Selected Topics in Vector spaces: Vectors, Vector norms, vector algebra, subspaces, basis vectors, Gramm-Schmidt orthonormalization. Matrices, matrix rank, matrix norms, determinant, inverse, condition number. Hermitian and symmetric matrices, positive definite matrices unitary matrices, projection matrices and other special matrices. LDU decomposition, QR decomposition, Eigenvalue decomposition, singular value decomposition. Solving linear system of equations using matrices. Least-Squares approach, total least squares approach. Numerical issues. Perturbation theory of matrices. Differentiation of scalar functions of vectors and matrices. Matrix functions of scalar variables, Kronecker product of matrices.

Analysis: Review of real and complex number systems, topology of metric spaces. Continuity and differentiability. Construction of the Lebesgue measure, measurable functions, limit theorems. Lebesgue integration. Different notions of convergence and convergence theorems. Product measures and Fubini’s theorem. Signed measure and the RadonNikodym theorem, change of variables.

Optimization Techniques: Need for unconstrained methods in solving constrained problems. Necessary conditions of unconstrained optimization, structure of methods, quadratic models. Methods of line search, Armijo-Goldstein and Wolfe conditions for partial line search. Global convergence theorem, steepest descent method. Linear and Quadratic Programming. Duality in optimization.

Stochastic Models: Review of Random variables, Stochastic processes, Markov chains, stationary distribution of Markov chains, Poisson and birth and death processes.