403/503 Linear Algebra – Syllabus

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.