TEW Linear Algebra, Analysis of several variables, Probability Theory (2016)

Syllabus Covered

(a) Linear Algebra (7 lectures)

Resource Person:  Prof. Sumitra Purakayastha, Applied Statistics Division, Indian Statistical Institute, Kolkata
Vector space over a field. Sub-spaces. Sum and intersection of two sub-spaces.Linearly independent set. Linear span. Basis of a vector space. Existence of 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. Linear homogeneous system of equations : Solution space as a subspace. Homogeneous system AX = 0 in n unknowns : Necessary and sufficient condition for the consistency of the 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 value and Eigen vector.
Inner Product Spaces. Norm. Euclidean vector spaces (EVS), Triangle Inequality and Cauchy-Schwarz Inequality in EVS. Orthogonality of vectors. Orthonormal basis, Gram-Schmidt process of orthonormalisation. Elementary matrices. Congruence of matrices. Real Quadratic form Reduction to Normal Form.
Texts followed:
(i) Rao, A.R. and Bhimasankaram P. (2000). Linear Algebra. Second edition. Hindustan Book Agency.
(ii) Kumaresan, K. (2000). Linear Algebra: A Geometric Approach. PHI.

(b) Analysis of Several Variables (6 lectures)

Resource Persons:  Prof. Swagato K. Ray (4lectures), Theoretical Statistics and Mathematics Division, Indian Statistical Institute, Kolkata
And
Prof. Rudra P. Sarkar (2 lectures), Theoretical Statistics and Mathematics Division, Indian Statistical Institute, Kolkata

Functions of several variables, Continuity, Partial derivatives, Differentiability, Chain Rule, Inverse function theorem and Implicit function theorem. Taylor’s theorem, Maxima and minima. Lagrange’s multiplier method.Multiple integrals, Repeated integrals, The Jacobian theorem, Line, surface and volume integrals.

(c) Probability (5 lectures)

Resource Person: Prof. KrishanuMaulik ,Theoretical Statistics and Mathematics Division, Indian Statistical Institute, Kolkata

Notion of random experiments, events and their probability. Probability of some basic models using combinatorial methods. Axiomatic definition of probability and study of conditional probability, Bayes theorem, independence. Univariate and multivariate distribution functions, their moments and their properties. Probability generating functions and their applications. Continuous distributions, examples and properties. Transformations of random variables with some interesting examples. A quick introduction to Poisson process. A brief review of different modes of convergence with reference to laws of large numbers and central limit theorem.

TimeTable

S. P. – Prof. Sumitra Purakayastha (Linear Algebra)

S. K. R. – Prof. Swagato K. Ray (Analysis of Several variables (Differentiation)

K. M. – Prof. Krishanu Maulik (Probability Theory)

R. P. S. – Prof. Rudra P. Sarkar (Analysis of Several variables (Integration))

How to reach

1. Bus: Get down at either Dunlop or ISI or Bon-Hooghly bus stop. Then take a cycle rickshaw from rickshaw stand to the college.
2. Local Trains:  Either of the following:
   • Sealdah –Dankuni local train to Baranagar Road Station.  Then take a cycle rickshaw from rickshaw stand to the college.
   • Any local train to Belghoria station. Any one of the bus numbers 201, 237 or 234/1 to either Dunlop or ISI or Bon-Hooghly bus stop. Then take a cycle rickshaw from rickshaw stand to the college.
3. Metro train: Get down at Dumdum station. Take bus number DN 9/1 to either Dunlop or ISI or Bon-Hooghly bus stop.
4. Car:  From Shyambazar take B. T. Road and turn right at Bon-Hooghly crossing. From Barrackpore take B. T. Road and turn left at ISI bus stop.