Math 403 Course Webpage

Advanced Linear Algebra

Spring 2026, Duke University

General information

Lectures: Tuesday and Thursday, 11:45 – 13:00, Physics Building 227

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:

• Climenhaga: Lecture notes for Advanced Linear Algebra
• Hefferon: Linear Algebra
• Treil: Linear Algebra Done Wrong
• Cherney, Denton, and Waldron: Linear Algebra
• Cornell: Honors Linear Algebra
• Gustafson: Advanced Topics in Linear Algebra (not a full text, but a few pages of notes)

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://scholars.duke.edu/person/ezra.miller
Course webpage: you're already looking at it... but it's https://sites.math.duke.edu/~ezra/403/403.html
Canvas site: https://canvas.duke.edu/courses/71537 (available to registered students via Duke NetID)
Office hours:   Tuesday, 15:50 – 17:00 in Physics 209 or outside
                          Thursday, 13:00 – 14:00 in Physics 209 or outside

Course description

• abstraction
• approximation
• variation
• positivity
• convexity
• application

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

Lecture notes and videos

• (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 GL_n
• (notes / video) Lecture 17: computations for O_n and SL_n, 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

• 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) is 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.
  • That said, there are valid reasons to turn your video off, particularly if you have issues with your internet connection, your working space, or other external factors. Turn off your video if you must, 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 and the written portions of two midterms and final projects this semester are listed in the table below.
• The midterms will be graded based on individual oral evaluations; see oral midterm exam evaluation.
• 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.  (Currently all due dates are Saturdays at 11:59pm.)
• 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 are completed in pairs for those registered for Math 403.  If you are unable to find a partner, then the instructor will assign you one.
• In contrast, grad students enrolled in Math 703 complete their final projects solo.

Assignment	Due Date	Problems
Homework #1	Sat.  24 January	in PDF or LaTeX
Homework #2	Sat.    7 February	in PDF or LaTeX
Midterm 1	Sat.  14 February
Homework #3	Sat.  28 February	in PDF or LaTeX
Homework #4	Sat.  21 March	in PDF or LaTeX
Homework #5	Sat.  11 April	in PDF or LaTeX
Midterm 2	Sat.  18 April
Final project	Wed. 22 April, 12:00

Policies regarding homework and midterms

• Late homework or (especially) exams will not be accepted.  If you fail to submit an assignment there will be no make-ups.  Some accommodation may be possible in one of the following situations: personal emergencies or tragedies, an incapacitating illness, a religious holiday, or varsity athletic participation.  These types of absences require various procedures.  For "unusual circumstances, such as illness", you have two options, both documented in Duke's attendance and missed work policy: submitting an incapacitation form or requesting a Dean's excuse.  Please follow the instructions relevant to your case when it arises—or before it arises, if possible (such as with athletics or religious holidays)—and in any case review these instructions now to familiarize yourself with the procedures.  According to Duke policy, submission of an incapacitation form does not guarantee that the absence or lateness will be excused.
• 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.
• 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) until after everyone has completed their oral evaluations.

AI and academic honesty policies

• Consulting generative AI is prohibited for all aspects of this course.
• The purpose of this course is to train your mind to think in certain ways.  Past experience has shown that this goal is impaired or circumvented entirely by AI consultation.  Instead of asking a computer, ask the instructor or (if the assignment allows) a classmate.  There is a place for AI in mathematics, but this is not it.
• It is beyond disrespectful to ask me or a grader to waste our valuable time providing feedback on machine-generated output.  You wouldn't ask a woodshop instructor to evaluate a store-bought birdhouse.  You wouldn't ask an athletic coach to evaluate artificially generated video of you taking foul shots. And in those cases you wouldn't learn anything about carpentry or basketball.  So why would you submit anything AI generated for math homework?
• You may at any time be required to defend your written work orally, with no advance warning.  Don't submit it in writing if you can't explain it orally.  This policy is being elevated to an official grading policy; see the section on oral midterm exam evaluation.
• 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.
• Students are expected to adhere to the Duke Community Standard more generally.  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.  The instructor has prosecuted a lot of AI usage violations.   Every single incident tagged by the instructor—concerning AI or any other infraction—has resulted in sanctions: I have never lost a case.  Don't test me.

Term projects

• 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.

• 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.

1. Pick partners; when you have a partner, each of you should email me by Friday, February 20.  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, February 27 (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 16, 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 25, 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 9, 14, 16, and 21 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 Wednesday, April 22, 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.

Oral midterm exam evaluation

• Each midterm exam grade will be determined by an oral evaluation.
  • Each student's oral evaluation will result in a letter grade.
  • The midterm score will be the product Y × Z, where Y is the number of points corresponding to the earned letter grade and Z is the fraction of problems not left blank in the written solutions.
  • For example, if a student answers 5 out of 7 midterm problems and has a perfect oral evaluation, then the final midterm score will be 72 = 100 - 2*14.  A lower oral evaluation letter grade scales that 72 accordingly.
  • The instructor reserves the right to alter a final midterm score by taking into account the written responses. This could raise your grade or lower it, compared to the letter grade earned on the oral evaluation.
• Oral responses are required to be generated by the student during the oral exam. The student is allowed to refer to their written solutions from time to time but is not allowed to read from their written solutions while presenting oral responses.
• The instructor will select which midterm problem(s) to probe during the oral exam after the written solutions are submitted. It should not be expected that the selection will be uniformly random.  However, note the next item.
• If a student's written solutions contain a blank response to a certain question, then the instructor will not ask about that question during oral evaluation.  A proportional number of points will be deducted from the midterm exam score for that problem, but there will be no further repercussions for a blank response.
• Two students will be present for each oral evaluation, to ensure fairness: every oral evaluation should have an observer.
• Students pairs are determined by the students. If you are unable to find an oral exam partner, then the instructor will assign you one.
• Each evaluation will run for 15 minutes.  That means that for each evaluation pair, the evaluation period will be 30 minutes.
• Each midterm evaluation will occur on the Monday following the Saturday submission deadline for the written responses.  For this reason, it is particularly crucial that your solutions be submitted on time.  Some kind of sign-up system will allow students to select available evaluation times on a first-come, first-served basis.
• This is the first semester that AI has resulted in the implementation of oral evaluations for midterms in this course.  Please be patient and forgiving.  The evaluation process is therefore experimental and may change over the course of the semester—for example, in response to feedback.  With that in mind, if you have feedback that you feel could improve the process, then please do speak up.  These are changing times for everyone.