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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