Integrated MSc-PhD Program

SYLLABUS

KSM1C01: Algebra I

Subspaces of , Fields, Vector Spaces, Bases and Dimension, Computing with bases, Change of basis, Direct sums, Infinite dimensional spaces, Linear transformations, The dimension formula, Matrix of a linear transformation, Linear operators, Eigenvalues and eigenvectors, The characteristic polynomial, Triangular and Diagonal forms, Jordan form and decomposition, Applications of Linear Operators, Symmetry of Plane, Isometries of the Plane, Finite Groups of Orthogonal Operators on the Plane, Discrete Groups of Isometries, Plane Crystallographic Group, Abstract Symmetry: Group Operations, The Operation on Cosets, The Counting Formula, Operations on Subsets, Permutation Representations, Finite Subgroups of the Rotation Group, Cayley’s Theorem, The Class Equation, p-Groups, Conjugation in the Symmetric Group, Normalizers, The Sylow Theorems, The Free Group, Generators and Relations, Bilinear Forms, Symmetric Forms, Hermitian Form, Orthogonality, Euclidean Spaces and Hermitian Spaces, The Spectral Theorem, Conics and Quadrics, Skew-Symmetric Form.

Suggested texts:

1. 1. 1. 1. 1. 1. Michael Artin, Algebra, Prentice Hall, Inc., Englewood Cliffs, NJ, 1991. (Chapters 3-8)
               2. Kenneth Hoffman and Ray Kunze, Linear algebra, Prentice-Hall Mathematics Series, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1961.
               3. Denis Serre, Matrices, second ed., Graduate Texts in Mathematics, vol. 216, Springer, New York, 2010, Theory and applications.
               4. Sheldon Axler, Linear algebra done right, third ed., Undergraduate Texts in Mathematics, Springer.