6.972 Linear Algebra and Functional Analysis for Signals & Systems (H)

L TR2:30-4, Room 26-168
Professor John Wyatt, jlw@mit.edu, Professor Alex Megretski, ameg@mit.edu
Prereq.: 6.011 and 18.06 and some mathematical maturity
3-0-9

This subject qualifies as a Communication, Control, and Signal Processing Engineering Concentration subject.

The class provides a rigorous introduction to core linear algebra and functional analysis concepts used in communication, control, signal processing and optimization. It teaches coordinate-free approach to linear algebra, normed vector spaces, quadratic optimization, convexity and duality, motivated by applications in linear systems, Fourier analysis, robustness assessment, and linear coding. The presentation employs geometric interpretation (on the plane and in space) of linear algebra and functional analysis statements, highlighting differences and similarities in finite and infinite dimensional constructions.

The course is aimed at filling the gaps in basic linear algebra/functional analysis background of students interested in communication, control, signal processing, optimization, and related areas.

The content is based partially on the book ``Optimization by Vector Space Methods'' by D. Luenberger, with additional material on Fourier transforms and linear algebra over finite fields.

Principal topics include:

(a) field-independent linear algebra (vector spaces, bases, dimensions, matrix algebra, linear transformations, linear equations, determinants, characteristic polynomials) emphasizing the coordinate-free approach with applications in linear systems and coding theory;

(b) quadratic optimization (positive definiteness, Gram-Schmidt orthogonalization, scalar products, projection theorem) with applications in signal processing and optimal control;

(c) convexity (convex functions and convex sets, convex optimization, cutting plane methods, Caratheodory theorem, Minkovsky functionals, Krein-Milman Theorem, Hahn-Banach theorem, minimax theorem) with applications in optimization;

(d) approximation and topology (norms, approximation, functional spaces, Fourier and Laplace transforms, compactness, fixed point theorems, differentiation, implicit mapping theorems, first and second order conditions of optimality) with applications in robustness analysis and optimization.