Unit 7
Two Applications of Linear Algebra

Introduction

In this final unit, we study two applications of linear algebra. The first is an elementary form of cryptography.

Cryptography is a major field of modern mathematics; indeed, the secure transfer of information over the Internet would be impossible without encoding methods that were extremely difficult to break. In this section, however, we consider only a very simple form of code.

The second application is in the field of iterated systems. This is an area of mathematics which has grown tremendously in significance with the advent of digital computers. It has applications in the new fields of chaos theory, fractal geometry, data compression, and image processing.

Objectives

When you have completed this unit, you should be able to

1. define the terms “cipher,” “plaintext,” “ciphertext,” “encryption,” “decryption” and “substitution cipher.”
2. define a polygraphic cipher system and a Hill cipher.
3. carry out simple operations in modular arithmetic, and state some simple theorems of modular arithmetic.
4. use simple methods of modular arithmetic and matrix algebra to encode and decode messages with the Hill cipher.
5. define the term “iterated system,” and give simple linear examples of such systems.
6. define the orbits, fixed points, attractors, and basins of attraction for iterated systems.
7. determine the long-term behaviour of simple linear iterated systems.