Introduction to Modern Algebra and Matrix Theory

539 Pages·2011·8.951 MB·English

by  Otto Schreier, Emanuel Sperner

INTRODUCTION TO
MODERN ALGEBRA
AND MATRIX THEORY
INTRODUCTION TO
MODERN ALGEBRA
AND MATRIX THEORY
Second Edition
O. SCHREIER
E. SPERNER
Translated by
M D
ARTIN AVIS
AND
M H
ELVIN AUSNER
DOVER PUBLICATIONS, INC.
Mineola, New York
Bibliographical Note
This Dover edition, first published in 2011, is an unabridged
republication of the work originally published in 1959 by Chelsea
Publishing Company, New York.
Library of Congress Cataloging-in-Publication Data
Schreier, O. (Otto), 1901–1929.
[Einführung in die analytische Geometeie und Algebra. English]
Introduction to modern algebra and matrix / O. Schreier and E.
Sperner. — Dover ed.
              p. cm.
Originally published: 2nd ed. New York : Chelsea Pub., 1959.
Summary: “This unique text provides students with a basic course
in both calculus and analytic geometry. It promotes an intuitive
approach to calculus and emphasizes algebraic concepts. Minimal
prerequisites. Numerous exercises. 1951 edition”—Provided by
publisher.
Includes bibliographical references and index.
ISBN-13: 978-0-486-48220-0 (pbk.)
ISBN-10: 0-486-48220-0 (pbk.)
  1. Geometry, Analytic. 2. Algebra. I. Sperner, E. (Emanuel), 1905-
II. Title.
    QA551.S433 2011
    516.3—dc22
2011000043
Manufactured in the United States by Courier Corporation
48220001
www.doverpublications.com
EDITOR’S PREFACE
Schreier and Sperner’s justly famous Einführung in die
Algebra und Analytische Geometrie was originally published
in two volumes. Both of these volumes (with the omission of
a final chapter on projective geometry) are here presented
in a one-volume English translation, under the title
Introduction to Modern Algebra and Matrix Theory.
It may be noted that Chapter V of the present translation
(Volume II, Chapter II of the original) incorporates the
material of the authors’ book Lectures on Matrices
(Vorlesungen über Matrizen).
F. S. A. G.
TRANSLATOR’S PREFACE
We wish to thank Dr. F. Steinhardt for his very valuable
help in the preparation of the present translation. Thanks
are due, also, to Miss Zelma Ann McCormick.
MARTIN DAVIS     
MELVIN HAUSNER
FROM THE PREFACE TO VOLUME I
OF THE GERMAN EDITION
Otto Schreier had planned, a few years ago, to have his
lectures on Analytic Geometry and Algebra published in
book form. Death overtook him in Hamburg on June 2, 1929,
before he had really begun to carry out his plan. The task of
doing this fell on me, his pupil. I had at my disposal some
sets of lecture notes taken at Schreier’s courses, as well as
a detailed (if not quite complete) syllabus of his course
drawn up at one time by Otto Schreier himself. Since then, I
have also given the course myself, in Hamburg, gaining
experience in the process.
In writing the book I have followed Schreier’s
presentation as closely as possible, so that the textbook
might retain the characteristics impressed on the subject
matter in Otto Schreier’s treatment. In particular, as regards
choice and arrangement of material, I have followed
Schreier’s outline faithfully, except for a few changes of
minor importance.
This textbook is motivated by the idea of offering the
student, in two basic courses on Calculus and Analytic
Geometry, all that he needs for a profitable continuation of
his studies adapted to modern requirements. It is evident
that this implies a stronger emphasis than has been
customary on algebra, in line with the recent developments
in that subject.
The prerequisites for reading this book are few indeed.
For the early parts, a knowledge of the real number system
—such as is acquired in the first few lectures of almost any
calculus course—is sufficient. The later chapters make use
of some few theorems on continuity of real functions and on
sequences of real numbers. These also will be familiar to the
student from the calculus. In some sections which give
intuitive interpretations of the subject matter, use is made
of some well-known theorems of elementary geometry,
whose derivation on an axiomatic basis would of course be
beyond the scope of this text.
What the book contains may be seen in outline by a
glance at the table of contents. The student is urged not to
neglect the exercises at the end of each section; among
them will be found many an important addition to the
material presented in the text.
To Messrs. E. Artin and W. Blaschke in Hamburg I wish to
extend my most cordial thanks for their manifold help. I also
wish to express my gratitude for reading the proofs, to Miss
A. Voss (Hamburg) and to Messrs. R. Brauer (Koenigsberg),
G. Feigl (Berlin), K. Henke (Hamburg), and E. Schubarth
(Basel).
EMANUEL SPERNER
FROM THE PREFACE TO VOLUME II
OF THE GERMAN EDITION
The second volume of Einführung in die Analytische
Geometrie und Algebra… is divided into three chapters, the
first two of which [Chapters IV and V of the present
translation] deal with algebraic topics, while the last, and
longest, [to be published as a separate volume.—Ed.] is an
analytic treatment of (n-dimensional) projective geometry.
The authors’ earlier monograph, Vorlesungen über
Matrizen (Hamburger Einzelschriften, Vol. 12) has been
incorporated into [Chapter V of] the present book, with a
few re-arrangements and omissions in order to achieve a
more organic whole.
•         •         •
To Mr. W. Blaschke (Hamburg) I owe a debt of gratitude
for his continuous interest and help. I also wish to thank
Messrs. O. Haupt (Erlangen) and K. Henke (Hamburg) for
many valuable hints and suggestions. In preparing the
manuscript, my wife has given me untiring assistance. For
reading the proofs I am indebted to Mr. H. Bueckner
(Koenigsberg), in addition to those named above.
Koenigsberg, October 1935.
EMANUEL SPERNER
TABLE OF CONTENTS
EDITOR’S PREFACE
TRANSLATORS’ PREFACE
AUTHORS’ PREFACE
C I
HAPTER
AFFINE SPACE; LINEAR EQUATIONS
§ 1. n-dimensional Affine Space
§ 2. Vectors
§ 3. The Concept of Linear Dependence
§ 4. Vector Spaces in R
n
§ 5. Linear Spaces
§ 6. Linear Equations
Homogeneous Linear Equations
Non-homogeneous Linear Equations
Geometric Applications
C II
HAPTER
EUCLIDEAN SPACE; THEORY OF
DETERMINANTS
§ 7. Euclidean Length
Appendix to § 7: Calculating with the Summation Sign
§ 8. Volumes and Determinants
Fundamental Properties of Determinants