Introduction to Linear Algebra, 2nd edition

Thomas A Whitelaw

• Publisher: Routledge
• Year: 2019
• ISBN:
  9781351436847
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

This popular textbook was thoughtfully and specifically tailored to introducing undergraduate students to linear algebra. The second edition has been carefully revised to improve upon its already successful format and approach. In particular, the author added a chapter on quadratic forms, making this one of the most comprehensive introductory te

CHAPTER ONE

A SYSTEM OF VECTORS

1. Introduction

This textbook provides an introduction to Linear Algebra—a vast subject in its own right, and one which is useful in many parts of pure and applied mathematics. Many of the major problems discussed in linear algebra arise from systems of simultaneous linear equations and the rectangular arrays (matrices, as we call them) of coefficients occurring in such systems. It is also true that several ideas of importance in linear algebra can be traced to geometrical sources.

The contents of this first chapter are largely geometrical. The chapter uses geometrical notions to introduce a system of mathematical objects called vectors, and it shows that this system can be used to deal with various geometrical problems. In fact, however, much of the importance of the ideas of this chapter lies beyond geometry, in the essentially algebraic generalizations which these ideas admit. In particular, a later chapter (chapter 5) will introduce the concept of a vector space. This is a generalization of the idea of a system of vectors (as encountered in this chapter), and it is a concept which is very powerful and illuminating in the discussion of many problems, including those that arise from systems of linear equations and matrices. Some of the details and emphases of this first chapter are designed to prepare the way for the generalizations that come later.

For the time being, though, those generalizations lie over the horizon, and our first concern is with the motivation for studying vectors of the kind considered in this chapter. Such motivation is to be found in physics as well as in geometry. In physics there are many things (e.g. displacements, forces, magnetic fields) that have both a magnitude and a direction: we call such things vectorial quantities. Even those who claim no knowledge of physics can understand that a walk (in a straight line) from point A to point B belongs in this category because it has a magnitude (its length) and a direction (from A to B). Correspondingly, in a geometrical discussion, we may well be concerned not only with the distance between the points A and B but also with the direction of the step from A to B.

For these and other rea...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Chapter One A SYSTEM OF VECTORS
• Chapter Two MATRICES
• Chapter Three ELEMENTARY ROW OPERATIONS
• Chapter Four AN INTRODUCTION TO DETERMINANTS
• Chapter Five VECTOR SPACES
• Chapter Six LINEAR MAPPINGS
• Chapter Seven MATRICES FROM LINEAR MAPPINGS
• Chapter Eight EIGENVALUES, EIGENVECTORS AND DIAGONALIZATION
• Chapter Nine EUCLIDEAN SPACES
• Chapter Ten QUADRATIC FORMS
• Appendix MAPPINGS
• ANSWERS TO EXERCISES
• INDEX