Introduction to Modern Algebra and Matrix Theory

434 Pages·2011·12.42 MB·English

by  O. Schreier

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
Existence and Uniqueness of Determinants
Volumes
§ 9. The Principal Theorems of Determinant Theory
The Complete Development of a Determinant
The Determinant as a Function of its Column Vectors
The Multiplication Theorem
The Development of a Determinant by Rows or Columns
Determinants and Linear Equations
Laplace’s Expansion Theorem
§ 10. Transformation of Coordinates
General Linear Coordinate Systems
Cartesian Coordinate Systems
Continuous Deformation of a Linear Coordinate System
§ 11. Construction of Normal Orthogonal Systems and Applications
§ 12. Rigid Motions
Rigid Motions in R
2
Rigid Motions in R
3
§ 13. Affine Transformations
C III
HAPTER
FIELD THEORY; THE FUNDAMENTAL
THEOREM OF ALGEBRA
§ 14. The Concept of a Field
§ 15. Polynomials over a Field
§ 16. The Field of Complex Numbers
§ 17. The Fundamental Theorem of Algebra
C IV
HAPTER
ELEMENTS OF GROUP THEORY
§ 18. The Concept of a Group
§ 19. Subgroups; Examples