Linear Algebra: A Course of Higher Mathematics, Volume III, Part I deals with linear algebra and the theory of groups that are usually found in theoretical physics.

This volume discusses linear algebra, quadratic forms theory, and the theory of groups. The properties of determinants are discussed for determinants offer the solution of systems of equations. Cramer's theorem is used to find the solution of a system of linear equations that has as many equations as unknowns. Linear transformations and quadratic forms, for example, coordinate transformation in three-dimensional space and general linear transformation of real three-dimensional space, are considered. The formula for n-dimensional complex space and the transformation of a quadratic form to a sum of squares are analyzed. The latter is explained by using Jacobi's formula to arrive at a significant form of the reduction of a quadratic form to a sum of squares. The basic theory of groups, linear representations of groups, and the theory of partial differential equations that is the basis of the formation of groups with given structural constants are explained.

This book is recommended for mathematicians, students, and professors in higher mathematics and theoretical physics.

IntroductionPreface 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 Seterminants 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 Contra-variant 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 Dj x Dj, of Linear Representations of the Rotation Group 76. Orthogonality 77. Characters 78.