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A Course of Higher Mathematics
Adiwes International Series in Mathematics, Volume 3, Part 1
- 1st Edition - January 1, 1964
- Author: V. I. Smirnov
- Editor: A. J. Lohwater
- Language: English
- eBook ISBN:9 7 8 - 1 - 4 8 3 1 - 5 2 5 9 - 2
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 di… Read more
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Request a sales quoteLinear 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.
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
- No. of pages: 334
- Language: English
- Edition: 1
- Published: January 1, 1964
- Imprint: Pergamon
- eBook ISBN: 9781483152592