Mir Books

Higher algebra—the subject of this text—is a far-reaching and natural generalization of the basic school course of elementary algebra. Central to elementary algebra is without doubt the problem of solving equations. The study of equations begins with the very simple case of one equation of the first degree in one unknown. From there on, the development proceeds in two directions: to systems of two and three equations of the first degree in two and, respectively, three unknowns, and to a single quadratic equation in one unknown and also to a few special types of higher-degree equations which readily reduce to quadratic equations (quartic equations, for example).

The second half of the course of higher algebra, called the algebra of polynomials, is devoted to the study of a single equation in one unknown but of arbitrary degree. Since there is a formula for solving quadratic equations, it was natural to seek similar formulas for higher-degree equations. That is precisely how this division of algebra developed historically. Formulas for solving equations of third and fourth degree were found in the sixteenth century. The search was then on for formulas capable of expressing the roots of equations of fifth and higher degree in terms of the coefficients of the equations by means of radicals, even radicals within radicals. It was futile, though it continued up to the beginning of the nine­ teenth century, when it was proved that no such formulas exist and that for all degrees beyond the fourth there even exist specific examples of equations with integral coefficients whose roots cannot be written down by means of radicals.

Translated from the Russian by George Yankovsky

Note: This is a new, hi-res scan.

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CONTENTS

Introduction

Chapter 1. Systems of Linear Equations. Determinants
1. The Method of Successive Elimination of Unknowns
2. Determinants of Second and Third Order
3. Arrangements and Permutations
4. Determinants of nth Order
5. Minors and Their Cofactors
6. Evaluating Determinants
7. Cramer’s Rule

Chapter 2. Systems of Linear Equations (General Theory)
8. n-Dimensional Vector Space
9. Linear Dependence of Vectors
10. Rank of a Matrix
11. Systems of Linear Equations
12. Systems of Homogeneous Linear Equations

Chapter 3. The Algebra of Matrices
13. Matrix Multiplication
14. Inverse Matrices
15. Matrix Addition and Multiplication of a Matrix by a Scalar
16. An Axiomatic Construction of the Theory of Determinants

Chapter 4. Complex Numbers
17. The System of Complex Numbers
18. A Deeper Look at Complex Numbers
19. Taking Roots of Complex Numbers

Chapter 5. Polynomials and Their Roots
20. Operations on Polynomials
21. Divisors. Greatest Common Divisor
22. Roots of Polynomials
23. Fundamental Theorem
24. Corollaries to the Fundamental Theorem
25. Rational Fractions

Chapter 6. Quadratic Forms
26. Reducing a Quadratic Form to Canonical Form
27. Law of Inertia
28. Positive Definite Forms

Chapter 7. Linear Spaces
29. Definition of a Linear Space. An Isomorphism
30. Finite-Dimensional Spaces. Bases
31. Linear Transformations
32. Linear Subspaces
33. Characteristic Roots and Eigenvalues

Chapter 8. Euclidean Spaces
34. Definition of a Euclidean Space. Orthonormal Bases
35. Orthogonal Matrices, Orthogonal Transformations
36. Symmetric Transformations
37. Reducing a Quadratic Form to Principal Axes. Pairs of Forms

Chapter 9. Evaluating Roots of Polynomials
38. Equations of Second, Third and Fourth Degree
39. Bounds of Roots
40. Sturm’s Theorem
41. Other Theorems on the Number of Real Roots
42. Approximation of Roots

Chapter 10. Fields and Polynomials
43. Number Rings and Fields
44. Rings
45. Fields
46. Isomorphisms of Rings (Fields). The Uniqueness of the Field of Complex Numbers
47. Linear Algebra and the Algebra of Polynomials Over an Arbitrary Field
48. Factorization of Polynomials into Irreducible Factors
49. Theorem on the Existence of a Root
50. The Field of Rational Fractions

Chapter 11. Polynomials in Several Unknowns
51. The Ring of Polynomials in Several Unknowns
52. Symmetric Polynomials
53. Symmetric Polynomials Continued
54. Resultant. Elimination of Unknowns. Discriminant
55. Alternative Proof of the Fundamental Theorem of the Algebra of Complex Numbers

Chapter 12. Polynomials with Rational Coefficients
56. Reducibility of Polynomials over the Field of Rationals
57. Rational Roots of Integral Polynomials
58. Algebraic Numbers

Chapter 13. Normal Form of a Matrix
59. Equivalence of λ-Matrices
60. Unimodular λ-Matrices. Relationship Between Similarity of Numerical Matrices and the Equivalence of Their Characteristic Matrices
61. Jordan Normal Form
62. Minimal Polynomials

