Syllabus

Linear Algebra:

(a) Vector Spaces: Definition of Vector space, Sub spaces, linearly independence and depend- ence,linear Span, Basis, Dimension

(b) System of linear equations: Range space and Null space of a matrix, Rank of a matrix, Ex- istence and uniqueness of solution of the system of linear equations, Dimension of the Solution Space associated with the system of linear equations

(c) Eigen values and Eigen vectors: Definition of Eigen values and Eigen vectors of a square matrix and their properties including similarity matrices

(d) Diagonalization and SVD: Diagonalization of a square matrix, Singular-Value-Decomposi- tion (SVD) and Pseudo-inverse of a matrix

Fourier Series and Transform:

(a) Fourier Series: Fourier Series of 2pi periodic functions, Cosine Series, Sine Series, Fourier series of a function defined on an interval [a,b] of length T=b-a, Point-wise Dirichlet conver- gence Theorem for Fourier Series.

(b)Fourier Transform: Representation of a function defined over R in Fourier Integral and rep- resentation of Fourier Integral as a pair of transformations: Fourier Transform and Fourier In- verse Transform, Properties of Fourier Transform

(c)Laplace transform: Definition and necessity of Laplace transform, Inverse Laplace transform, Properties of Laplace Transform

Introductory Complex Analysis

(a) Complex Differentiation: Definition of Continuity and Differentiability-Cauchy-Riemann Equation -Analytic function

(b) Complex Integration: Defintion of Contour-Contour Integration (Complex Line Integration)

Introductory Probability Theory:

Random variables, probability distribution functions, discrete and continuous distributions. If time permits, multivariate distribution to be added.