Term Project
Colorado State University, Pueblo
Math 307 — Introduction to Linear Algebra — Spring 2011

The goal of these project is to give students an opportunity to work a little more deeply in one area than we usually do during the term.

The expectation is that your work will result in some concrete product. This can be a short written explanation accompanying a piece of software or some written calculations, or could be a slightly longer, stand-alone report.

Whatever you do, it should take about as much time as a small multiple of homework sets (the multiple being at least two and less than five).

Your first step should be to choose a topic and to discuss it with me. Feel free to come up with your own idea, or to do one of (or inspired by) the following:

1. Write a computer program which demonstrates graphically and/or numerically the results of various kinds of matrix multiplications on various vectors (also given as input) in R² or some other particular Rⁿ.
2. Write a version of a number of our definitions and results which generalizes the approach from class to complex vector spaces and Hermitian inner products; go as far as describing the spectral theorem for self-adjoint operators (and perhaps an application of this theorem, say to observations in quantum mechanics). [A place to get started on this is the exploration Vectors and Matrices with Complex Entries in the book on pp. 552-556.
3. A report on the "eigenfaces" approach to facial recognition; some software would be nice (but fairly hard). There are many references about this topic which can be easily found on the Internet.
4. A report — and some software? &mdash on the "Digital Image Compression" topic in the book on pp. 616, and in many other references.
5. Read up and then write a description of Jordan Cannonical Form and/or Schur's Triangularization Theorem — both are results which give a particularly simple form for a general matrix (a simple form, like a diagonal matrix, which works even if the matrix is not diagonalizable).
6. Write up solutions to the Explorations on pp. 393-396 of our textbook. Some examples of computations with the method for approximating eigenvalues would be nice; a computer tool to implement this algorithm for a general input matrix would be fantastic.
7. Write a brief summary of the Perron-Frobenius Theorem and of one of its applications.
8. Write a report on Lagrange interpolation, including careful definitions and solutions of problems 59-62 of §6.2.
9. Write a report on Google's PageRank algorithm including theoretical description and practical applications. Some worked-out examples would be nice; even nicer would be a general computer tool to perform a similar calculation.
10. Write a computer tool to calculate and display (graphically, if possible) the kinds of discrete linear dynamical systems discussed in §4.6 of the textbook.
11. Write a computer tool to implement the iterative approaches to finding eigenvalues discussed in §4.5 of the textbook.
12. Write a solution/explanation/generalization of the curve fitting application of determinants on pp. 287-288 of the textbook.
13. Write an explanation, including some carefully worked-out examples, of the kinds of applications of matrix operations to robotics as described on pp. 224-227 of the textbook.
14. For students willing to work with scalars in the more exotic collections Z/nZ -- the integers modulo n: write a report (with worked-out examples and/or an accompanying computer tool) about any of the recurring applications in the textbook to computer science, such as error-correcting or -detecting codes.
15. Write a report about the Magic Squares exploration on pp. 464-466 of the textbook; again, a computer tool to generate magic squares with certain properties would be nice.
16. Write a report about the Tilings, Lattices and the Crystallographic Restriction exploration on pp. 519-521 of the textbook; a relatex computer tool would be nice, although would require somewhat sophisticated computer graphics skills. [There already exist many such computer tools; also, there are many good books on this subject.]
17. Something else entirely you think of and discuss with me....