Math 403 (Spring 2024)

Math 403 Course Webpage

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

Textbooks:

• Linear Algebra and its Applications, by Peter D. Lax, second edition (book; available electronically)
• Kristopher Tapp: Matrix Groups for Undergraduates (book; available electronically)
• G. W. Stewart and Ji-guang Sun: Matrix Perturbation Theory (book; not electronic)

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
Email: ezramath.duke.edu
Webpage: https://math.duke.edu/people/ezra-miller
Course webpage: you're already looking at it... but it's https://services.math.duke.edu/~ezra/403/403.html
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
• approximation
• variation
• positivity
• convexity
• application
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:

• basic algebra: arbitrary fields, quotients, exact sequences, duality
• normed linear spaces
• convexity
• grassmannians and flags
• Hermitian, positive (semi-)definite, and normal matrices
• principal component analysis (PCA) and singular value decomposition (SVD)
• classical matrix groups
• perturbation of eigenvalues
• Perron–Frobenius theory: matrices with positive entries
• tensors and exterior algebra
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.
• (notes / video) Lecture 1: basic algebra of groups, arbitrary fields, vector spaces, homomorphisms
• (notes / video) Lecture 2: quotients, exact sequences, homology, Euler characteristic
• (notes / video) Lecture 3: duality, complex numbers, Hermitian inner products
• (notes / video) Lecture 4: Jordan form, Fundamental Theorem of Algebra, direct sum, minimal polynomial, Cayley–Hamilton Theorem
• (notes / video) Lecture 5: Banach spaces: norms, topology, equivalence
• (notes / video) Lecture 6: convex set, polytope, interior, boundary, separating hyperplane
• (notes / video) Lecture 7: supporting hyperplane, extreme point, Krein–Milman Thm
• (notes / video) Lecture 8: grassmannians as quotients and as manifolds
• (notes / video) Lecture 9: manifolds (recap), flag varieties
• (notes / video) Lecture 10: isometries of spheres and real inner product spaces
• (notes / video) Lecture 11: unitary matrices, upper-triangular representations, spectral theorem, normal operators
• (notes / video) Lecture 12: positive (semi)definite operators, singular values, polar and Schmidt decompositions, singular value decomposition (SVD)
• (notes / video) Lecture 13: ellipsoids, operator & Frobenius norms, principal component analysis (PCA)
• (notes / video) Lecture 14: matrix norms, consistency, general perturbation theorems
• (notes / video) Lecture 15: theorems of Elsner, Bauer–Fike, Gerschgorin
• (notes / video) Lecture 16: Lie algebra as tangent space, matrix exponentials, computation for GLn
• (notes / video) Lecture 17: computations for On and SLn, Lie groups and quotients as manifolds
• (notes / video) Lecture 18: entrywise positive matrices, dominant eigenvalue, Perron's theorem
• (notes / video) Lecture 19: directed graphs, stochastic matrices, Markov chains, Frobenius theorem
• (notes / video) Lecture 20: multilinear algebra, tensor product, universal property, category
• (notes / video) Lecture 21: alternating operator, exterior power, volume, determinant, functor
• (notes / video) Lecture 22: cross product, Laplace expansion, minor, r-dimensional volume, Plücker coordinate

Lecture and video policies

• (This item is a hold-over from the pandemic. But the overarching message of respect for health and safety is still valid.)
Everyone attending class in person, including students and the lecturer, must comply with the Duke Compact regarding health and safety of yourself and others.
• Failure to comply will result in immediate dismissal from the classroom. In particular,
• kindly remind the lecturer if they are not in compliance, and dismiss them if they refuse to comply!
• Recordings of lectures from Spring 2021 are posted on the Math Department video server. Each video has been edited to delete identifying information of students whose faces (and, alas, names) occasionally appeared onscreen; if you detect identifying information that accidentally remains, please inform the instructor.
• Students are expected to attend lecture live, in person, unless circumstances require lecture to be delivered virtually, in which case attendance online is expected. There is currently no plan to enable hybrid lectures.
• Class participation can enhance or detract from homework grades. Attendance in lecture is a baseline for participation, which more substantially requires contribution to in-class discourse.
• Occasional absence for expected sorts of circumstances (e.g., illness, athletic participation, religious holidays) are not penalized.
• Lectures might sometimes need to be held online or with a recording.
• There will usually be days of advance warning if this step needs to be taken, although in case of instructor illness advance warning might not be possible.
• Links for these online sessions will be available on the course Canvas page.
• These links are not to be shared. Sharing can result in class disruptions from unwanted third parties and can constitute a breach of class participants' privacy.
• Your video should remain on by default during any virtual synchronous lecture. The classroom experience is a communal one. It is dependent on multi-way communication. You should be able to see the lecturer and your classmates, and they should be able to see you.
• There are reasons to turn your video off, particularly if you have issues with your internet connection, your working space, or other external factors. But in general everyone should do their best to participate visually.

