Math 8530, Spring 2021

Math 8530, Spring 2021


"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

Links to the individual lectures are listed below. Or, you can view the full YouTube playlist here. To avoid blurriness, these are best viewed by changing the settings to 720p (High Definition) rather than the default of 240p. This can be easily done by clicking the "wheel" on the lower right corner; right next to the "cc" button.

Section 1: Vector spaces (6 lectures, 3 hrs 2 min)

Section 2: Linear maps (7 lectures, 4 hrs 13 min) Section 3: Multilinear forms (7 lectures, 4 hrs 15 min) Section 4: Spectral theory Section 5: Inner product spaces Section 6: Self-adjoint mappings 5 lectures, 3 hr 28 min Section 7: Positive linear maps

Homework