Linear Algebras

Table of Contents

• Preface
• Part I. Definitions, Illustrations and Elementary Theorems:
• 1-2. Ordinary complex numbers, number fields
• 3-4. Matrices
• matric algebra as a linear algebra
• 5. Definition of linear algebras, coordinates, units
• 6-8. Division
• principal unit
• transformation of units
• 9-10. Equations and polynomials in a single number
• 11. Unique place of quaternions among algebras
• 12-13. Properties of quaternions, relation to matrices
• 14. Cayley's generalization of real quaternions
• 15-17. Characteristic determinanats
• invariance
• 18. Binary algebras
• 19. Rank and rank equations of an algebra
• 20. Ternary algebras with a modulus
• 21-22. Reducibility, direct sum, direct product
• 23-24. Units normalized relative to a number
• example
• Part II. Revision of Cartan's General Theory of Complex Linear Associative Algebras with a Modulus:
• 25-27. Units having a character
• example
• 28-34. Nilpotent numbers, normalized units
• examples
• 35. Separation of algebras into two categories
• 36-38. Algebras A1 of first category, nilpotent algebras
• 39. Normalized units for A1
• 40-45. Algebras A2 of the second category
• 46. Any A2 has a quaternion sub-algebra
• 47-48. Normalized units for A2
• determinant
• 49. Invariant sub-algebra, simple algebras
• 50-51. Main theorem
• commutative case
• Part III. Relations of Linear Algebras to Other Subjects:
• 52. Linear associative algebras and linear groups
• 53. Linear associative algebras and bilinear forms
• 54. Relations of linear algebras and finite groups
• 55. Dedekind's view for commutative associative algebras
• Part IV. Linear Algebras over a Field F:
• 56. Statement of main theorem
• real simple algebras
• 57. Algebras of Weierstrass
• 58-60. Division algebras
• 61. Statement of further results
• 62. Analytic functions of hypercomplex numbers.
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