I.
Preliminaries 

24. Rank of a product 
1. Sum/product symbols 

25. Orthogonal
bases 
2. Mathematical induction 

VI
Linear Spaces

3. Polynomials 

26. Homogeneous/ nonhomogeneous
equations 
4.
Permutations 

27. General solution of homogeneous
system 
II. Determinants


28. Solubility of nonhomogeneous
systems of equations 
5. Determinants of second and
third order 

29. Homogeneous
variable 
6. Determinants of order n 

30. Numerical solution of linear
equations 
III. The most important
properties of Determinants


VII Hermitian/Quadratic forms

7. Determinants, most important
properties 

31. Transformation of Hermitian forms 
8. Description of determinants according to
Weierstrass 

32. Characteristic roots. Eigen
vectors. 
9. Determinant of transposed matrix 

33. Principal axes
transformation 
10. Expansion formulae 

34. Definite Hermitian form 
11. Evaluation of determinants 

VIII. More about
determinants and matrices

12. Cramer's Rule 

35. Vandermonde determinants 
13. Multiplication of determinants 

36. Hadamard's determinant
estimate 
IV.
Matrices 

37. Laplace's expansion rule

14. Multiplication of matrices 

38. Partial matrices 
15. Inverse Matrix 

39. Characteristic roots 
16. Group of regular matrices 

40. Kronecker product 
17. Addition of matrices 

41. CayleyHamilton relations 
18. Contragredient and orthogonal
matrices 

X. Similarity

V. Vector spaces. Rank of a matrix


^{42. Classes of} ^{similarity} 
19. Vector spaces 

43.
Linear mapping 
20. Linear dependence 

44. Decomposition into
components with a single characteristic root. 
21. Relationship between different
bases 

45. Decomposition into
elementary components 
22. Dimensions of partial spaces 

46. Jordan's normal form 
23. Rank of matrix 

47. Similarity to diagonal
matrices 