Course Details
Subject {L-T-P / C} : CS6108 : Linear Algebra and Random Processes { 3-0-0 / 3}
Subject Nature : Theory
Coordinator : Prof. Pankaj Kumar Sa
Syllabus
Linear Algebra
Vector spaces: Column and row spaces, Solving Ax=0 and Ax=b, Independence, basis, dimension, linear transformations.
Orthogonality: Orthogonal vectors and subspaces, projection and least squares, Gram-Schmidt orthogonalization
Determinants: Determinant formula, cofactors, inverses and volume
Eigenvalues and Eigenvectors: Characteristic polynomial, Diagonalization, Hermitian and Unitary matrices, Spectral theorem, Change of basis.
Positive definite matrices and singular value decomposition
Random processes
Preliminaries: Events, probability, conditional probability, independence, product spaces
Random Variables: Distributions, law of averages, discrete and continuous RV, random vectors, Monte Carlo simulation
Discrete Random Variables: Probability mass functions, independence, expectation, conditional expectation.
Continuous Random Variables: Probability density functions, independence, expectation, conditional expectation, functions of RV, sum of RV, multivariate normal distribution, sampling from a distribution.
Convergence of Random Variables: Modes of convergence, Borel-Cantelli lemmas, laws of large numbers, central limit theorem, tail inequalities
Course Objectives
- To present the central methods and ideas of linear algebra and random process with a unified approach.
- To understand the linear operators on finite-dimensional vector spaces.
- To comprehend the complexity of describing random, time-varying functions.
Course Outcomes
(a) Students would get a better perspective of AI/ML instead of thinking of it as magic. <br />(b) Students would be able to abstract the data and model. <br />(c) Equip the students with uncertainty and stochasticity.
Essential Reading
- Gilbert Strang, Linear Algebra and its Applications, Cengage , 2014
- Geoffrey Grimmett, Probability and Random Processes, Oxford University Press , 2006
Supplementary Reading
- Athanasios Papoulis and S. Unnikrishna Pillai, Probability, Random Variables and Stochastic Processes, McGraw Hill , 2002
- Peter J. Olver, Applied Linear Algebra, Pearson Education , 2006