A Course of Higher Mathematics

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.

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

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