A Course of Higher Mathematics

A Course of Higher Mathematics

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

1st Edition - January 1, 1964

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  • Author: V. I. Smirnov
  • eBook ISBN: 9781483140131

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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


    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


    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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