Chapter 14. Groups
63. Definition of a Group
64. Subgroups
65. Normal Divisors, Factor Groups, Homomorphisms
66. Direct Sums of Abelian Groups
67. Finite Abelian Groups

Bibliography

Index

Screenshot

This book has been written on the basis of lectures which I delivered at the department of mathematics and mechanics of Moscow State Uni­ versity. In drawing up the program for my lectures, I proceeded on the belief that the selection of material must not be random nor must it rest exclusively on established tradition. The most important and interesting applications of ordinary differential equations to engineering are found in the theory of oscillations and in the theory of automatic control. These applications were chosen to serve as guides in the selection of material. Since oscillation theory and automatic control theory without doubt also play a very important role in the development of our contemporary tech­ nical culture, my approach to the selection of material for the lecture course is, if not the only possible one, in any case a reasonable one. In attempting to give the students not only a purely mathematical tool suitable for engineering applications, but also to demonstrate the appli­ cations themselves, I included certain engineering problems in the lectures. In the book they are presented in §13, 27, and 29. I consider that these problems constitute an integral organic part of the lecture course and, accordingly, of this book.

We received a mail with the title “Pontryagin – Ordinary Differential Equations (retyped in LaTeX)” from Althea Sindy

inside it was this gem of book typeset in LaTeX with this lovely message

“I have given this book the love it deserves. :3”

You sure have, thanks a ton!

All credits to Althea Sindy for reviving this gem of book and giving it a new life.

You can get the book here and here

Contents

1 INTRODUCTION 1

1 First-order differential equations . . . . . . . . . . . . . . 1

2 Some elementary integration methods . . . . . . . . . . . 6

3 Formulation of the existence and uniqueness theorem . . . 20

4 Reduction of a general system of differential equations to a

normal system . . . . . . . . . . . . . . . . . . . . . . . . 5 Complex differential equations . . . . . . . . . . . . . . . 6 Some properties of linear differential equations . . . . . . 28

36

42

2 LINEAR EQUATIONS WITH CONSTANT COEFFI-

CIENTS 45

7 The linear homogeneous equation with constant coefficients.

Case of simple roots . . . . . . . . . . . . . . . . . . . . . 46

8 The linear homogeneous equation with constant coefficients.

Case of multiple roots . . . . . . . . . . . . . . . . . . . . 9 Stable polynomials . . . . . . . . . . . . . . . . . . . . . . 55

62

10 The linear nonhomogeneous equation with constant coeffi-

cients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 13 14 Method of elimination . . . . . . . . . . . . . . . . . . . . 12 The method of complex amplitudes . . . . . . . . . . . . . Electrical circuits . . . . . . . . . . . . . . . . . . . . . . . 68

73

82

87

The normal linear homogeneous system with constant coef-

ficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

15 Autonomous systems of differential equations and their

phase spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 113

16 The phase plane of a linear homogeneous system with con-

stant coefficients . . . . . . . . . . . . . . . . . . . . . . . 127

3 LINEAR EQUATIONS WITH VARIABLE COEFFI-

CIENTS 143

17 The normal system of linear equations . . . . . . . . . . . 143

vvi CONTENTS

18 19 The linear equation of nth order . . . . . . . . . . . . . . The normal linear homogeneous system with periodic coef-

ficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

161

4 EXISTENCE THEOREMS 169

20 Proof of the existence and uniqueness theorem for one

equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

21 Proof of the existence and uniqueness theorem for a normal

system of equations . . . . . . . . . . . . . . . . . . . . . . 179

22 23 Local theorems of continuity and differentiability of solutions 190

First integrals . . . . . . . . . . . . . . . . . . . . . . . . . 203

24 Behavior of the trajectories on large time intervals . . . . 211

25 Global theorems of continuity and differentiability . . . . 214

5 STABILITY 223

26 27 28 29 30 Lyapunov’s theorem . . . . . . . . . . . . . . . . . . . . . 225

The centrifugal governor and the analysis of Vyshnegradskiy 237

Limit cycles . . . . . . . . . . . . . . . . . . . . . . . . . . 245

The vacuum-tube oscillator . . . . . . . . . . . . . . . . . 265

The states of equilibrium of a second-order autonomous

system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Stability of periodic solutions . . . . . . . . . . . . . . . . 273

293

6 LINEAR ALGEBRA 311

32 33 34 The minimal annihilating polynomial . . . . . . . . . . . . Matrix functions . . . . . . . . . . . . . . . . . . . . . . . The Jordan form of a matrix . . . . . . . . . . . . . . . . 311

318

326