Template for Courses

SC116 Algebraic Structures 3-1-0-4 Winter Semester – 2007-08

Instructor: Prof. Samaresh Chatterji

Objective: This course helps students understand algebraic structures as underlying specific objects, computations, and systems, develops familiarity with the key algebraic structures which are most frequently encountered: groups, rings, fields, vector spaces, both abstractly in terms of axioms and concretely in terms of the most important examples. It also makes them acquainted with the concept of homomorphisms of algebraic structures in general and in its specific manifestation in the context of the different examples, knowledge of specific applications of the above understanding, both in attacking mathematical problems and in ICT.

Contents: Groups (Subgroups, Isomorphism and Homomorphism, Cosets, Product of Groups, Quotient Groups), Vector Spaces (Fields, Vector Spaces, Subspaces, Bases and Dimension, Co-ordinates), Linear Transformations (Algebra of Linear Transformations, Isomorphism, Matrix Representations), Linear Equations (System of Linear Equations, Elementary Row Operations, RREF, Invertible Matrices), Linear Functionals (including the double dual and the transpose), Eigenvalues and Eigenvectors (Characteristic Polynomial, Orthogonal and Unitary Matrices, Diagonalisation, Systems of Differential Equations, Matrix Exponential) and Polynomials (Algebra of Polynomials, Irreducible polynomials, Prime Factorization).

Teaching Method: Lecture cum Tutorial: Lecture – three lectures per week, Tutorial – one tutorial per week (students divided into four groups). During the tutorial session, the Tutor will clarify concepts, and the TA’s will assist students in solving exercises.

Evaluation: Continuous evaluation will be carried out with weightage as follows (may be slightly modified later): Assignments and Quizzes – 15%, Class Test (two) – 35%, Final Examination – 50%. Total marks out of 100 will be converted to letter grade point using a curve (modified normal distribution).

Books: Artin: Algebra, 7th ed, Prentice-Hall, 2001 Datta: Matrix and Linear Algebra, Prentice Hall Herstein: Topics in Algebra, Indian edition, John Wiley Hoffman and Kunze: Linear Algebra, 2nd ed, Prentice-Hall Lay: Linear Algebra and Its Applications, 2nd ed, Addison-Wesley Scheinerman: Mathematics: A Discrete Introduction, Brooks /Cole (This list may be modified later)