Math 202A is the first quarter of a three-quarter applied algebra class. It is a PhD level course on linear algebra, and the assumption is that you know linear algebra at the undergraduate level. Furthermore, the way in which the material is delivered will probably cause Math 202 to feel quite different from a standard undergraduate lecture course. Below is a lecture schedule.
| Date | Modality | Topic |
| Sep 26 | Asynchronous | Course information |
| Sep 29 | In Person | Categories I |
| Oct 1 | In Person | Categories II |
| Oct 3 | In Person | Categories III |
| Oct 6 | In Person | Lengths |
| Oct 8 | In Person | Coordinates |
| Oct 10 | Asynchronous | Transformations |
| Oct 13 | None | Columbus Day |
| Oct 15 | In Person | Enrichment |
| Oct 17 | In Person | Covectors |
| Oct 20 | In Person | Direct sums |
| Oct 22 | In Person | Subspaces |
| Oct 24 | Asynchronous | Projections |
| Oct 27 | Asynchronous | Combinatorial SVD I |
| Oct 29 | Asynchronous | Combinatorial SVD II |
| Oct 31 | None | Halloween |
| Nov 3 | In Person | Geometric SVD I |
| Nov 5 | In Person | Geometric SVD II |
| Nov 7 | In Person | Geometric SVD III |
| Nov 10 | In Person | Adjoint and transpose |
| Nov 12 | In Person | Projection, inversion |
| Nov 14 | In Person | More SVD review |
| Nov 17 | None | Travel |
| Nov 19 | Field Trip | Weingarten Calculus |
| Nov 21 | Asynchronous | Review notes |
| Nov 24 | In Person | Singular value geom. |
| Nov 26 | In Person | Low rank approx. |
| Nov 28 | None | Thanksgiving |
| Dec 1 | In Person | Normality |
| Dec 3 | In Person | Selfadjointness |
| Dec 5 | In Person | Spectrum |
Here is a bit more detail on the topics we will cover. This is not exhaustive and is intended only as a broad outline of some of the main themes.
We will begin by introducing the rudiments of category theory and discuss its usefulness in organizing our existing knowledge and future endeavors. For us category theory is an organizational device, but it has applications in the sciences and is a branch of applied algebra in its own right.
We will discuss the categories of sets, groups, posets, and vector spaces in some detail, and give a category-theoretic definition of group representation theory. We will then consider the “quantization” functor which transports us from the category of finite sets to the category of finite-dimensional complex vector spaces, which is where we shall stay for most of the course.
We will study finite-dimensional complex vector spaces with a scalar product, which we call Hilbert spaces. Morphisms in this category are linear functions, which for historical reasons are called linear transformations. Any function between finite sets can be encoded as a matrix with entries in 
A nice feature of the set of all linear transformations from one Hilbert space to another is that it is also a Hilbert space with a canonically defined scalar product called the Frobenius form. The Hilbert space structure on linear transformations plays a significant role in applications of linear algebra since it provides a way to discuss the magnitude of a matrix. There are many other useful matrix norms, and we will discuss a few of these from the point of view of the SVD.
After concluding our discussion of homomorphisms between Hilbert spaces we will concentrate on endomorphisms of a Hilbert space, which are called linear operators. Linear operators on a Hilbert space quantize self-maps on a finite set. For example, the quantization of an indicator function is an orthogonal projection and the quantization of a permutation is a unitary operator. Using the SVD, we establish the polar decomposition of an arbitrary operator and the spectral decomposition of a normal operator.
We then move on to a detailed study of the spectral decomposition of selfadjoint operators: the theory of eigenvalues. We will obtain many interesting results about eigenvalues, including results on dominance, interlacing, and perturbation. A very interesting class of selfadjoint operators are the adjacency operators of finite graphs, and we will discuss these in some detail and prove a few famous results on their eigenvalues, such as the Perron-Frobenius theorem on the largest eigenvalue and the Alon-Boppana theorem on the spectral gap.
It is natural to wonder what the spectrum of a typical selfadjoint operator looks like — what are the expected features of its eigenvalue distribution? This question motivates the subject of random matrix theory. We will prove the famous Wigner semicircle law which describes the expected eigenvalue distribution of large random Hermitian matrices.
Not all vector spaces which arise in practice have complex scalars. In particular, real coefficients seem more natural in applications though one should keep in mind that quantum mechanics tells us this is not really so. Throughout the course, we will regularly comment on when our Hilbert space results remain valid for real finite-dimensional vector spaces with scalar product, which are called Euclidean spaces. Typically this is quite straightforward. More significant differences occur when we move to the category of finite-dimensional vector spaces defined over a finite field. This category also has many important applications, and can be thought of as a different kind of quantization of the category of finite sets. We will spend a few lectures discussing this interesting category.
We will end with a discussion of the most important special function of a linear operator: the exponential function. For operators on a Hilbert space of dimension greater than one, it is not in general true that the exponential function converts addition into multiplication. The correct statement is the Campbell-Baker-Hausdorff formula, which is the first step down the road to Lie theory. We will present an elementary proof of this result due to Eichler.
This completes our rough outline of topics. There is no official textbook for the class and I will post typed notes following each lecture. However, while the course is running I recommend consulting the following books:
- Seven Sketches in Compositionality: An Invitation to Applied Category Theory, by Brendan Fong and David Spivak. Just browse this one.
- Lectures on Linear Algebra, by Israel Gelfand. Perhaps the best book on linear algebra ever written; I recommend reading the whole thing.
- Matrix Analysis, by Charles Johnson and Roger Horn. Use this as a comprehensive reference.
- Groups, Matrices, and Vector Spaces, by James B. Carrell. A nice book worth consulting regularly to broaden your horizons.
- All the Mathematics You Missed but Need to Know for Graduate School, by Thomas Garrity. Read the chapter on linear algebra as a refresher. This book can be useful for qualifying exam study.
Your grade in Math 202A will be based on weekly problem sets (70%) and a final exam (30%). There is no midterm exam. Problem sets will typically be due on Sunday at 23:59, and will be submitted and returned via GradeScope (the course TA, Runqiu Xu, will send out an email during the first week of classes explaining this protocol in more detail). Your lowest problem set score will be dropped. The final exam is scheduled for 12/09/2025 at 15:00 in our regular lecture room, B412. I have no control over these coordinates, and if you cannot sit for the exam you should not take the course.
