Introduction to Abstract and
Linear Algebra: Sourav Mukhopadhyay
LECTURE 
TOPIC 
DESCRIPTION 
1. 
Definition of Set, Representation of set, empty set, universal set, subset. 

2. 
Union, Intersection, properties of union and intersection, complementation. 

3. 
De Morgan's law, set difference, symmetric difference, Venn diagram. 

4. 
Power set, cardinality of the power set, partition of a set, cartesian product, relation. 

5. 
Definition of binary relation, examples, relfextive, symmetric, transitive and equivalence relation. 

6. 
Equivalence classes, partition, Zn. 

7. 
Definition of mapping, domain, codomain, image, range, onetoone, onto, bijective function, composition of mapping. 

8. 
Bijective mapping, cycle, equipotent set, equivalence relation, denumerable (enumerable), countable set. 

9. 
Binary operator, closure, commutative, associative, example of binary composition. 

10. 
Algebric structure, Groupoid, commutative groupoid, identity and inverse element, semigroup, monoid, quasigroup, group. 

11. 
Group, finite groups, abelian group, uniqueness of identity and inverse. 

12. 
Integral power of an element, order of an element, properties of order. 
13. 
Induced binary composition, subgroups, unique identity, example of subgroups, necessary and sufficient conditions for subgroups. 

14. 
Sufficient condition for subgroup, centre of a group, centraliser of an element, cyclic subgroup generated by an element, cyclic group, generator. 

15. 
Intersection and union of two subgroups, product of two subgroups, definition of left cosets. 

16. 
Left cosets of a subgroup, disjoint left cosets, left cosets as equivalence classes, any two left cosets are having same cardinality, lagrange theorem on order of a subgroup. 

17. 
Definition, equivalence classes of right cosets, number of distinct left cosets and number of distinct right cosets are same, index of a subgroup. 

18. 
Definition of normal subgroups, necessary and sufficient condition for normal subgroups, Quotient group, homomorphism, isomorphism. 

19. 
Definition of rings, commutative rings, rings with identity, examples of rings, polynomial rings. 

20. 
Divisors of zeros, Integral Domain, Field. 

21. 
External composition, definition of vector spaces, vector spaces over real field 

22. 
Properties of vector spaces, vector subspaces, necessary and sufficient condition for sub spaces, linear sum of subspaces. 

23. 
Linear combination of vectors, linear span, linear dependency and independency. 

24. 
Finite dimensional vector space, definition of basis, Replacement theorem, any two bases have same number of vectors. 
25. 
Definition of rank of a vector space, dimension of null space is zero, extension theorem, co ordinate of a vector with respect to a ordered basis. 

26. 
Dimension of subspaces, sum of two subspaces, direct sum of two subspaces, dimension of sum of two subspaces. 

27. 
Definition of linear mapping, examples, properties of linear mapping, kernel of a linear mapping is a subspace, onetoone linear mapping. 

28. 
Image of a linear mapping, subspace, dimension of kernel and image of a linear mapping, nullity and rank theorem of a linear mapping. 

29. 
Composition of linear mapping, inverse of linear mapping, isomorphism, isomorphic, two same dimensional (finite) vector spaces (U and V say) are isomorphic (U ~ V). 

30. 
V ~ F^n, addition of two linear mapping, scalar multiplication of two linear mapping, linear space of linear mappings, matrix representation of linear mapping. 

31. 
definition of rank of a matrix, maximum order of nonzero minor, elementary row and column operations, row reduced echelon form, row equivalence 

32. 
Elementary matrices, row and columns operations is same as multiplying with elementary matrices, rank of a matrix is same as rank of the row reduced or columns reduced matrix, normal form. 

33. 
augmented matrix, rank of augmented matrix, consistency, inconsistencies, no solution, unique solution and infinitely many solution 

34. 
Row space and Column space of a matrix, row rank, column rank 

35. 
Characteristic equation, CayleyHamilton theorem, eigen value of a matrix, product of eigen values 
36. 
Eigen value of a non singular matrix, eigen vectors, eigen vector corresponding to unique eigen value, independence of eigen vectors 

37. 
Algebric and geometric multiplicity of eigen value, simple eigen value, regular eigen value 

38. 
Eigen values of a real symmetric matrix are real, eigen values of a orthogonal martix 

39. 
Eigen value of real orthogonal matrix, similar relation between matrices, two similar matrix have same eigen values 

40. 
Definition of diagonalisable, conditions of diagonalisable, examples 