You should soon form groups to work on projects. I want from each group
the names of the members and the project title with a
brief abstract by mid October. Here I will collect my ideas for your
final projects. These are my suggestions but you don't have to stick
to this list if you have other ideas. Please get my approval before
working on a project. The list below will be updated frequently.
Lights off: This is a beautiful application of solving linear equations over the field of two elements (0 and 1).
This game can be easily generlaised to 3 dimensions. In fact, if you analyze this carefully you should be able to develop a theory of "commutative games." Here is a nice wikipedia link which is useful [ Lights out ]. See variations at the bottom. Also, the applications of linear algebra link also has this game.
Error correcting codes: You can learn about error correcting codes from this page Linear codes. This is fun, you can come up with nice games using linear algebra in Hamming codes. The book on Elementary asbtract algebra by Hungerford is good for this. That is where I learned this subject. You can download this book for free CLICK HERE. If you can't download or get it from the library, you can borrow my personal copy.
Game theory and the min-max theorem: Check out the book Linear Algebra and applications by Gilbert and Strang. If you can't get it from the library, you can borrow my personal copy.
Markov chains: These are radom processes in which the next state of a system depends only on its current stage. A simple example is the sequence of moves in the popular children's game called Snakes and Ladders. Linear algebra plays a nice role in the analysis of such process which
arise in numerous fields. Read this wikipedia article for more information.
Applications of linear algebra in Crypography: I explained some of this in class. Plain English text is chopped into vectors which live in R^n for some n and these vectors are encripted using an invertible matrix (known beforehand to both parties). The receiver decripts the message by applying the inverse of the matrix to the vector he receives. There are all kinds of variations which start from this simple idea. You can explore more; click here for some ideas.
Geometry: Rotations in the 3-space: I don't think I have any reference for this but it is an absolutely beautiful application of linear algebra. The point is to show carefully that the composition of two rotations is again another rotation. See the first day's class notes where
I explained what this problem is about. If you are interested in this, please come and talk to me. I will tell you more about it.
Applications of the Singular value decomposition: I have been very curious about this for many years. Andy Schultz (a freind and collaborator of mine) have worked on this. See his homepage . See how his image transforms when you place the cursor on his image. He explained very well how this work. Image compression is truly fascinating. Explore this beautiful world.
Linear programming: This is another beautiful theory which is an extension of linear algebra. Here is a nice reference for this. The applications are very
pretty cool CLICK HERE.
Applications of linear algebra in differential equations: There
is a very rich theory of systems of linear ODEs. Suppose there is an interaction of n variablesx_1, x_2, ... x_n and this interactions can be
modeled by a system of 1st order linear differential equations as x' = Ax where A is an n x n matrix and x is a column vector of varibles in the problem. Then one can use tools from linear algebra to solve this system and this has brilliant applications in the real world.
For me, the most interesting ones come from Physics, spring-mass system, double pendulum etc. A good reference is Differential Equations -- An introduction to modern methods and applications. James R. Brannan and William E. Boyce. You can borrow this book from me.
Advanced topics in linear algebra: If you are not interested in applications, you can work on something totally pure. You can work on more
advanced topics of linear algebra such as quadratic forms, generalised inverses and least squares method, theory of unimodular matrices, rational canonical forms and jordan types to mention a few. These can be found on advanced books on linear algebra. Talk to me about them.
Paper submission: Each student should write a paper on the
topic of their project work. I want a paper from
each student, not one from each group. The deadline for Paper submission is Monday, December 6th. Here are some important things to keep in mind when you write your paper.
The quality of the paper is more important than its length. You can aim for a 5 page paper.
Your paper should have a title, abstract, introduction, the main
body, conclusion (optional), and references.
You are welcome to use material on the web or from the library but
please be sure to give references. Your own input is very essential.
The paper should be written in some form of Tex or Microsoft Word.
It should be well-written, and should aim at a level of mathematical maturity that is clearly beyond a typical homework assignment.
You should have an extremely good introduction. Give background information on the topic, provide motivation for the results in the paper if appropriate, and examples.
In my opinion a good Mathematical paper is one that satisfies the C^5 condition: Clear, Concise, Coherent, Creative, and Cool. That is the key to learning the art of writing good mathematics. I will keep this factors in mind as I grade your submissions. This also applies for presentations
Here is a tentative schedule of presentations. It is essential that you attend ALL talks and not just your own talk. There is a lot you (and I) can learn from these talks. More information about these talks will be announced later.
Monday, Nov. 29th
Applications of Linear Algebra
I will talk about some applications of linear algebra with particular emphasis on games.
Wednesday, Dec. 1st
Game Theory and the Nash Equilibrium (Honors Project Presentation)
We will show how linear algebra can be used to study equilibrium strategies in games. This will be illustrated with a specific example from Poker.
Thursday, Dec 2nd
Steve Dudley Andrew Krolman Ashok Raghavan
We show how linear algebra can be used to create basic
algorithms to solve Sudoku type puzzles from the most basic
(circular) to the more complex (regular or extreme) ones.
Friday, Dec. 3th
Using Leopoult's Input-Output Model to Solve Economic Questions
Andrew Gillespie Nathan Johnson Justin McGetrick Laura Prochot Michael Soto Andrew Walter
Using Leontief input-output model we will solve the required
inputs and outputs of a closed market. Furthermore, by using the
demand vector, we will be able to solve for gross production along
with net production. Thus, seeing if a company is making a profit.
Monday, Dec. 6th
Applications of Linear Algebra in Crypotography
Kate Cross Sean Higgins Katie Jean Elizabeth Kuntz Mike Suess
There are many fascinating methods of crypography using Linear
Algebra. Historical uses are shared all the way to the most current
uses. Examples of each type of crypography are shown using the
class slogan, "MATRICES ACT ON VECTORS."
Wednesday, Dec. 8th
Error Correcting Codes
Mel Chwee Andrew Kampwerth Ryan Korth Brittany Priest Alyson Weber
We will explore the different applications of error correction
and error detection codes including but not limited to credit
cards, bar codes, postal codes, and Hamming codes.
Thursday, Dec. 9th &
Friday, Dec. 10th
Applications of Eigenvalues
Adam Bourjaily Bonnie Klauber Drew McGary Jordan Morris Chris Nielson Yuto Tsukida
We will disucss the eigen value problem along with its theory
using examples from the Fibonacci sequence and other recursive
sequences, the Markov processes, differential equations and
applications in Physics.
VERY VERY IMPORTANT NOTE: You have to read the material in the book (both the
theory and examples) before you start doing the homework problems from
Week 12 (Nov 8 - Nov 14)
Sec 4.3 SKIP Sec 4.4 SKIP Sec 5.3 1, 3, 9, 13, 17, 23, 27, 29--48, 49, 50, 53, 55, 57, --
--61, 65, 71, 75, 81, 83, 84,85, 86,
87, 88 Sec 5.4 SKIP Sec 5.5 TBA (Very important section for those students who are
working on the project on Fibonacci sequences and related things. )
Week 11 (Nov 1 - Nov 7)
No new homework assigned this. Catch up on the HW from previous
sections and complete the worksheet.