Projects

I believe heavily in writing and speaking across the curriculum. Students need to be able to communicate technical ideas and solutions to problems clearly, confidently, and concisely. It was with this in mind, and the fact that I also like the idea of alternative assessment, that the projects came about. Students, in the third phase of the course, are required to give an oral presentation and a written report. I give a description of the projects below.

Projects give students to go in a direction that they want to explore. Giving students options and guidance, to complete a project in their realm of interest, allows them to bring their own enthusiasm and creativity to the table. It makes me smile.

Fundamental Theorem of Algebra: The Fundamental Theorem of Algebra has a fascinating history, as there are well over 100 proofs that use distinct methods. This project was based on the paper:

Derksen, Harm, The Fundamental Theorem of Algebra and Linear Algebra, Amer. Math Monthly 110(2003), 620-623
Keith Conrad's distillation of the paper can be found here .

This made a great way to study the theorem as:

It's a proof by induction with nontrivial base case
It's an application of the properties of the space of Hermitian matrices.
Commuting linear maps between complex symmetric matrices.

These are all interesting and exciting topics that sometimes get shorted in linear algebra courses.

Manifolds: The student who completed this project knew that he wanted to study theoretical mathematics in college. He had taken multivariable calculus, and we discussed the notions behind rings and modules earlier in the course. In the study of manifolds, one runs into modules over the ring of smooth functions rather regularly. The plan of study for this project is here.

Iterative Methods: Python implementations of iterative methods. These iterative methods are used to solve systems of linear equations and to find the eigenvalues and eigenvectors associated to a particular matrix. The QR-factorization and Cholesky factorization play a role in these methods. The results that he will be talking about, and the algorithms he’s implemented, can be found in the following two papers:

Iterative Methods for Finding Eigenvalues and Eigenvectors – Maysum Panju
Singular Values Using Cholesky Decomposition – Aravindh Krishnamoothy, Kenan Kocagoez

Least-Squares Problems: Working through aspects of least squares, which has applications in statistics and curve fitting data. Least squares also give a framework for doing approximations in a finite-dimensional inner-product space. The presentation consisted of the exercises and computational details to a chapter from Exercises in Computational Mathematics with MATLAB by Lyche and Merrien.