ABSTRACT ALGEBRA & LINEAR ALGEBRA SUCCESS KEY-(IN HINDI)

1. Groups and Their Basic Properties

2. Subgroups

3. Cyclic Groups

4. Coset Decomposition, Lagrange’s and Fermat’s Theorem

5. Normal Subgroups

6. Quotient Groups

Unit-II

7. Homomorphism and Isomorphism of Groups, Fundamental 

Theorem of Homomorphism

8. Transformation and Permutation Group Sn (n < 5), Cayley’s

Theorem

9. Group Automorphism, Inner Automorphism, Group of 

Automorphisms

Unit-III

10. Definition and Basic Properties of Rings, Subrings

11. Ring Homomorphism, Ideals, Quotient Ring

12. Polynomial Ring

13. Integral Domain

14. Field

Unit-IV

15. Definition and Examples of Vector Space, Subspaces, Sum and

Direct sum of Subspaces, Linear Span, Linear Dependence,

Linear Independence and Their basic Properties

16. Basis, Finite Dimensional Vector Space and Dimension 

(Existence Theorem, Extension Theorem, Inoriance of the

number of Elements), DImension of sum of Subspaces,

Quonient Space and It Dimension

Unit-V

17. Linear Transformation and Its Representation as a Matrix

18. Algebra of Linear transformations, Rank-Nullity Theorem,

Change of basis, Dual Space, Bi-dual Space and Natural

Isomorphism Adjoint of a Linear Transformation

19. Eigen-Values and Eigen-Vectors of a Linear Transformation,

Diagonalization