Name of the Speaker with affiliation

No. of Lectures

Detailed Syllabus

Paramita Das, ISI Kolkata

6

Vector space over a field. Sub-spaces. Sum and intersection of two sub-spaces. Basis, Replacement Theorem. Extension theorem. Extraction of basis from a set of generators. Linear transformation. Rank and Nullity theorem.

Matrix of a Linear transformation. Row and column rank of a matrix. Necessary and sufficient condition for the consistency of an inhomogeneous system. Solution of the system of equations (Matrix method, Cramer's Rule).

Characteristic equation of a square matrix. Eigen value and Eigenvector. Cayley-Hamilton Theorem. Simple properties of Eigen values and Eigen vectors.

Inner Product Spaces. Cauchy-Schwarz Inequality. Orthogonality of vectors. Orthonormal basis, Gram-Schmidt process of orthonormalisation. Elementary matrices. Congruence of matrices. Real Quadratic form; Reduction to Normal Form.

Partha Sarathi Chakraborty and Samik Basu

ISI Kolkata

12

Topology of R^n. Functions of several variables, Continuity. (2 lectures)

Partial derivatives, Differentiability, Chain Rule, Inverse function theorem and Implicit function theorem. Taylor's theorem, Maxima and minima. Lagrange's multiplier method. Picard's theorem. (6 lectures)

Multiple integrals, Repeated integrals, Fubini's theorem, The Jacobian theorem, Line, surface and volume integrals, Green's theorem, Gauss theorem. (4 lectures)