## 403/503 Linear Algebra – Syllabus

January 12, 2015

Math 403/503: Linear algebra.

Andrés E. Caicedo.
Contact Information: See here.
Time: TTh 10:30 – 11:45 am.
Place: Mathematics Building, Room 124.
Office Hours: T, 1:30 – 3:00 pm.

Textbook: Our two main references will be:

In both texts, linear algebra is always carried out on (finite dimensional) vector spaces over $\mathbb R$ or $\mathbb C$. Although it will not be our emphasis, linear algebra over finite fields plays an increasingly important role in a variety of subjects. For an introduction, see

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.

My goal is to present the basic theoretical prerequisites to use linear algebra in a fruitful way (be it in applications or in a mathematical context). In particular, the field of convex optimization has gained significance and is the object of intense current research. Probably we will not have time to cover as much of this topic as I would like, but we should at least discuss linear optimization and some generalizations.

Roughly, I expect to cover at least the following topics:

• (Finite dimensiona) vector spaces over $\mathbb R$ and $\mathbb C$, and subspaces.
• Bases, spanning sets, linear independence.
• Linear transformations and matrices.
• Inner products and norms.
• Eigenvalues and eigenvectors.
• Symmetric, hermitian, normal, unitary transformations.
• Jordan canonical form.
• Singular values decomposition.
• Roger-Penrose pseudoinverse.
• Numerical computation of eigenvalues.
• Linear optimization.
• Convex sets and functions.

Although we will discuss numerical methods and algorithms, the emphasis will be theoretical. Detailed proofs of the main results will be provided, and a level of rigor and mathematical sophistication compatible with a beginning graduate course are expected. In particular, students are expected to be familiar with the contents, results, and proofs, of a first course on linear algebra.

Grading: Based on homework. Some routine problems will be assigned frequently, to be turned in from one lecture to the next, and some more challenging problems will be posted periodically, and more time will be provided for those. 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. Even if you collaborate with others and work in groups, 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. Failure to provide credit or to indicate these sources will affect your grade. Your homework should be typed in LaTeX.

Please visit Sharelatex or Overleaf, where you can start practicing right away. LaTeX has been the primary tool for the dissemination of mathematics (and many other sciences, take a look at the ArXiv to get an idea of how widely used the program is), and it has been so for almost 35 years, even though it has changed very little in that time. It is important to master the LaTeX system, since the language it provides for expressing mathematics will certainly be the standard for many years to come. MathJax and other technologies are expected to eventually replace LaTeX as the standard, but for the time being, knowing it is essential. for instance, Scott Aaronson lists as the first of his Ten Signs a Claimed Mathematical Breakthrough is Wrong that the authors do not use (La)TeX.

LaTeX is available as free software, and abundant documentation exists. A few useful references are The (not so) short introduction to LaTeX, the NASA guide to LaTeX commands, and The comprehensive LaTeX symbol list. I recommend that you also bookmark and visit frequently the Q&A site on Stack Exchange.

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

## 403/503 – Advanced linear algebra – Syllabus

January 19, 2014

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.