Problems in Higher Algebra

512 Pages·1965·24.345 MB·English

by  D. K. Faddeev, I. S. Sominskii

Problems in Higher Algebra
A SERIES OF BOOKS IN MATHEMATICS
R. A. Rosenbaum, Editor
Daniel T. Finkbeiner, II Introduction to Matrices and Linear Transformations
Henry L. Alder Introduction to Probability and Statistics (Third Edition)
and Edward B. Roessler
D. O. Shklarsky, The USSR Olympiad Problem Book: Selected Problems and Theorems
N. N. Chentzov, of Elementary Mathematics
and I. M. Yaglom
Sherman K. Stein Mathematics: The Man-made Universe,
Robert R. Stoll Set Theory and Logic
J. L. Brenner Problems in Differential Equations
A. I. Mal'cev Foundations of Linear Algebra
D. K. Faddeev Computational Methods of Linear Algebra
and V. N. Faddeeva
C. E. Springer Geometry and Analysis of Projective Spaces
Jack R. Britton, University Mathematics. I and II
R. Ben Kriegh,
and Leon W. Rutland
D. K. Faddeev Problems in Higher Algebra
and I. S. Sominskii
GOLDEN GATE EDITIONS
W. W. Sawyer A Concrete Approach to Abstract Algebra
Leonard M. Blumenthal A Modern View of Geometry
Robert R. Stoll Sets. Logic. and Axiomatic Theories
A. O. Gelfond The Solution of Equations in Integers
B. V. Gnedenko An Elementary Introduction to the Theory of Probability
and A. Ya. Khinchin
J. B. Roberts The Real Number System in an Algebraic Setting
Peter van de Kamp Elements of Astromechanics
Problems in Higher Algebra
D. K. Faddeev, I. S. Sominskii
Translated by
J. L. BRENNER
STANFORD RESEARCH INSTITUTE
W. H. FREEMAN AND COMPANY
SAN FRANCISCO AND LONDON
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@ Copyright 1965 by J. L. Brenner.
The publisher reserves all rights to reproduce this book,
in whole or in part, with the exception of the right to
use short quotations for review of the book.
Printed in the United States of America.
Library of Congress Catalog Card Number: 65-18946
PREFACE
Selections from this fine collection of problems
have been quoted ever since its first appearance in 1954.
It is time to make it more widely available to the
western world.
The section on linear equations will be interesting
because of its systematic coverage, and because it uses
the vector-space of then-tuples of coefficients to
expound the principal results. Calculation of determinants
has been lifted from the sterile to the fruitful by means
of imaginative problems, many of them solved by use of
mathematical induction. At the same time, there are
plenty of simple problems, but written by a master who
can make the material interesting.
The section on Sturm sequences includes applications
to the classical polynomials--the polynomials of Legendre,
of Hermite, of Laguerre; and to the truncated exponential
series. A feature of the book is that every problem,
or at the most every group of two or three problems, is
independent, and sections of the book can be sampled or
omitted at the user's discretion.
I have added a few problems to the section on matrices;
I learned them at the 1964 Gatlinburg conference. I hope
the mathematical public will find these and other
additions attractive.
This book is recommended as an adjunct text, as a
problem book, and for self study. It is a pleasure to
acknowledge the splendid cooperation of the publisher,
and the consummate skill of Shirley DeMeo, the
typist-compositor.
Joel Brenner
Menlo Park, California January, 1965
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TABLE OF CONTENTS
Preface
First Part .
PROBLEMS
Chapter I. Complex Numbers .. 1
1. Reckoning with Complex Numbers 1
2. Complex Numbers in Polar Form 4
3. Equations of the Third and Fourth
Degree 13
4. Roots of Unity . 15
Chapter II. Computation of Determinants 22
1. Second and Third Order Determinants 22
2. Permutations •...•...•... 23
3. Definition of Determi.nants .•.•. 25
4. Fundamental Properties of Determinants 28
5. Evaluation of Determinants ..• 32
6. Multiplication of Determinants . 53
7. Miscellaneous Problems ••..• 60
Chapter III. Systems of Linear Equations 66
. .
1. Cramer's Rule 66
.
2. The Rank of a Matrix 70
3. Systems of Linear Forms 73
4. Systems of Linear Equations 75
Chapter IV.· Matrices 87
1. Operations with Square Matrices 87
2. Rectangular Matrices. Certain
Inequalities .... 98
Chapter V. Polynomials and Rational Functions
of a Single Indeterminate . . . 106
1. Properties of Polynomials. Taylor's
Formula, Multiple Roots 106
2. Proof of the Fundamental Theorem of
Algebra and Related Questions 112
3. Factoring into Linear Factors.
Factoring into Factors that are
Irreducible in the Field of Real
Numbers. Formulas Connecting
Coefficients and Roots ... . 114
4. Euclidean Algorithm · ... . 122
5. The Interpolation Problem; Rational
Functions .......... . 126
6. Rational Roots of Polynomials.
Reducibility and Irreducibilitj in
the Field of Rational Numbers 132
7. Bounds for the Roots of Polynomials 137
8. Sturm's Theorem .......•... 140
9. Miscellaneous Methods for the Location
of Roots of Polynomials 145
10. Approximate Calculation of the Roots
of Polynomials .... 151
Chapter VI. Symmetric Functions 154
1. Symmetric Functions in Terms of the
Elementary Symmetric Functions.
Calculation of the Symmetric Functions
of the Roots of an Algebraic Equation. 154
2. Power Sums 162
3. Transformation of Equations 165
4. Resultants and Discriminants 168
5, Tschirnhaus Transformation and the
Elimination of Irrationalities in the
Denominator ..••.•.... 176
6. Polynomials that are Invariant under
Even Permutation of the Variables.
Polynomials that are Invariant under
Circular Permutation of the
Variables . . . . . . . . . . . . . . 178
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Page
Chapter VII. Linear Algebra . 182
1. Subspaces and Linear Varieties.
Coordinate Subspaces. 182
2. Elementary Geometry of n-Dimensional
Euclidean Space . . 186
3. Proper Values and Proper Vectors of a
Matrix . 193
4. Quadratic Forms and Symmetric Matrices 195
5. Linear Spaces. The Jordan Canonical Form 203
Second Part
HINTS
Chapter I. Complex Numbers 211
Chapter II. Computation of Determinants 216
Chapter IV. Matrices . 230
Chapter V. Polynomials and Rational Functions
of a Single Indeterminate 233
Chapter VI. Symmetric Functions 241
Chapter VII . Linear Algebra . . . .. 244
Third Part
SOLUTIONS
Chapter I. Complex Numbers 249
Chapter II. Computation of Determinants 274
Chapter I II . Systems of Linear Equations 291
Chapter IV. Matrices . . . . . . . . . . 303
Chapter V. Polynomials and Rational Functions
of a Single Indeterminate 340
Chapter VI. Symmetric Functions 415
Chapter VI I . Linear Algebra ...... . 453
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