ABSTRACT ALGEBRA AND LINEAR ALGEBRA

1.1 Historical background :

1.1.1 A brief historical background of the Algebra in the context of India and Indian heritage and culture

1.1.2 A brief biography of Brahmagupta

1.2 Groups, Subgroups and their basic properties

1.3 Cyclic groups

1.4 Coset decomposition

1.5 Lagrange’s and Fermat’s theorem

1.6 Normal subgroups

1.7 Quotient groups

– Unit-II –

2.1 Homomorphism and Isomorphism of groups

2.2 Fundamental theorem of homomorphism

2.3 Transformation and Permutation group Sn(n < 5)

2.4 Cayley’s theorem

2.5 Group automorphism

2.6 Inner automorphism

2.7 Group of automorphisms

– Unit-III –

3.1 Definition and basic properties of rings

3.2 Ring homomorphism

3.3 Subring

3.4 Ideals

3.5 Quotient ring

3.6 Polynomial ring

3.7 Integral domain

3.8 Field

– Unit-IV –

4.1 Definition and examples of Vector space

4.2 Subspaces

4.3 Sum and direct sum of subspaces

4.4 Linear span, Linear dependence, Linear independence and Their basic properties

4.5 Basis

4.6 Finite dimensional vector space and dimension

4.6.1 Existence theorem

4.6.2 Extension theorem

4.6.3 Invariance of the number of elements

4.7 Dimension of sum of subspaces

4.8 Quotient space and its dimension

– Unit-V –

5.1 Linear transformation and its representation as a matrix

5.2 Algebra of linear transformation

5.3 Rank-Nullity theorem

5.4 Change of basis, dual space, bi-dual space and natural isomorphism

5.5 Adjoint of a linear transformation

5.6 Eigenvalues and Eigenvectors of a linear transformation

5.7 Diagonalization