Math 403 (Spring 2024)

Math 403 Course Webpage

Advanced Linear Algebra

Spring 2024, Duke University

General information | Course description | Lecture notes and videos | Lecture and video policies | Assignments | Homework schedule | Grading | Links | Fine print

General information

Lectures: Tuesday and Thursday, 11:45 – 13:00, Physics Building 047


Online resources:

Contact information for the Instructor

Name: Professor Ezra Miller
Address: Mathematics Department, Duke University, Box 90320, Durham, NC 27708-0320
Office: Physics 209
Phone: (919) 660-2846
Course webpage: you're already looking at it... but it's
Canvas site: available to registered students via Duke NetID
Office hours:   Tuesday, 13:00 – 14:00 in Physics 209 or outside
                          Thursday, 16:00 – 17:00 in Physics 209 or outside

Course description

This course covers topics in linear algebra beyond those in a first course. 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.

Course content:

Prerequisites: Fluency with a first course (Math 218 or 221) will be assumed. Notions from calculus will also be crucial: limits, continuity, derivatives, and so on, in arbitrary dimension; thus solid knowledge of multivariable calculus (Math 211 or 222) will also be required. No other mathematical content is assumed, but the more you know, the easier the course is. Students who take this as their first course beyond the 200-level report that it is quite difficult. In contrast, those who have already taken some upper-division courses find Math 403 to be quite manageable, in part because not everything is completely new and in part because they already have experience writing proofs.

Lecture notes and videos

All lectures in one PDF file

The lecture notes can and may be updated or corrected. If you think you have found an error, check that you have the latest version before sending a correction.

All videos in a Math Department server Collection

The video player on the Math Department video server have some nice features. For example, it is possible to zoom in on the video to make the writing on the board bigger.

Lecture and video policies


Check here two weeks before each homework is due, or one week before each exam is due, for the specifics of the assignments. If an assignment hasn't been posted and you think it should have been, then please do email the instructor. Sometimes I encounter problems (such as, for example, the department's servers going down) while posting assignments; other times, I might simply have neglected to copy the assignment into the appropriate directory, or to set the permissions properly. (I do try to check these things, of course, but sometimes web pages act differently for users inside and outside the Math department so I don't notice.)

Assignment Due Date Problems
Homework #1
Sat. 27 January in PDF or LaTeX  
Homework #2
Sat. 10 February in PDF or LaTeX  
Midterm 1 Sat. 17 February  
Homework #3
Sat.   2 March in PDF or LaTeX  
Homework #4
Sat. 23 March in PDF or LaTeX  
Homework #5
Tue.   9 April in PDF or LaTeX  
Midterm 2 Sat.  20 April
Final project Tue. 23 April, 12:00  

Policies regarding homework and midterms

Term projects involve choices of partners and topics (exception: grad students enrolled in Math 703 complete their term projects solo); ideally, you should be excited about both. The project itself consists of a The precise length of each presentation will depend on scheduling issues. (The presentations need to fit into four lecture slots.)

Action items and timeline.
  1. Pick partners; when you have a partner, each of you should email me by Friday, February 23. That way, I not only get notified of the pairs, but I also get independent confirmation from each member of the pair. After everyone who is able to choose a partner has done so, I will assign any remaining Math 403 students to pairs. Those enrolled in Math 703 should take no action here; you will work solo. The total number of students enrolled often remains in flux by this time in the semester (believe it or not). If you are willing to be in a group of three, who will write a longer paper and deliver a longer presentation, then say so, but keep in mind that it might not be necessary.
  2. By Friday, March 1 (a week before Spring break), post to Canvas Assignments a couple of paragraphs outlining your chosen topic. Before submitting those paragraphs, I encourage you to talk with me for a few minutes about your topic to ensure that it is at least in principle suitable.
  3. By Monday, March 18, post to Canvas Assignments a preliminary outline of your term paper. (You should consider posting this before Spring break instead of after.) It should be detailed enough for someone (such as me) to be able to determine whether its scope is realistic and its depth sufficient.
  4. By Wednesday, March 27, post to Canvas Assignments a rough draft or highly detailed outline of your term paper. You should have completed all of your research by this time.
  5. The oral presentation slots on April 16 and 18 need to be allotted. If you have a preference, then tell me so, but be sure to give at least one backup option. First come, first served for these; you can email me about this (long) before you submit your topic selection paragraphs. After preferences are taken into account, the instructor will assign the remaining slots randomly.
  6. Final complete term papers are due on Tuesday, April 23, which is the last day of undergraduate and graduate classes. There is no final exam in this course, so Math 403 will be done (for you, not for me) on that date.

Grading scheme

Final course grades: Participation in class discussion and office hours can contribute to your homework score.


University academic links Departmental links

The fine print

I will do my best to keep this web page for Math 403 current, but this web page is not intended to be a substitute for attendance. Students are held responsible for all announcements and all course content delivered in class.
Many thanks are due to Jeremy Martin and Vic Reiner, who provided templates for this webpage many years ago.

The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by Duke University.
Mon Jan 29 15:21:27 EST 2024