Abstract Linear Algebra Winter 2019

Instructor: Danny Calegari, email: dannyc

Course Assistant/Grader: Pallav Goyal, email: pallav

TR 11-12:20 Eckhart 312

Description of course:

The official course description (which I didn't write) is this:

This is a theoretical course in linear algebra intended for students taking higher level mathematics courses. Topics include vector spaces and linear transformations, matrices and the algebra of matrices, determinants and their properties, the geometry of Rn and Cn, bases, coordinates and change of basis, eigenvalues, eigenvectors, characteristic polynomial, diagonalization, special forms including QR factorization and Singular Value Decomposition, and applications.
It's a good course description, and broadly in line with how I think the class will go. But in fact, we might diverge from this to some extent (I'm not 100% sure what QR factorization and SVD are, for instance). Actually, I don't like the word "Abstract" in the course title. Abstraction is sometimes necessary, and sometimes not; sometimes useful, and sometimes not. We'll pursue abstraction when we need to or want to, and not because the course catalog tells us to.

Here's an unofficial course description, which might be a bit closer to how the class actually runs:

Linear algebra is approximately 50% of mathematics; 9 most of the remaining 50% is calculus. It's more or less about two things: vector spaces, and their symmetries and transformations. In most serious applications the vector spaces we actually care about have very large dimensions: hundreds, millions, or infinitely many. Our intuitions about these vector spaces are typically grounded in vector spaces of dimension two or (more rarely) three. It's a remarkable fact that these intuitions are usually solid and reliable. But to be sure of this, we'll need to figure out how to abstract our intuitions and formalize them in a way that we can check them rigorously.
We'll see.


Class canceled Thursday January 31 because of weather; due date for homework 3 postponed until Friday noon.


None yet.

Notes from class:

Notes from class will be posted online here and updated as we go along. The main purpose of these notes is to be some sort of record of what I did --- or meant to do --- in class each day. That's the goal anyway; the notes will inevitably lag behind reality. (notes updated 1/9/2019)

Office Hours:

My office hours are 1-2pm on Mondays in my office, which is Ryerson 353. I'm also happy to meet by appointment, or (preferably) correspond by email.

Pallav's office hours are 2-3:30pm Monday and 11-12:30pm Friday in Eckhart 14, and he'll hold problem sessions on Tuesdays 6-7pm in Ryerson 358.


There will be one midterm, and one final exam. The midterm(s) will be taken in class on Thursday February 7. The final will be take-home.

The midterm accounts for 25% of the grade in total, and the final for another 25%. The remaining 50% comes from homework, which is assigned weekly. If you have any issues with homework, exams, grades, whatever, please bring them up with me as soon as possible. Also: please come to class. My sympathy with regard to any issues you bring up will be almost perfectly correlated with your class attendance record.


Collaboration on homework is allowed, and even encouraged. With that said, please be sure that you write up your own solutions independently in order to be able to benefit fully from the exercise.

The homework should take approximately 3 hours per week. If your experience seriously deviates from this in either way, please let me know by email as soon as possible so we can do something about it.

Homework is posted online every week by Thursday at the latest, and is due before class the following Thursday in Pallav's mailbox in the basement of Eckhart.

Most homework will be taken from Sergei Treil's book, linked below. The notation A:B.C means exercise C from section B, chapter A.


The main reference is: Another nice online reference is: For an introduction to the theory of linear algebra applied to graphs (via the adjacency matrix), see: