Introduction to Linear Algebra

A Primer for Social Scientists

Gordon Mills

• Publisher: Routledge
• Year: 2017
• ISBN:
  9781351511568
• Edition: 1
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

This is the first book on linear algebra written specifically for social scientists. It deals only with those aspects of the subject applicable in the social sciences and provides a thorough understanding of linear algebra for those who wish to use it as a tool in the design, execution, and interpretation of research. Linear mathematical models play an important role in all of the social sciences. This book provides a step-by-step introduction to those parts of linear algebra which are useful in such model building. It illustrates some of the applications of linear analysis and helps the reader learn how to convert his formulation of a social science problem into algebraic terms. The author covers matrix algebra, computational methods, linear models involving discrete variables, and clear, complete explanations of necessary mathematical concepts. Prior knowledge of calculus is not required since no use is made of calculus or of complex numbers. A novel feature of the mathematical content of the book is the treatment of models expressed in terms of variables which must be whole numbers (integers). The book is distinguished by a step-by-step exposition that allows the reader to grasp quickly and fully the principles of linear algebra. All of the examples used to illustrate the text are drawn from the social sciences, enabling the reader to relate the subject to concrete problems in his field. Exercises are included as a necessary part of the text to develop points not covered in the text and to provide practice in the algebraic formulation of applied problems. An appendix gives solutions (or hints) for selected exercises.

CHAPTER 1

Introduction

1.1 Linear and non-linear systems

The preface to this book contains some important remarks on how to use the book; if you have not already studied these remarks, you should do so now. The book deals with the algebra of linear systems ; it also illustrates the use of these concepts and techniques in economics and in other social sciences. The algebra to be studied is called linear⁵ because graphs and sketches giving geometrical interpretation or illustration are composed of straight lines. In algebraic language, attention is confined to expressions in which each term is linear, i.e. each variable is raised to the power one, and any one term in the expression contains only one variable (unless it happens to be a constant term which, of course, does not contain any variable at all). The following example of a linear expression has four terms and involves three variables;

3+2x1+x2+4x3

Thus the book is not concerned with expressions having non-linear terms such as

√x3,x12,x1−1,7x1x2,log⁡x1or2x2ex3

The mathematical analysis of linear expressions is old-established. In the great bulk of it, it is supposed that the variables x₁,x₂,x₃ etc. are continuous; that is to say, each variable may take on any value, including not only whole numbers (or integers, as they are often called) such as 2, 23, 175 and so forth, but also fractions such as 12 and decimal expressions such as 2⋅353˙. In intuitive language, a variable may take on any value within a specified range or interval. It is usually supposed that the relevant range is all positive or negative numbers, however large or small. Similarly, in this book, except where otherwise indicated, the mathematical analysis is developed on the assumption that we are dealing with continuous variables which can take on positive or negative values. Of course, in applying the analysis to social science contexts, negative values for variables often have to be ruled out as inadmissible; there is no sense in predicting that next year’s rainfall will be − 37.3 cm. or that national income will be − £247.80 per head. Thus in considering the solution of a set of linear equations (for example) the applied context may make it necessary to ask not merely ‘Is there a solution?’ but rather Ts there a non-negative solution?’ On some occasions, the context may require the variables to take on integer values only. Often the usual mathematical analysis in terms of continuous variables may give an adequate approximation; at the end of the calculation, any non-integer values can be rounded to the nearest integer. But sometimes it is necessary to recognize the integer requirement (i.e. to treat the variables as discrete rather than continuous variables) right at the outset, in which case a very different type of mathematical analysis is required. On the whole, the analysis of linear systems involving discrete variables is not as well developed as that for continuous variables; but a short introduction to the subject is included in Chapter 7.

Linear systems are very useful in analysing many problems in economics and other social sciences, and in carrying out statistical estimation. In other words, many phenomena can be represented (at least to a good approximation) by linear functions. This is very fortunate because it turns out that both theoretical analysis and numerical calculation are much simpler for linear than for nonlinear models. But in any given problem context, we must not automatically assume that a linear mod...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Contents
• Preface
• 1 Introduction
• 2 Vectors
• 3 Matrices
• 4 Elementary operations and the rank of a matrix
• 5 The inverse of a square matrix
• 6 The solution of simultaneous linear equations
• 7 Integer variables and other topics
• Appendix A: Determinants
• Appendix B: Solutions and hints for some of the exercises
• Appendix C: Suggestions for further reading
• Index