Paper II Total Marks: 100
Table of Contents
Section A
Groups: Definition and examples of Groups, Order of a Group, Order of an element of a Group, Abelian and non-Abelian Groups and Cyclic groups. Lagrange theorem and applications, Normal subgroups, Characteristic Subgroups of a group, Normalizer in a group, Centralizer in a group. Fundamental Theorem of Homomorphism, Isomorphism theorems of groups, Automorphisms Rings, Fields and Vector Spaces: Examples of Rings, Subrings, Integral domains, Fields, Vector spaces, Linear independence/ dependence, Basis and dimension of finitely generated spaces, Examples of Field extension and finite fields, Examples of finite and infinite dimensional vector spaces.
Section B
Metric Spaces and Topological Spaces: Definition and Examples of Metric spaces and topological spaces, Closed and Open Spheres, Interior, Exterior and Frontier of a Set, Sequences in Metric Spaces, Convergence of Sequences. Definition and examples of Normed Spaces. Inner product spaces, Gram-Schmidt Process of Orthonormalization Matrices and Linear Algebra: Linear transformations, Matrices and their algebra, Reduction of matrices to Echelon and Reduced Echelon form. Solution of a system of homogenous and Non-Homogenous equations, Numerical methods of solving equations (Gauss-Siedal method, Jaccobi method) Properties of Determinants,
Eigenvalues and Eigenvectors and the Diagonalization of the Symmetric Matrices.
Recommended Books:
1. Nicholson. W.K., Elementary Linear Algebra with Applications, Ed. 2, Prentice Hall, Englewood, USA.
2. Herstein, I.N., Topics in Algebra, John Willey and Sons (New York) 1964.
3. Rowen, L., Rings (I & II). Academic Press, Ins.
4. Dar, K.H., First Step to Abstract Algebra, Feroz Sons Publishers, Lahore (1996).
5. Yusuf, S.M., Majeed A., Amin. M., Mathematical Methods, Ilmi Kitab Khana, Lahore.
6. Atkinson, K. E., An Introduction to Numerical Analysis, Ed. 2 John Willey, New York, 1989.
7. Ahmad. F and Afzal. M, Numerical Analysis, National Book Foundation, Islamabad.
8. Simmons, F.J., Topology, McGraw Hill Company, New York.
9. Kreyszig, E., Introductory Functional Analysis with Applications, John Willey and Sons, New York, 1978.
10. Majeed, A. Elements of Topology and Functional Analysis, Ilmi Kitab Khana, Lahore.