A Course of Higher Mathematics
International Series of Monographs In: Pure and Applied Mathematics, Volume 3, Part 1
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International Series of Monographs in Pure and Applied Mathematics, Volume 59: A Course of Higher Mathematics, III/I: Linear Algebra focuses on algebraic methods. The book first ponders on the properties of determinants and solution of systems of equations. The text then gives emphasis to linear transformations and quadratic forms. Topics include coordinate transformations in three-dimensional space; covariant and contravariant affine vectors; unitary and orthogonal transformations; and basic matrix calculus. The selection also focuses on basic theory of groups and linear representations of groups. Representation of a group by linear transformations; linear representations of the unitary group in two variables; linear representations of the rotation group; and Abelian groups and representations of the first degree are discussed. Other considerations include integration over groups, Lorentz transformations, permutations, and classes and normal subgroups. The text is a vital source of information for students, mathematicians, and physicists.
Table of Contents
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 Determinants
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 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 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 DjXdj,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
85.1nfinitesimal Transformations and Representations of the Rotation Group
86. Representations of The Lorentz Group
87. Auxiliary Formulae
88. The Formation of Groups with Given Structural Constants
89. Integration Over Groups
90. Orthogonality. Examples
Index
Volumes Published in This Series
Product details
- No. of pages: 336
- Language: English
- Copyright: © Pergamon 2013
- Published: January 1, 1964
- Imprint: Pergamon
- eBook ISBN: 9781483140131
About the Author
V. I. Smirnov
About the Editors
I. N. Sneddon
M. Stark
S. Ulam
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