Linear Algebra and Complex Analysis

Linear Algebra: Vector spaces, subspaces, span, Linear dependence, independence of vectors, basis, dimension, linear transformations, range, kernel, rank, nullity of linear transformation, space of all linear transformations, Operator equations, matrix associated with a linear map, linear map associated with a matrix, elementary row operations, solution of algebraic equations, consistency conditions. Matrix inversion by row operations, Eigenvalues and eigenvectors, Hermitian and skew Hermitian matrices, orthogonal and unitary matrices, application to reduction of quadrics.

Complex Analysis: Limit, continuity, differentiability and analyticity of functions Cauchy-Riemann equations (cartesian and polar), Harmonic functions, Elementary complex functions, Line integrals, upper bounds for moduli of contour integrals, Cauchy’s integral theorem, Cauchy’s integral formula, derivatives of analytic functions, Power series, Taylor’s series, Laurent’s series, Zeros and singularities, Residue theorem, evaluation of improper integrals by residue theorem.