Syllabus
Unit 1
Gaussian elimination – LU decomposition – Vector spaces associated with Matrices- Special orthogonal matrices
– Fourier Series and Fourier Transform and its properties – Convolution – Projection matrix and Regression – Convolution sum – Convolution Integral – Eigenvalues and Eigenvectors of Symmetric matrices – Eigenvalues and Eigen vectors of ATA, AAT – Relationship between vector spaces associated with A, ATA, AAT- Singular Value Decomposition – Concept of Pseudoinverse- Computational experiments using MATLAB/Excel/Simulink
Unit 2
Taylor series expansion of multivariate functions-conditions for maxima, minima and saddle points-Concept of gradient and Hessian matrices- Impulse Response computations- converting higher order into first order equations – concept of eAT- Multivariate regression and regularized regression -Theory of convex and non- convex optimization-Newton method for unconstrained optimization- Signal processing with regularized regression- Computational experiments using MATLAB/Excel/Simulink
Unit 3
Random variables and distributions – Expectation, Variance, Moments, Cumulants- Moment generating functions
– Sampling from univariate distribution- various methods – Bayes theorem, Concept of Jacobian, and its use in finding pdf of functions of Random variables (RVs), Box-muller formula for sampling normal distribution – Concept of correlation and Covariance of two linearly related RVs.
Unit 4
Introduction to quantum computing–Introduction to spin – state vectors – Qubits – Entanglement. Measurement in Quantum Mechanics.
Objectives and Outcomes
Course Objectives
- To introduce students to the fundamental concepts of linear algebra, differential equations, optimization, and probabilistic modelling
- To enable students to apply the concepts they learn in practical situations by using analytical and numerical methods to model real-world problems.
- To expose students to the wide range of applications of linear algebra, ordinary differential equations, probability theory, and quantum computing within the scientific field and to inspire them to pursue further study or research in these areas.
- To equip students with advanced mathematical knowledge and problem-solving skills highly valued in various industries and research fields.
Course Outcomes
After completing this course, students will be able to
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Implement matrix decomposition techniques to solve linear systems of equations.
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Formulate optimization problems and solve them using gradient based and Newton’s
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Analyse data using fundamental techniques of probability.
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Explain quantum entanglement, qubits and state vectors
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Text Books / References
Gilbert Strang, Linear Algebra and Learning from Data, Wellesley, Cambridge press, 2019.
William Flannery, Mathematical Modelling and Computational Calculus, Vol-1, Berkeley Science Books, 2013.
Stephen Boyd and Lieven Vandenberghe, Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares, 2018.
Douglas C. Montgomery and George C. Runger, Applied Statistics and Probability for Engineers, (2005) John Wiley and Sons Inc
Bernhardt, Chris. Quantum computing for everyone. Mit Press, 2019. (From pages 37 to 70).
