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Skip to main content# A Course of Higher Mathematics

## International Series of Monographs In: Pure and Applied Mathematics, Volume 3, Part 1

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1st Edition - January 1, 1964

Author: V. I. Smirnov

Editors: I. N. Sneddon, M. Stark, S. Ulam

Language: EnglisheBook ISBN:

9 7 8 - 1 - 4 8 3 1 - 4 0 1 3 - 1

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… Read more

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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.

ContentsIntroductionPreface to the Fourth Russian EditionChapter 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 FunctionsChapter 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 OperatorsChapter 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. ExamplesIndexVolumes Published in This Series

- No. of pages: 336
- Language: English
- Edition: 1
- Published: January 1, 1964
- Imprint: Pergamon
- eBook ISBN: 9781483140131