triangulation and diagonalization theorems and then build our way

up to the general theory. We have split Chapter 8 so that the basic material on

inner product spaces and unitary diagonalization is followed by a Chapter 9 which

treats sesqui-linear forms and the more sophisticated properties of normal opera-

tors, including normal operators on real inner product spaces.

We have also made a number of small changes and improvements

from the

first edition. But the basic philosophy behind the text is unchanged.

We have made no particular concession to the fact that the majority of the

students may not be primarily interested in mathematics. For we believe a mathe-

matics course should not give science, engineering, or social science students a

hodgepodge of techniques, but should provide them with an understanding

basic mathematical

concepts.

. . .

Preface

On the other hand, we have been keenly aware of the wide range of back-

grounds which the students may possess and, in particular, of the fact that the

students have had very little experience with abstract mathematical reasoning.

For this reason, we have avoided the introduction of too many abstract ideas at

the very beginning of the book. In addition, we have included an Appendix which

presents such basic ideas as set, function, and equivalence relation. We have found

it most profitable not to dwell on these ideas independently,

but to advise the

students to read the Appendix when these ideas arise.

Throughout the book we have included a great variety of examples of the

important concepts which occur. The study of such examples is of fundamental

importance and tends to minimize the number of students who can repeat defini-

tion, theorem, proof in logical order without grasping the meaning of the abstract

concepts. The book also contains a wide variety of graded exercises (about six

hundred), ranging from routine applications to ones which will extend the very

best students. These exercises are intended to be an important part of the text.

Chapter 1 deals with systems of linear equations and their solution by means

of elementary row operations on matrices. It has been our practice to spend about

six lectures on this material. It provides the student with some picture of the

origins of linear algebra and with the computational technique necessary to under-

stand examples of the more abstract ideas occurring in the later chapters. Chap-

ter 2 deals with vector spaces, subspaces, bases, and dimension. Chapter 3 treats

linear transformations, their algebra, their representation by matrices, as well as

isomorphism, linear functionals, and dual spaces. Chapter 4 defines the algebra of

polynomials over a field, the ideals in that algebra, and the prime factorization of

a polynomial. It also deals with roots, Taylor’s formula, and the Lagrange inter-

polation formula. Chapter 5 develops determinants of square matrices, the deter-

minant being viewed as an alternating n-linear function of the rows of a matrix,

and then proceeds to multilinear functions on modules as well as the Grassman ring.

The material on modules places the concept of determinant in a wider and more

comprehensive setting than is usually found in elementary textbooks. Chapters 6

and 7 contain a discussion of the concepts which are basic to the analysis of a single

linear transformation on a finite-dimensional vector space; the analysis of charac-

teristic (eigen) values, triangulable and diagonalizable transformations; the con-

cepts of the diagonalizable and nilpotent parts of a more general transformation,

and the rational and Jordan canonical forms. The primary and cyclic decomposition

theorems play a central role, the latter being arrived at through the study of

admissible subspaces. Chapter 7 includes a discussion of matrices over a polynomial

domain, the computation of invariant factors and elementary divisors of a matrix,

and the development of the Smith canonical form. The chapter ends with a dis-

cussion of semi-simple operators, to round out the analysis of a single operator.

Chapter 8 treats finite-dimensional inner product spaces in some detail. It covers

the basic geometry, relating orthogonalization to the idea of ‘best approximation

to a vector’ and leading to the concepts of the orthogonal projection of a vector

onto a subspace and the orthogonal complement of a subspace. The chapter treats

unitary operators and culminates in the diagonalization of self-adjoint and normal

operators. Chapter 9 introduces sesqui-linear forms, relates them to positive and

self-adjoint operators on an inner product space, moves on to the spectral theory

of normal operators and then to more sophisticated results concerning normal

operators on real or complex inner product spaces. Chapter 10 discusses bilinear

forms, emphasizing canonical forms for symmetric and skew-symmetric forms, as

well as groups preserving non-degenerate forms, especially the orthogonal, unitary,

pseudo-orthogonal and Lorentz groups.

We feel that any course which uses this text should cover Chapters 1, 2, and 3

Preface

thoroughly, possibly excluding Sections 3.6 and 3.7 which deal with the double dual

and the transpose of a linear transformation. Chapters 4 and 5, on polynomials and

determinants, may be treated with varying degrees of thoroughness. In fact,

polynomial ideals and basic properties of determinants may be covered quite

sketchily without serious damage to the flow of the logic in the text; however, our

inclination is to deal with these chapters carefully (except the results on modules),

because the material illustrates so well the basic ideas of linear algebra. An ele-

mentary course may now be concluded nicely with the first four sections of Chap-

ter 6, together with (the new) Chapter 8. If the rational and Jordan forms are to

be included, a more extensive coverage of Chapter 6 is necessary.

Our indebtedness remains to those who contributed to the first edition, espe-

cially to Professors Harry Furstenberg, Louis Howard, Daniel Kan, Edward Thorp,

to Mrs. Judith Bowers, Mrs. Betty Ann (Sargent) Rose and Miss Phyllis Ruby.

In addition, we would like to thank the many students and colleagues whose per-

ceptive comments led to this revision, and the staff of Prentice-Hall for their

patience in dealing with two authors caught in the throes of academic administra-

tion. Lastly, special thanks are due to Mrs. Sophia Koulouras for both her skill

and her tireless efforts in typing the revised manuscript.

K. M. H. / R. A. K.

Linear Algebra - K.Hoffman, R.Kunze, Prentice-Hall, 1971.pdf

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