, 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
All rights reserved. No part of this book may be
reproduced in any form or by any means without
permission in writing from the publisher.
London
Sydney
PRENTICE-HALL INTERNATIONAL, INC.,
PRENTICE-HALL OF CANADA, LTD., Toronto
PRENTICE-HALL OF AUSTRALIA, P'l'Y. W'D.,
PRENTICE-HALL OF INDIA PRIVATE LIMITED, New Delhi
PRENTICE-HALL OF JAPAN, INC., Tokyo
Current printing (last digit):
10 9 8 7 6
Library of Congress Catalog Card No. 75-142120
Printed in the United States of America
, Preface
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.LT. 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
up to the general theory. We have split Chapter 8 so that the basic material on
in relation to triangulation and diagonalization theorems and then build our way
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 of
basic mathematical concepts.
iii
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
All rights reserved. No part of this book may be
reproduced in any form or by any means without
permission in writing from the publisher.
London
Sydney
PRENTICE-HALL INTERNATIONAL, INC.,
PRENTICE-HALL OF CANADA, LTD., Toronto
PRENTICE-HALL OF AUSTRALIA, P'l'Y. W'D.,
PRENTICE-HALL OF INDIA PRIVATE LIMITED, New Delhi
PRENTICE-HALL OF JAPAN, INC., Tokyo
Current printing (last digit):
10 9 8 7 6
Library of Congress Catalog Card No. 75-142120
Printed in the United States of America
, Preface
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.LT. 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
up to the general theory. We have split Chapter 8 so that the basic material on
in relation to triangulation and diagonalization theorems and then build our way
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 of
basic mathematical concepts.
iii
