Paper IITotal Marks: 100Section AGroups: Definition and examples <strong>of</strong> Groups, Order <strong>of</strong> a Group, Order <strong>of</strong> an element <strong>of</strong>a Group, Abelian and non-Abelian Groups and Cyclic groups. Lagrange <strong>the</strong>orem andapplications, Normal subgroups, Characteristic Subgroups <strong>of</strong> a group, Normalizer in agroup, Centralizer in a group. Fundamental Theorem <strong>of</strong> Homomorphism, Isomorphism<strong>the</strong>orems <strong>of</strong> groups, AutomorphismsRings, Fields and Vector Spaces: Examples <strong>of</strong> Rings, Subrings, Integral domains,Fields, Vector spaces, Linear independence/ dependence, Basis and dimension <strong>of</strong>finitely generated spaces, Examples <strong>of</strong> Field extension and finite fields, Examples <strong>of</strong>finite and infinite dimensional vector spaces.Section BMetric Spaces and Topological Spaces: Definition and Examples <strong>of</strong> Metric spacesand topological spaces, Closed and Open Spheres, Interior, Exterior and Frontier <strong>of</strong> aSet, Sequences in Metric Spaces, Convergence <strong>of</strong> Sequences. Definition and examples<strong>of</strong> Normed Spaces. Inner product spaces, Gram-Schmidt Process <strong>of</strong> OrthonormalizationMatrices and Linear Algebra: Linear trans<strong>for</strong>mations, Matrices and <strong>the</strong>ir algebra,Reduction <strong>of</strong> matrices to Echelon and Reduced Echelon <strong>for</strong>m. Solution <strong>of</strong> a system <strong>of</strong>homogenous and Non-Homogenous equations, Numerical methods <strong>of</strong> solvingequations (Gauss-Siedal method, Jaccobi method) Properties <strong>of</strong> Determinants,Eigenvalues and Eigenvectors and <strong>the</strong> Diagonalization <strong>of</strong> <strong>the</strong> Symmetric Matrices.
Recommended Books:1. Nicholson. W.K., Elementary Linear Algebra with Applications, Ed. 2, PrenticeHall, 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., Ma<strong>the</strong>matical Methods, Ilmi Kitab Khana,Lahore.6. Atkinson, K. E., An Introduction to Numerical Analysis, Ed. 2 John Willey, NewYork, 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 andSons, New York, 1978.10. Majeed, A. Elements <strong>of</strong> Topology and Functional Analysis, Ilmi Kitab Khana,Lahore.
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