The goal of this course is to develop within the student a degree of mathematical maturity. This maturity will be cultivated by working problems, following and constructing proofs, and writing programs. The main topics covered in this course include Banach and Hilbert spaces, orthogonal projections, representations and approximations, linear operators, pseudoinverses, characteristic values, and the singular value decomposition. Applications of the mathematical tools presented in class to the fields of communications and signal processing will be provided.
- Explore fundamental concepts of logic including sets, axioms, quantifiers, implications, necessary and sufficient conditions. Illustrate valid proof methods such as proofs by contradiction, proofs by contrapositive, the principle of mathematical induction and counter examples.
- Establish basic notions of topology in the context of metric spaces. Study formal definitions for open sets, closed sets, convergence, limit points, completeness and continuous functions.
- Review linear algebra, combinations of vectors, independence, bases and dimensions. Distinguish between vector spaces, normed spaces and inner-product spaces. Go over the projection theorem and illustrate some of its applications.
- Introduce the notions of linear operators, fundamental subspaces, matrix representations, inverses and pseudoinverses. Examine the properties of characteristic polynomials, eigenvalues, eigenvectors and eigenfunctions. Develop the theory of the singular value decomposition. Survey special matrices and important matrix factorizations.
- Apply vector space methods to signal processing, optimization, least-squares filtering and minimum mean-square error estimation. Analyze iterative methods and the least mean squares algorithm.
- Acquire the ability to recognize, formulate and solve pertinent engineering problems using vector space methods and Hilbert spaces. Gain proficiency at using high-level programming languages such as C++ or Python.
- Engage the student in an active learning experience. Expose the student to search engines, scholastic resources, research tools, indexes and databases. Prepare the student to become an active contributor to the common body of knowledge.