Description
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.
Chapter
§ 2. The solution of systems of equations
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.
24. Examples of affine orthogonal tensors.
25· The case of n-dimeneional 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.
35. Classification of quadratic forms.
37. The simultaneous reduction of two quadratic forms to sums of squares.
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.
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.
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.
58. Classes and normal subgroups.
60. Isomorphic and homomorphic groups.
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.
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 Dj X Dy of linear representations of the rotation group·
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.
85. Infinitesimal transformations and representations of the rotation group·
86. Representations of the Lorentz group·
88. The formation of groups with given structural constants.
89. Integration over groups.
90. Orthogonality. Examples.
VOLUMES PUBLISHED IN THIS SERIES
A Course of Higher Mathematics
：International Series of Monographs In: Pure and Applied Mathematics, Volume 3P1
Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.
