MATH 107-01/02, Linear Methods, Spring 2011

Abstract

Linear algebra is a set of powerful techniques with applications to many problems that arise in human endeavors. We will study some methods that have been developed to address these problems and examine some of their theoretical foundations. The problems and examples we will consider come from subjects like business, economics, and politics. We will use Excel from time to time. We will begin with a brief review of (systems of) linear equations. We will learn how to solve linear systems using Gauss-Jordan elimination. We will discuss matrix algebra with applications to input-output analysis and Markov processes. Input-output analysis is used to determine how interdependent producers should behave. Markov processes are used to predict long term values of interdependent quantities that vary probabilistically. Next, we will talk about linear programming (LP) problems. An LP problem is one in which you seek to maximize some linear function, such as profit, subject to certain linear constraints, such as budgetary or workforce limitations. You will learn to solve LP problems geometrically, using the simplex method, and by computer. Integer and 0-1 programming problems, which are closely related to LP problems, are too labor-intensive to solve by hand. We will learn how to solve them using Excel. We will touch briefly on the subject of computational complexity. We will conclude with a look at game theory, the study of strategic interactions between competing interests. We will explore the connection between game theory and LP problems.