403/503 – Advanced linear algebra – Syllabus

Math 403/503: Advanced linear algebra.

Andrés E. Caicedo.
Contact Information: See here.
Time: TTh 10:30 – 11:45 am.
Place: Mathematics Building, Room 124.
Office Hours: Th, 1:30 – 3:00 pm, or by appointment. (If you need an appointment, email me a few times/dates that may work for you, and I’ll get back to you).

Textbook: Although it is not a textbook per se, our main reference will be

• Jiří Matoušek. Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra. American Mathematical Society, Student Mathematical Library, vol. 53, 2010; 182 pp. ISBN-10: 0-8218-4977-8. ISBN-13: 978-0-8218-4977-4.

Here is the publisher’s page. A preliminary version is available from the author’s page. Review (MR2656313 (2011f:15002)) by Torsten Sander at MathSciNet.

Another useful reference in the same spirit is

• László Babai, and Péter Frankl. Linear Algebra Methods in Combinatorics. With Applications to Geometry and Computer Science. Preliminary Version 2 (September 1992), 216 pages.

This book is unpublished. A copy can be obtained from the Department of Computer Science at the University of Chicago, or elsewhere. There are also several sets of lecture notes on this topic available online. See for example here or here.

We will not restrict our lectures to topics related to these applications, and also cover some more traditional material, and some numerical aspects of the theory. A pdf of Uwe Kaiser‘s notes can be found here and the TeX source here. I will provide handouts of additional material as needed.

Contents: The department’s course description reads:

Concepts of linear algebra from a theoretical perspective. Topics include vector spaces and linear maps, dual vector spaces and quotient spaces, eigenvalues and eigenvectors, diagonalization, inner product spaces, adjoint transformations, orthogonal and unitary transformations, Jordan normal form.

The way we will develop the theory is by studying examples of some of its typical applications, and then covering the topics needed to understand these examples.

Please bookmark this post. I update it frequently with detailed week-to-week descriptions.

Detailed day to day description and homework assignments:

• January 21. Review of basic linear algebra: Vector spaces, fields. On “field” vs “body”, see here.
• January 23. Linear transformations, matrices, bases. Quick overview of topics. Solving linear recurrences (Matoušek’s lectures 1 and 2).
• January 28. Recurrent sequences. A useful reference (unfortunately in Spanish): Sucesiones recurrentes, by A. I. Markushevich. The goal is to use these results to motivate the study of the Jordan canonical form. Office hours this week will be on Friday, 11:45-1:15.
• January 30. Recurrent sequences. Diagonalizable and non-diagonalizable matrices.
• February 4. Ideals. Minimal polynomials. Homework 1, due February 18.
• February 6. Jordan blocks. Recurrent sequences.
• February 11. The Jordan form theorem.
• February 13. Odds and ends. Other possible approaches to solving linear recurrences. (We will revisit generalized eigenspaces, direct sums, and the Jordan form theorem later on.) Next topic: Parity (including Matoušek’s lectures 3, 4, 17).
• February 18. Parity. Oddtown. (Matoušek’s lecture 3).
• February 20. The Eventown theorem. Bilinear forms, inner products, isotropic vectors, singular spaces. (Section 2.3 in the Babai-Frankl notes.)
• February 25. Singular spaces (continued).
• February 27. The Berlekamp-Graver strong Eventown theorem.
• March 4. The generalized Fisher inequality and additional results on set systems with forbidden intersections (Matoušek’s lecture 4). Finite projective planes. Homework 2, due March 18. Next topic: Finding eigenvalues.
• March 6. Finite projective planes (continued). Isbell’s theorem.
• March 11. Isbell’s theorem (continued). Matoušek’s lecture 17.
• March 13. Wilkinson’s polynomial, Geršgorin’s circles theorem.
• March 18. Geršgorin’s theorem (continued), Lévy’s theorem.
• March 20. Taussky’s extension of Lévy’s theorem, irreducible matrices, strongly connected graphs.
• March 22–30. Spring break. See here for some references on Geršgorin’s theorem, and an opportunity for extra credit, due April 8.
• April 1. The equivalence of irreducibility and strong connectedness. Ostrowski’s theorem.
• April 3. Orthogonal matrices have real eigenvalues. The power method for computing eigenvalues.
• April 8. Orthogonal matrices are diagonalizable. Francis’s QR algorithm.
• April 10. Francis’s algorithm, continued.
• April 15. Reflectors. Francis’s algorithm (conclusion). Homework 3, due May 8.
• April 17. Geometric approach to determinants.
• April 22. Weierstrass axiomatization.
• April 24. Determinants (continued). Vandermonde determinant.
• April 29. Vandermonde determinant (continued). Matoušek’s lecture 21.
• May 1. Matoušek’s lecture 21 (continued). Algebraic graph theory. Homework 4, due May 15.

I expect to complete a draft of a set of notes based on our lectures during the Summer. Contact me if you are interested in a copy.

Grading: Based on homework. No extensions will be granted, and no late work will be accepted. I expect there will be no exams, but if we see the need, you will be informed reasonably in advance. You are encouraged to work in groups and to ask for help. However, the work you turn in should be written on your own. Give credit as appropriate: Make sure to list all books, websites, and people you collaborated with or consulted while working on the homework. If relevant, indicate what software packages you used, and include any programs you may have written, or additional data.

I may ask you to meet with me to discuss details of sets, and I suggest that before you turn in your work, you make a copy of it, so you can consult it if needed.

I post links to supplementary material on Google+. Circle me and let me know if you are interested, and I’ll add you to my Linear Algebra circle.

Twitter.

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