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.
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978-1-4831-4013-1 (9781483140131)
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ContentsIntroductionPreface to the Fourth Russian EditionChapter I Determinants. The Solution of Systems of Equations§ 1. Properties of Determinants1. Determinants2. Permutations3. Fundamental Properties of Determinants4. Evaluation of Determinants5. Examples6. Multiplication of Determinants7. Rectangular Arrays§ 2. The Solution of Systems of Equations8. Cramer's Theorem9. The General Case of Systems of Equations10. Homogeneous Systems11. Linear Forms12. N-Dimensional Vector Space13. Scalar Product14. Geometrical Interpretation of Homogeneous Systems15. Non-Homogeneous Systems16. Gram's Determinant. Hadamard's Inequality.17. Systems of Linear Differential Equations with Constant Coefficients.18. Functional Determinants19. Implicit FunctionsChapter II Linear Transformations and Quadratic Forms20. Coordinate Transformations in Three-Dimensional Space21. General Linear Transformations of Real Three-Dimensional Space22. Covariant and Contravariant Affine Vectors23. Tensors. 24. Examples of Affine Orthogonal Tensors25. The Case of N-Dimensional Complex Space26. Basic Matrix Calculus27. Characteristic Roots of Matrices and Reduction to Canonical Form28. Unitary and Orthogonal Transformations29. Buniakowski's Inequality30. Properties of Scalar Products and Norms31. Orthogonalization of Vectors32. Transformation of A Quadratic Form To A Sum of Squares33. The Case of Multiple Roots of The Characteristic Equation34. Examples35. Classification of Quadratic Forms36. Jacobi's Formula37. The Simultaneous Reduction of Two Quadratic Forms To Sums of Squares38. Small Vibrations39. Extremal Properties of The Eigenvalues of Quadratic Forms40. Hermitian Matrices and Hermitian Forms41. Commutative Hermitian Matrices42. The Reduction of Unitary Matrices to The Diagonal Form43. Projection Matrices44. Functions of Matrices45. Infinite-Dimensional Space46. The Convergence of Vectors47. Complete Systems of Mutually Orthogonal Vectors48. Linear Transformations with An Infinite Set of Variables49. Functional Space50. The Connection Between Functional and Hilbert Space51. Linear Functional OperatorsChapter III. The Basic Theory of Groups and Linear Representations of Groups52. Groups of Linear Transformations53. Groups of Regular Polyhedra54. Lorentz Transformations55. Permutations. 56. Abstract Groups57. Subgroups. 58. Classes and Normal Subgroups59. Examples60. Isomorphic and Homomorphic Groups61. Examples62. Stereographic Projections63. Unitary Groups and Groups of Rotations64. The General Linear Group and the Lorentz Group65. Representation of A Group By Linear Transformations66. Basic Theorems67. Abelian Groups and Representations of the First Degree68. Linear Representations of the Unitary Group In Two Variables69. Linear Representations of The Rotation Group70. The Theorem On the Simplicity of the Rotation Group71. Laplace's Equation and Linear Representations of the Rotation Group72. Direct Matrix Products. 73. The Composition of Two Linear Representations of A Group74. The Direct Product of Groups and Its Linear Representations75. Decomposition of the Composition DjXdj,of Linear Representations of the Rotation Group76. Orthogonality77. Characters78. Regular Representations of Groups79. Examples of Representations of Finite Groups80. Representations of A Linear Group In Two Variables81. Theorem On The Simplicity of the Lorentz Group82. Continuous Groups. Structural Constants83. Infinitesimal Transformations84. Rotation Groups85.1nfinitesimal Transformations and Representations of the Rotation Group86. Representations of The Lorentz Group87. Auxiliary Formulae88. The Formation of Groups with Given Structural Constants89. Integration Over Groups90. Orthogonality. ExamplesIndexVolumes Published in This Series
