An Introduction to Matrices, Sets and Groups for Science Students

G. Stephenson

• Publisher: Dover Publications
• Year: 2016
• ISBN:
  9780486809168
• Edition: 1
• Topic: Mathématiques
• Subtopic: Mathématiques appliquées

This outstanding text offers undergraduate students of physics, chemistry, and engineering a concise, readable introduction to matrices, sets, and groups. Concentrating mainly on matrix theory, the book is virtually self-contained, requiring a minimum of mathematical knowledge and providing all the background necessary to develop a thorough comprehension of the subject.
Beginning with a chapter on sets, mappings, and transformations, the treatment advances to considerations of matrix algebra, inverse and related matrices, and systems of linear algebraic equations. Additional topics include eigenvalues and eigenvectors, diagonalisation and functions of matrices, and group theory. Each chapter contains a selection of worked examples and many problems with answers, enabling readers to test their understanding and ability to apply concepts.

CHAPTER 1

Sets, Mappings and Transformations

1.1 Introduction

The concept of a set of objects is one of the most fundamental in mathematics, and set theory along with mathematical logic may properly be said to lie at the very foundations of mathematics. Although it is not the purpose of this book to delve into the fundamental structure of mathematics, the idea of a set (corresponding as it does with our intuitive notion of a collection) is worth pursuing as it leads naturally on the one hand into such concepts as mappings and transformations from which the matrix idea follows and, on the other, into group theory with its ever growing applications in the physical sciences. Furthermore, sets and mathematical logic are now basic to much of the design of computers and electrical circuits, as well as to the axiomatic formulation of probability theory. In this chapter we develop first just sufficient of elementary set theory and its notation to enable the ideas of mappings and transformations (linear, in particular) to be understood. Linear transformations are then used as a means of introducing matrices, the more formal approach to matrix algebra and matrix calculus being dealt with in the following chapters.

In the later sections of this chapter we again return to set theory, giving a brief account of set algebra together with a few examples of the types of problems in which sets are of use. However, these ideas will not be developed very far; the reader who is interested in the more advanced aspects and applications of set theory should consult some of the texts given in the list of further reading matter at the end of the book.

1.2 Sets

We must first specify what we mean by a set of elements. Any collection of objects, quantities or operators forms a set, each individual object, quantity or operator being called an element (or member) of the set. For example, we might consider a set of students, the set of all real numbers between 0 and 1, the set of electrons in an atom, or the set of operators ∂/∂x₁, ∂/∂x₂, …, ∂/∂x_n. If the set contains a finite number of elements it is said to be a finite set, otherwise it is called infinite (e.g. the set of all positive integers).

Sets will be denoted by capital letters A, B, C, …, whilst the elements of a set will be denoted by small letters a, b, … x, y, z, and sometimes by numbers 1, 2, 3, … .

A set which does not contain any elements is called the empty set (or null set) and is denoted by ø. For example, the set of all integers x in 0 < x < 1 is an empty set, since there is no integer satisfying this condition. (We r...

Table of contents

• Cover
• Title page
• Copyright
• Contents
• Preface
• 1. Sets, Mappings and Transformations
• 2. Matrix Algebra
• 3. The Inverse and Related Matrices
• 4. Systems of Linear Algebraic Equations
• 5. Eigenvalues and Eigenvectors
• 6. Diagonalisation of Matrices
• 7. Functions of Matrices
• 8. Group Theory
• Further Reading
• Answers to Problems
• Index