# A Course of Higher Mathematics

## 1st Edition

### Adiwes International Series in Mathematics, Volume 3, Part 1

**Author:**V. I. Smirnov

**Editor:**A. J. Lohwater

**eBook ISBN:**9781483152592

**Imprint:**Pergamon

**Published Date:**1st January 1964

**Page Count:**334

## Description

Linear Algebra: A Course of Higher Mathematics, Volume III, Part I deals with linear algebra and the theory of groups that are usually found in theoretical physics.

This volume discusses linear algebra, quadratic forms theory, and the theory of groups. The properties of determinants are discussed for determinants offer the solution of systems of equations. Cramer's theorem is used to find the solution of a system of linear equations that has as many equations as unknowns. Linear transformations and quadratic forms, for example, coordinate transformation in three-dimensional space and general linear transformation of real three-dimensional space, are considered. The formula for n-dimensional complex space and the transformation of a quadratic form to a sum of squares are analyzed. The latter is explained by using Jacobi's formula to arrive at a significant form of the reduction of a quadratic form to a sum of squares. The basic theory of groups, linear representations of groups, and the theory of partial differential equations that is the basis of the formation of groups with given structural constants are explained.

This book is recommended for mathematicians, students, and professors in higher mathematics and theoretical physics.

## Table of Contents

Introduction

Preface to the Fourth Russian Edition

Chapter I. Determinants. The Solution of Systems of Equations

§ 1. Properties of Determinants

1. Determinants

2. Permutations

3. Fundamental Properties of Determinants

4. Evaluation of Determinants

5. Examples

6. Multiplication of Determinants

7. Rectangular Arrays

§ 2. The Solution of Systems of Equations

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 Homogeneous Systems

15. Non-homogeneous Systems

16. Gram's Determinant. Hadamard's Inequality

17. Systems of Linear Differential Equations with Constant Coefficients

18. Functional Seterminants

19. Implicit Functions

Chapter II. Linear Transformations and Quadratic Forms

20. Coordinate Transformations in Three-dimensional Space

21. General Linear Transformations of Real Three-dimensional Space

22. Covariant and Contra-variant 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 Reduction 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 Quadratic Form to a Sum of Squares

33. The Case of Multiple Roots of the Characteristic Equation

34. Examples

35. Classification of Quadratic Forms

36. Jacobi's Formula

37. The Simultaneous Reduction of Two Quadratic Forms to Sums of Squares

38. Small Vibrations

39. Extremal Properties of the Eigenvalues of Quadratic Forms

40. Hermitian Matrices and Hermitian Forms

41. Commutative Hermitian Matrices

42. The Reduction of Unitary Matrices to the Diagonal Form

43. Projection Matrices

44. Functions of Matrices

45. Infinite-dimensional 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

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. Isomorphic and Homomorphic Groups

61. Examples

62. Stereographic Projections

63. Unitary Groups and Groups of Rotations

64. The General Linear Group and the Lorentz Group

65. Representation 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 Representations 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 Dj x Dj, of Linear Representations of the Rotation Group

76. Orthogonality

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. Representations of the Lorentz Group

87. Auxiliary Formula

88. The Formation of Groups with Given Structural Constants

89. Integration over Groups

90. Orthogonality. Examples

Index

## Details

- No. of pages:
- 334

- Language:
- English

- Copyright:
- © Pergamon 1964

- Published:
- 1st January 1964

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483152592