Math 136 project: applications of linear algebra

This quarter, you need to do a project involving an application of linear algebra. You must work in groups of either two or 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 study the problems posed in the project. Mark and I will be available to help you find references and to answer any other questions.

It may be that you'd like to choose something that's not on the list below. If so, make an appointment to see me before Friday, April 29 so that you can get started on a more detailed outline of your proposed project.

Project descriptions

Cryptography

Cryptography is the science of encoding and decoding messages and has existed almost since humans first started writing things down. More recently, cryptography is of interest to computer scientists and the NSA. The particular application studied here is pretty basic in cryptography, but uses some interesting 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 study 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.

Acknowledgement. This material has been compiled by various members of the math department, including Robin Graham, Tom Duchamp and one of our former graduate students, Rebekah Hahn.

Due dates

In general, these projects will take some time and effort on your part. So, to make sure you don't fall behind, there are several due dates for the project. Missing any of the due dates can affect your final project grade.

5:00pm, Friday, April 29: Send me email specifying your topic and the names of the people in your group. To avoid unnecessary duplication, please send me only one message per group.

5:00pm, Friday, May 6: You need to turn in (by email or by hand) citations for two references that your group has found and a summary (roughly a paragraph in length) of what you've learned about your topic so far. 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, Tuesday, May 17: A rough draft of your report is due. Each draft will be "peer reviewed" by two other groups.

Beginning of class, Monday, May 23: Peer reviews due.

5:00pm, Tuesday, May 31: The final draft of your report is due.

Grading and other information

Each project should include the following sections.

Background 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 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 List the references you used to complete this report. You don't need to get out your Strunk & White or anything, just list title, author, and year published 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.

Projects are worth 25 points, and of those 25 points, you get 5 points for meeting all of the due dates. The remaining 20 points are divided evenly into two criteria: the mathematics and your presentation of it.

Mathematics. Obviously you should avoid mathematical errors. Your project should use linear algebra in an interesting way: it's not good enough to just suddenly multiply a few matrices for no apparent reason. You also need to cover the material described above – background, solutions to the problems, bibliography. For full credit, you should perhaps go beyond just the description in the handout.

Presentation. There are "local" and "global" writing issues. Local ones: Have you chosen good notation? Are you using (mathematical) language well and appropriately? Is everything you're written relevant? Have you included a good level of detail: not too much, not too little? Do you have good transitions? Are there grammatical errors or misspellings? Does your paper sound good when read aloud?

Global writing issues: Have you organized the project well? Note that you don't need to have sections labeled "Background" and "Solutions" – you can organize the paper however you think makes sense. One model could be: first state the goal of the project in general terms, then give necessary background, then discuss the project in detail, interweaving solutions to the problems as applications or examples. Alternatively, you could pose some of the problems as motivating questions at the beginning as part of the overview of the project.