Advanced Linear Algebra
Topics in linear algebra beyond those in a first course. For example: principal component analysis and other decompositions (singular value, Cholesky, etc.); Perron-Frobenius theory; positive semi-definite matrices; linear programming and more general convexity and optimization; basic simplicial topology; Gerschgorin theory; classical matrix groups. Applications to computer science, statistics, image processing, economics, or other fields of mathematics and science. Prerequisites: (1) Math 212 or 222, and (2) Math 218 or 221.
The main themes of the course are
Abstraction refers to the setting of general vector spaces, with finite dimension or not, with given basis or not, over an arbitrary field. Approximation asks for best answers to linear systems when exact solutions either don't exist or are not worth computing to arbitrary precision. This is related to variation of subspaces or of entries of matrices: what kind of geometric space is the set of all $k$-dimensional subspaces? And what happens to the eigenvalues of a matrix when the entries of the matrix are wiggled? Positivity refers to the entries of a matrix or to the eigenvalues of a symmetric matrix; both have interesting, useful consequences. Convexity stems from the observation that a real hyperplane H splits a real vector space into two regions, one on either side of H. Intersections of regions like this yield familiar objects like cubes, pyramids, balls, and eggs, the geometry of which is fundamental to many applications of linear algebra. Throughout the course, motivation comes from many sources: statistics, computer science, economics, and biology, as well as other parts of mathematics. We will explore these applications, particularly in projects (paper plus oral presentation) on topics of the students' choosing.
Students will continue to develop their skills in mathematical exposition, both written and oral, including proofs.
Every student will complete a project, to include a paper and an oral presentation, on a topic of their choice. Some topics will be suggested, but students can formulate their own, with approval of the instructor.