Matrices and Transformations

Anthony J. Pettofrezzo

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486636344
• eBook ISBN: 9780486151809
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

This book presents an elementary and concrete approach to linear algebra that is both useful and essential for the beginning student and teacher of mathematics. Here are the fundamental concepts of matrix algebra, first in an intuitive framework and then in a more formal manner. A Variety of interpretations and applications of the elements and operations considered are included. In particular, the use of matrices in the study of transformations of the plane is stressed. The purpose of this book is to familiarize the reader with the role of matrices in abstract algebraic systems, and to illustrate its effective use as a mathematical tool in geometry.
The first two chapters cover the basic concepts of matrix algebra that are important in the study of physics, statistics, economics, engineering, and mathematics. Matrices are considered as elements of an algebra. The concept of a linear transformation of the plane and the use of matrices in discussing such transformations are illustrated in Chapter #. Some aspects of the algebra of transformations and its relation to the algebra of matrices are included here. The last chapter on eigenvalues and eigenvectors contains material usually not found in an introductory treatment of matrix algebra, including an application of the properties of eigenvalues and eigenvectors to the study of the conics. Considerable attention has been paid throughout to the formulation of precise definitions and statements of theorems. The proofs of most of the theorems are included in detail in this book. Matrices and Transformations assumes only that the reader has some understanding of the basic fundamentals of vector algebra. Pettofrezzo gives numerous illustrative examples, practical applications, and intuitive analogies. There are many instructive exercises with answers to the odd-numbered questions at the back. The exercises range from routine computations to proofs of theorems that extend the theory of the subject. Originally written for a series concerned with the mathematical training of teachers, and tested with hundreds of college students, this book can be used as a class or supplementary text for enrichments programs at the high school level, a one-semester college course, individual study, or for in-service programs.

chapter 1

Matrices

1-1 Definitions and Elementary Properties

In many branches of the physical, biological, and social sciences it is necessary for scientists to express and use a set of numbers in a rectangular array. Indeed, in many everyday activities it is convenient, if not necessary, to use sets of numbers arranged in rows and columns for keeping records, for purposes of comparison, and for a variety of other reasons.

Consider a company that manufactures three models of typewriters: an electric model, a standard model, and a portable model. If the company wishes to compare the units of raw material and labor involved in one month’s production of each of these models, an array may be used to present the data:

The units used are not intended to be realistic but merely to illustrate an oversimplified application of an array of real numbers. Units of material for the three models comprise the first row of the array, units of labor the second row, and units of production for each model (material and labor) the columns of the array. If the pattern in which the units are to be recorded is clearly defined in advance, this rectangular array may be presented simply as:

(1-1)

A second example of the use of rectangular arrays of real numbers is one that might be used by a basketball coach who wishes to keep a record of the scoring performances of three of his players. Consider the following array:

or simply

(1-2)

Rectangular arrays of elements a_ij such as

(1-3)

are called matrices (singular: matrix). Each element a_ij has two indices: the row index, i, and column index, j. The elements a_i1, a_i2, . . . , a_in are the elements of the ith row, and the elements a_1i, a_2j, . . . , a_mj are the elements of the jth column. The element a_ij is the element contained simultaneously in the it...

Table of contents

• DOVER BOOKS ON MATHEMATICS
• Title Page
• Copyright Page
• Dedication
• Preface
• Table of Contents
• chapter 1 - Matrices
• chapter 2 - Inverses and Systems of Matrices
• chapter 3 - Transformations of the Plane
• chapter 4 - Eigenvalues and Eigenvectors
• Bibliography
• Answers to Odd-Numbered Exercises
• Index