Assignments

• Reading assignments are included at the top of each homework and midterm.
• Due dates for the five homework assignments, two midterms, and final projects this semester are listed in the table below. The final project also includes an oral presentation near the end of the semester, during a class period to be determined.
• All homework and midterm assignments will be take-home. Each is due at the time of day specified on the assignment.
• All solutions you turn in, including midterms, homework, and term projects, must be typewritten using the provided LaTeX template. Communicating your ideas is an integral part of mathematics. In addition to the usual PDF files, LaTeX source files for each of the homework assignments as well as each of the midterms will be provided. You are required to use these as LaTeX templates for your solutions, by filling in your responses in those files. I am happy to answer any questions you might have about LaTeX, though you should consider asking your classmates first. The reason why it is crucial for you to use the provided LaTeX templates is because they contain margins settings and commands required for grading.
• Submit your homework and midterm solutions using Canvas "Assignments":
Math 403 Canvas –> Assignments
Submit both your .tex file (the grader or I may comment on your TeX usage) and your .pdf file, which can serve as verification that your system produces the same output as ours do. (There have been cases where .tex files upload improperly; the .pdf file then serves as proof of on-time submission.)
• The final project consists of an oral presentation and a written paper of 10 – 12 pages in length. The length of the oral presentation will depend on how many students are registered once the semester is fully under way. More details about term projects are included at the end of this "Assignments" section.
• The final projects might need to be completed in pairs if too many students are registered. If you are unable to find a partner, then the instructor will assign you one.
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

• Late homework will not be accepted.
• The logic of a proof must be completely clear for full credit.
• You must cite sources in your solutions. If you rely on so-and-so's theorem, then you must say where you found it. Be specific: "the dual rank theorem" is not precise; in contrast, "[Climenhaga, Theorem 5.10]" is. Theorems are often known by many names, so anyone grading your work could fail to recognize any given theorem by a name you might attach.
• Students are expected to adhere to the Duke Community Standard. Students affirm their commitment to uphold the values of the Duke University community by signing a pledge that states:
• I will not lie, cheat, or steal in my academic endeavors;
• I will conduct myself honorably in all my endeavors;
• I will act if the Standard is compromised.
The instructor has a long record of detecting and convicting violations, resulting in a wide range of sanctions, from grade changes to removal from the University. Every case brought by the instructor has resulted in sanctions: I have never lost a case. Don't test me.
• Collaboration on homework is encouraged while you discuss the search for solutions, but when it comes time to write them down, the work you turn in must be yours alone: you are not allowed to consult anyone else's written solution, and you are not allowed to share your written solutions. (It is very easy to tell when solutions have been copied or written together.) If you collaborate while searching for solutions, you must indicate—on the homework page—who your collaborators were. Failure to identify your collaborators is a breach of the Duke Community Standard.
• In contrast, no collaboration or consultation of human or electronic sources—except for the "Textbooks" and specific "Online resources" listed on this syllabus—is allowed for either of the two exams. You must work completely independently, without giving or receiving help from others or from the internet (beyond those sources listed above or on the homework). It bears repeating that the instructor has a long record of detecting and convicting violations of the Duke Community Standard, resulting in a wide range of sanctions, from grade changes to removal from the University. Again, don't test me.
• Consulting generative AI is prohibited for all assignments in this course, including homework, midterms, and term projects. Instead of asking a computer, ask a classmate or the instructor.
• If a student is responsible for academic dishonesty on a graded item in this course, then the student will have an opportunity to admit the infraction and, if approved by the Office of Student Conduct, resolve it directly through a faculty-student resolution agreement; the terms of that agreement would then dictate the grading response to the assignment at issue. If the student is found responsible through the Office of Student Conduct and the infraction is not resolved by a faculty-student resolution agreement, then the sanctions can be severe and, depending on the nature of the infraction, might be out of the hands of the instructor, being instead determined by the Office of Student Conduct or another University body.

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 topic should deal with substantial mathematics traceable back to linear algebra.
• You aren't expected to prove new theorems, but you are expected to present mathematics that isn't covered elsewhere in this course or in your other courses.
• You are encouraged to find your own topic! Be sure to clear it with me, either in office hours or by email. I can help you determine how to bring your topic to sufficient depth and manageable scope.
• A file with potential topics is posted on Canvas:
Math 403 Canvas –> Announcements –> Term project topic suggestions
The list is not at all meant to be exhaustive, and the topics there are merely suggestions.
The project itself consists of a
• paper (of maybe a dozen pages or so) and
• presentation (of roughly 20 minutes to half an hour) to your classmates near the end of the semester.
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.

• 40% Homework and participation
• 15% Midterm #1
• 15% Midterm #2
• 30% Term project
Participation in class discussion and office hours can contribute to your homework score.