The Teachers’ Enrichment Workshop is a programme funded by the National Centre for Mathematics (A joint centre of TIFR & IIT Bombay). The TEW is meant for college teachers to revisit and update their content-knowledge. The lectures in this workshop covers specific topics which are relevant for the teachers' classroom instructions. An important component of this programme is the discussion hour during which the teachers will have opportunities to get their doubts cleared and work-out routine to advanced exercises.

 
Syllabus:

Detailed Syllabus:

• Linear Algebra :

• (Sumitra Purakayastha - 6 lectures):  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 basisfrom 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.

  Applications of Linear algebra in L.P.P and Game Theory: Hyperplane, Convex set, Cone, Extreme points, convex hull and convex polyhedron, Supporting and Separating hyperplane. Feasible solutions of an L.P.P. constitutes a convex set. The extreme points of the convex set of feasible solutions correspond to its Basic feasible solutions (B.F.S.) and conversely. Extreme points of the convex polyhedron generated by the set of feasible solutions and Optimal value of objective function. Reduction of a F.S. to a B.F.S.

•  Analysis of several variables

• (Swagato K. Ray - 4 lectures): 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.

• (Rudra P. Sarkar – 2 lectures): Multiple integrals, Repeated integrals, The Jacobian theorem, Line, surface and volume integrals,

• Probability Theory

• (Krishanu Maulik – 6 lectures) 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.

 References:

1. K. Hoffman and R. Kunze – Linear Algebra
2. W. Rudin – Principles of Mathematical Analysis
3. M. Spivak – Calculus on Manifolds
4. S. Axler – Linear Algebra done right
5. Sheldon Ross -Introduction to Probability Models
6. Casella and Berger -Statistical Inference

Time-table (with names of speakers and tutors ):

Krishanu Maulik, KM ||  Sumitra Purkayastha, SP || Swagato K. Ray, SKR || Rudrapada Sarkar, RPS

Name of the tutors:

Mithun Bhowmick, Indian Statistical Institute, Kolkata