Math 136 Project: Applications of Linear Algebra

The projects below are meant to expose you an application of linear algebra. You must work groups of between two and three people. Once your group has chosen a topic, your first task will be to locate one or two references that tell you how to use linear algebra to approach solving the problem posed in the project. Austin and I will be available to help you find references, etc.

It may be that you'd like to choose something that's not on this list. If so, make an appointment to see me before Monday, May 2 so that you can get started on a more detailed outline of your proposed project.

Each group should send me an e-mail message specifying your topic and the names of everyone in your group. The deadline for doing this is 5:00pm, Thursday, May 5. To avoid unnecessary duplication, please send me one message per group (be sure to include the names of everone in the group).

Acknowledgement. I obtained the material below from Rebekah Hahn, one of our former graduate students.

Project Descriptions

Cryptography: Cryptography is the science of encoding and decoding messages and has existed almost since man first started writing things down. More recently, Cryptography is of interest to Computer Scientists and the NSA. The idea behind this application is fairly basic in cryptography, but already requires some good mathematics.
Electrical Networks: Electrical Networks are used by Electrical Engineers to model circuits. This application gives an introduction to modeling circuits and how to use linear algebra to determine current or voltage at a particular point in the circuit.
Equilibrium Temperature Distributions: Equilibrium temperature distribution comes up in Physics and Chemistry in trying to study how heat will eventually be distributed across a thin metal plate, given the temperatures at the edge of the plate. This application explores this idea, and introduces iterative methods for finding solutions.
Genetics: Genetics is the study of inheritance. In this application, you will look at autosomal inheritance, the ideas behind recessive and dominant traits, and how linear algebra can be used to determine how a trait will be distributed in future generations.
Linear Programming: Linear Programming is a technique used in Operations Research and has applications to many different fields. In this application, you will learn about the simplex method for maximizing or minimizing a certain function, subject to a system of linear (in)equalities.
Markov Chains: Markov Chains are used to model systems, like weather, which change periodically. Typically, these changes are dependent on the immediate history of the system, in the way that the weather one day depends on conditions present the day before. This application nails down the definition of Markov Chain and explores a few areas that they are used in.
Population Growth: Exponential functions aren't the only mathematical tools for studying population growth. In this application, you will learn how linear algebra can be applied to studying the growth of a female population which has been divided according to age.
Theory of Games: You may have heard of the infamous prisoner's dilemma. Game Theory is the area of mathematics devoted to studying problems of this type, which often come up in Economics. In this application, you'll find out exactly what a game is and how to evaluate your chances of winning based on the strategy you choose.

General Information

Each of you has to chose a topic for the project you will be doing this quarter. As mentioned previously, you must work together in groups of 2 or 3 people.

Due Dates

In general, these projects will take some time and effort on your part. So, to keep you motivated and to help you get started early, there will be several due dates for the project:

5:00 pm Thursday, May 5: Send me an email from one member of your group giving the names of the members of the group and the topic you'll research.

5:00pm Thursday, May 12: You need to turn in two references that your group has found and a summary (roughly a paragraph in length) of what you've learned about your topic up to this point. This might include definitions of words related to your topic or a discussion of how you might start approaching one of the problems you've been assigned.

Beginning of class, Monday, May 23: A rough draft of your report (two copies) is due. Not getting it in on time will affect your final project grade. Each draft will be "peer reviewed" by two other groups.

Beginning of class, Thursday, May 26: Peer reviews due.

5:00pm, Thursday, June 2: Your report, in its final form, is due. Not getting it in on time will affect your final project grade.

Reports and Grading

Projects are worth 60 points, and each project should include the following sections.

Background (15 points) This is mostly a discussion of how the non-mathematical and mathematical portions of your topic fit together. In other words, you need to talk about what you needed to know about your topic in order to do the associated problems and how linear algebra fits into the picture. So you might include the definitions of the words I've given you, the linear algebra ideas you used (e.g. matrix multiplication, solving linear systems, etc), and some explanation about why these ideas were useful.
Solutions (35 points) You need to include solutions to the problems included in this packet. Don't just give the answers, however. Include a full, detailed explanation of what you're doing at each step. You'll want to use words and write in full sentences, though you can also have the occasional formula or sequence of equalities.
Bibliography (5 points) List the references you used to complete this report. You don't need to get out your Strunk & White or anything, just list title and author for any books you used. You should also include a list of people that you consulted or any other form of help that you received. For example, you might obtain some of your information from the internet; in this case, you could include the website. You'll need at least one book as a reference, preferably two, and a total of at least two references.

You'll notice that there are still 5 points unaccounted for. The remaining 5 points are for style: clarity, neatness, flow, design, organization and creativity--it's important to be able to communicate your ideas.

Note: you don't have to put your report in the precise order given above. You may prefer to use the assigned problems to illustrate how the ideas of the subject fit together with the mathematical ideas that you will be using, in which case Background and Solutions would be interwoven. Just make sure that these aspects appear in your report.