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Linear Algebra and Group Theory

Publisher: 
Dover Publications
Number of Pages: 
464
Price: 
29.95
ISBN: 
9780486482224

This is a very traditional, not to say old-fashioned, text in linear algebra and group theory, slanted very much towards physics. The present volume is a 2011 unaltered reprint of the 1961 McGraw-Hill edition, which was in turn extracted, translated, and edited from Smirnov’s 6-volume Russian-language work by Richard A. Silverman. The Russian work had no exercises, and Silverman added about 400 exercises from a variety of sources. Most of these have brief hints and answers in the back of the book.

The linear algebra portion focuses on matrices and their manipulation and use for solving systems of linear equations. This includes some discussion of vectors and tensors and of covariant and contravariant transformations. The book ignores numerical issues, and gives only Cramer’s rule for solving systems of equations. There is also a chapter on infinite-dimensional spaces, dealing with the Hilbert spaces l2 and L2. The treatment is very concrete and these spaces are treated only as collections of sequences or functions, not as complete normed vector spaces.

The group theory portion takes the last one-third of the book. It is not the abstract group theory we see in courses today, but is a detailed look at several particular types of concrete groups, in particular rotation and symmetry groups. There is also a fairly long chapter on group representations, that builds on the earlier work on matrices.

The teaching of linear algebra has received much attention in the curriculum in recent years, and I believe that today we have much better linear algebra books than this one. I especially like Strang’s Introduction to Linear Algebra. The treatment of group theory has evolved less, and this book is still useful as a brief introduction to groups in physics. There are a number of more recent and more detailed books on portions of this subject, in various degrees of abstraction. The comprehensive 1962 text Group Theory and its Application to Physical Problems by Morton Hamermesh (Addison-Wesley, reprinted in 1989 by Dover) covers the same ground as the last third of Smirnov, but in much greater depth.


Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

Date Received: 
Tuesday, August 2, 2011
Reviewable: 
Yes
Include In BLL Rating: 
No
V. I. Smirnov
Publication Date: 
2011
Format: 
Paperback
Category: 
Textbook
Allen Stenger
05/3/2012
  • Preface
  • PART I: DETERMINANTS AND SYSTEMS OF EQUATIONS
    • Chapter 1. Determinants and Their Properties
      • 1. The Concept of a Determinant
      • 2. Permutations
      • 3. Basic Properties of Determinants
      • 4. Calculation of Determinants
      • 5. Examples
      • 6. The Multiplication Theorem for Determinants
      • 7. Rectangular Matrices
      • Problems
    • Chapter 2. Solution of Systems of Linear Equations
      • 8. Cramer’s Rule
      • 9. The General Case
      • 10. Homogeneous Systems
      • 11. Linear Forms
      • 12. n-Dimensional Vector Space
      • 13. The Scalar Product
      • 14. Geometrical Interpretation of Homogeneous Systems
      • 15. Inhomogeneous Systems
      • 16. The Gram Determinant. Hadamard’s Inequality
      • 17. Systems of Linear Differential Equations with Constant Coefficients
      • 18. Jacobians
      • 19. Implicit Functions
      • Problems
  • PART II: MATRIX THEORY
    • Chapter 3. Linear Transformations
      • 20. Coordinate Transformations in Three Dimensions
      • 21. General Linear Transformations in Three Dimensions
      • 21. Covariant and Contravariant Affine Vectors
      • 23. The Tensor Concept
      • 24. Cartesian Tensors
      • 25. The n-Dimensional Case
      • 26. Elements of Matrix Algebra
      • 27. Eigenvalues of a Matrix. Reduction of a Matrix to Canonical Form
      • 28. Unitary and Orthogonal Transformations
      • 29. Schwarz’s Inequality
      • 30. Properties of the Scalar Product and Norm
      • 31. The Orthogonalization Process for Vectors
      • Problems
    • Chapter 4. Quadratic Forms
      • 32. Reduction of a Quadratic Form to a Sum of Squares
      • 33. Multiple Roots of the Characteristic Equation
      • 34. Examples
      • 35. Classification of Quadratic Forms
      • 36. Jacobi’s Formula
      • 37. Simultaneous Reduction of Two Quadratic Forms to Sums of Squares
      • 38. Small Oscillations
      • 39. Extremal Properties of the Eigenvalues of a Quadratic Form
      • 40. Hermitian Matrices and Hermitian Forms
      • 41. Commuting Hermitian Matrices
      • 42. Reduction of Unitary Matrices to Diagonal Form
      • 43. Projection Matrices
      • 44. Functions of Matrices
      • Problems
    • Chapter 5. Infinite-Dimensional Spaces
      • 45. Infinite-Dimensional Spaces
      • 46. Convergence of Vectors
      • 47. Complete Systems of Orthonormal Vectors
      • 48. Linear Transformations in Infinitely Many Variables
      • 49. Function Space
      • 50. Relation between the Spaces F and H
      • 51. Linear Operators
      • Problems
    • Chapter 6. Reduction of Matrices to Canonical Form
      • 52. Preliminary Considerations
      • 53. The Case of Distinct Roots
      • 54. The Case of Multiple Roots. First Step in the Reduction
      • 55. Reduction to Canonical Form
      • 56. Determination of the Structure of the Canonical Form
      • 57. An Example
      • Problems
  • PART III: GROUP THEORY
    • Chapter 7. Elements of the General Theory of Groups
      • 58. Groups of Linear Transformations
      • 59. The Polyhedral Groups
      • 60. Lorentz Transformations
      • 61. Permutations
      • 62. Abstract Groups
      • 63. Subgroups
      • 64. Classes and Normal Subgroups
      • 65. Examples
      • 66. Isomorphic and Homomorphic Groups
      • 67. Examples
      • 68. Stereographic Projection
      • 69. The Unitary Group and the Rotation Group
      • 70. The Unimodular Group and the Lorentz Group
      • Problems
    • Chapter 8. Representations of Groups
      • 71. Representation of Groups by Linear Transformations
      • 72. Basic Theorems
      • 73. Abelian Groups and One-Dimensional Representations
      • 74. Representations of the Two-Dimensional Unitary Group
      • 75. Representations of the Rotation Group
      • 76. Proof That the Rotation Group Is Simple
      • 77. Laplace’s Equation and Representations of the Rotation Group
      • 78. The Direct Product of Two Matrices
      • 79. The Direct Product of Two Representations of a Group
      • 80. The Direct Product of Two Groups and its Representations
      • 81. Reduction of the Direct Product Dj × Dj' of Two Representations of the Rotation Group
      • 82. The Orthogonality Property
      • 83. Characters
      • 84. The Regular Representation of a Group
      • 85. Examples of Representations of Finite Groups
      • 86. Representations of the Two-Dimensional Unimodular Group
      • 87. Proof That the Lorentz Group Is Simple
      • Problems
    • Chapter 9. Continuous Groups
      • 88. Continuous Groups. Structure Constants
      • 89. Infinitesimal Transformations
      • 90. The Rotation Group
      • 91. Infinitesimal Transformations and Representations of the Rotation Group
      • 92. Representations of the Lorentz Group
      • 93. Auxiliary Formulas
      • 94. Construction of a Group from its Structure Constants
      • 95. Integration on a Group. The Orthogonality Property
      • 96. Examples
      • Problems
  • Bibliography
  • Hints and Answers
  • Index
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