A Course of Higher Mathematics: Linear Algebra

336 Pages·1964·8.084 MB·English

by  V. I. Smirnov

A COURSE OF
HIGHER
MATHEMATICS
V. I. Smirnov
Volume III Part 1
LINEAR
ALGEBRA
I N T E R N A T I O N A L S E R I E S OF M O N O G R A P H S IN
PURE AND APPLIED MATHEMATICS
General Editors: I. N. Sneddon, M. Stark and S. Ulam
Volume 59
A COURSE OF
HIGHER MATHEMATICS
III/I
LINEAR ALGEBRA
A COURSE OF
Higher Mathematics
VOLUME III
PART ONE
V. I. SMIRNOV
Translated by
D. E. BROWN
Translation edited by
I. N. SNEDDON
Simson Professor in Mathematics
University of Glasgow
PERGAMON PRESS
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1964
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Copyright @ 1964
Pebgamon Press Ltd.
Library of Congress Catalog Card Number 63-10134
This translation has been made from the Russian Edition of
I. Smirnov’s book Kypc mcuteu MameMamuKU (Kurs vysshei matemaliki),
published in 1957 by Fizmatgiz, Moscow
MADE IN GBEAT BRITAIN
CONTENTS
vu
Introduction
Preface to the Fourth Russian Edition ix
CHAPTER I
DETERMINANTS. THE SOLUTION OF SYSTEMS OF EQUATIONS
§ 1. Properties of determinants 1
1. Determinants. 2. Permutations. 3. Fundamental properties of
determinants. 4. Evaluation of determinants. 3. Examples. 6. Multipli­
cation of determinants. 7. Rectangular arrays.
§ 2. The solution of systems of equations 30
8. Cramer’s theorem. 9. The general case of systems of equations.
10. Homogeneous systems. 11. Linear forms. 12. n-dimensional vector
space. 13. Scalar product. 14. Geometrical interpretation of homo­
geneous systems. 15. Non-homogeneous systems. 16. Gram’s deter­
minant. Hadamard’s inequality. 17. Systems of linear differential equa­
tions with constant coefficients. 18. Functional determinants. 19.
Implicit functions.
CHAPTER II
LINEAR TRANSFORMATIONS AND QUADRATIC FORMS 70
20. Coordinate transformations in three-dimensional space. 21. General
linear transformations of real three-dimensional space. 22. Covariant
and contravariant affine vectors. 23. Tensors. 24. Examples of affine
orthogonal tensors. 25. The case of n-dimensional complex space. 26.
Basic matrix calculus. 27. Characteristic roots of matrices and reduc­
tion to canonical form. 28. Unitary and orthogonal transformations.
29. Buniakowski’s inequality. 30. Properties of scalar products and
norms. 31. Orthogonalization of vectors. 32. Transformation of a quad­
ratic form to a sum of squares. 33. The case of multiple roots of the cha­
racteristic equation. 34. Examples. 35. Classification of quadratic
forms. 36. Jacobi’s formula. 37. The simultaneous reduction of two quad­
ratic forms to sums of squares. 38. Small vibrations. 39. Extremal
properties of the eigenvalues of quadratic forms. 40. Hennitian
matrices and Hermitian forms. 41. Commutative Hermitian matrices.
42. The reduction of unitary matrices to the diagonal form. 43. Pro­
jection matrices. 44. Functions of matrices. 45. Infinite-dimensional
V
VI CONTESTS
space. 46. The convergence of vectors. 47. Complete systems of
mutually orthogonal vectors. 48. Linear transformations with an
infinite set of variables. 49. Functional space. 50. The connection
between functional and Hilbert space. 51. Linear functional operators.
CHAPTER III
THE BASIC THEORY OF GROUPS AND LINEAR REPRESENTATIONS
OF GROUPS 188
52. Groups of linear transformations. 53. Groups of regular polyhedra.
54. Lorentz transformations. 55. Permutations. 56. Abstract groups.
57. Subgroups. 58. Classes and normal subgroups. 59. Examples.
60. Isomorphio and homomorphic groups. 61. Examples. 62. Stereo-
graphic projections. 63. Unitary groups and groups of rotations.
64. The general linear group and the Lorentz group. 65. Represen­
tation of a group by linear transformations. 66. Basic theorems.
67. Abelian groups and representations of the first degree. 68. Linear
representations of the unitary group in two variables. 69. Linear repre­
sentations of the rotation group. 70. The theorem on the simplicity of
the rotation group. 71. Laplace’s equation and linear representations of
the rotation group. 72. Direct matrix products. 73. The composition of
two linear representations of a group. 74. The direct product of groups
and its linear representations. 75. Decomposition of the composition
DjXDy, of linear representations of the rotation group. 76. Ortho­
gonality. 77. Characters. 78. Regular representations of groups.
79. Examples of representations of finite groups. 80. Representations
of a linear group in two variables. 81. Theorem on the simplicity of
the Lorentz group. 82. Continuous groups. Structural constants.
83.Infinitesimal transformations. 84. Rotation groups. 85.Infinitesimal
transformations and representations of the rotation group. 86. Repre­
sentations of the Lorentz group. 87. Auxiliary formulae. 88. The forma­
tion of groups with given structural constants. 89. Integration over
groups. 90. Orthogonality. Examples.
Index 323
Volumes Published in this Series 326
INTRODUCTION
A brief account of the history of this five-volume course of higher
mathematics has been given in the Introduction to Vol. I of the
present English edition. This volume and the subsequent ones were,
from the first Russian edition (1933), entirely the responsibility of
Professor Smirnov.
In most texts on the methods of mathematical physics algebraic
methods play a minor role compared with methods based on the
theory of functions. This is not so in Professor Smirnov’s scheme. In
this first part of Vol. Ill a full account is given of the two branches
of modern algebra — linear algebra and the theory of groups — which
are most frequently used in theoretical physics. There is a detailed
treatment of the theory of determinants and matrices and of quadra­
tic forms including all the results necessary for an understanding of
the concepts of functional and Hilbert space. The second part is devoted
to a full account of the basic theory of groups and of the linear repre­
sentations of groups. Novel, in a first course on algebra, is the inclusion
of the elements of the theory of continuous groups.
This volume is quite obviously of interest to applied mathemati­
cians and theoretical physicists but its claims as providing material
for a first course in abstract algebra for students of pure mathematics
should not be disregarded.
I. N. Sneddon
vii

PREFACE TO THE FOURTH RUSSIAN EDITION
In the present edition the third volume has been divided into two
parts in connection with the addition of new material. The first part
contains all material referring to linear algebra, to the theory of
quadratic forms, and to the theory of groups. I was greatly assisted
in compiling the additional material by D. K. Faddeyev. He was
partly responsible for the clarification of the simplicity of rotation
and Lorentz groups, for the presentation of the material referring
to the formation of groups with given structural constants and to
integration over groups [70, 81, 87, 88, 89, 90]. I am very grateful
to him for his assistance.
V. Smirnov