Linear Algebras

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, mormalized 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.