Math 8530, Fall 2023

Math 8530, Fall 2023


"We share a philosophy about linear algebra: we think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury." --Irving Kaplansky, writing of himself and Paul Halmos

About the class

This course will be a comprehensive survey of finite dimensional vector spaces and linear mappings at the graduate level. The goals are two-fold -- to prepare students for the linear algebra part of the prelim, and to build up a solid linear algebra foundation necessary for other classes and research, as linear algebra is a beautiful subject that appears in nearly all areas of mathematics, especially in applied fields such as statistics, operations research, and computational mathematics. Some advanced linear algebra classes, especially those geared to engineers and often called "matrix analysis," focus on computations involving matrices and under-emphasize the theory of vector spaces. Yet many other treatments take the opposite approach by presenting the theory of modules, which is too far detached from most applications that an applied mathematician, statistician, or optimizer will need. I will try to take a "Goldilocks" approach, and emphasize the theory of vector spaces (not modules), while being grounded in applications. The book that I will loosely follow, despite being quite theoretical, was written by a foremost expert on computational PDEs, containing that material that thought was most important for his applied mathematics graduate students to learn over the course of a year-long class.

Essentials

Lectures notes, slides, and videos

Links to the individual lectures are listed below. Or, you can view the full YouTube playlist here.

Section 1: Vector spaces (6 lectures, 3 hrs 2 min) [full slides; 27 pages, last updated 8/18/23]

Section 2: Linear maps (7 lectures, 4 hrs 13 min) [full slides; 39 pages,last updated 9/3/21] Section 3: Multilinear forms (7 lectures, 4 hrs 15 min) [full slides; 32 pages, last updated 10/27/21] Section 4: Spectral theory [full slides; 36 pages, last updated 10/27/21] Section 5: Inner product spaces [full slides; 50 pages, last updated 10/27/21] Section 6: Self-adjoint mappings 5 lectures, 3 hr 28 min [full slides; 37 pages, last updated 11/15/21] Section 7: Positive linear maps [full slides; 19 pages, last updated 12/1/21]

Homework