M314 Fall 2005 Project

Due Dates: Completed Project due Monday, November 21 (or earlier)

Introduction

Sometimes, linear systems of equations involve real-life data. Such data often involve measurements which can have errors in them, due, if for no other reason, to unavoidable statistical variability. Thus systems that you might expect to have a solution still might be inconsistent. Although exact solutions might not exist, you nonetheless might be interested in approximate solutions, particularly best possible approximate solutions. In this project you will study a method for getting best possible approximate solutions, in the context of predicting, based on early season results, how the Husker football team will do in the North Division of the Big Twelve.

Some Background

Here's some background. Consider a system Ax = b of equations. If it is consistent, then you can go ahead and solve it using the methods you've learned in this class. But suppose it's not consistent. Then look instead at ATAx = ATb. It turns out this new system always has a solution, and any solution x = (x1, ... , xn) of this new system has the property that [a1-b1]2 + ... + [an-bn]2 is as small as possible, where a = (a1, ... , an) is just Ax and (b1, ... , bn) is just b. Thus this method is called the ``method of least squares''.

The Project

Here's the project. It has two parts. Make sure your project write up includes the following: Here is a link to the Big Twleve football schedule, with scores of games played.

Grading

You may work on this project in groups. All members of a given group will get the same grade. Groups of up to 4 are fine. Please check with me if you wish to form a group of more than 4. Your project grade will depend partly on spelling and grammar, in addition to the correctness of the mathematical results and the clarity of your exposition, so I recommend carefully proofreading your write up and using a spell checker.