Mathematics 136A | Spring 2012 |
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## Math 136 project: applications of linear algebraThis 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. Ian 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 27 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.
## Due datesIn 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.
## Grading and other informationEach project should include the following sections.
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.
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. |