Introduction to Matrices and Vectors

Jacob T. Schwartz

• Publisher: Dover Publications
• Year: 2012
• eBook ISBN: 9780486143705
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

Realizing that matrices can be a confusing topic for the beginner, the author of this undergraduate text has made things as clear as possible by focusing on problem solving, rather than elaborate proofs. He begins with the basics, offering students a solid foundation for the later chapters on using special matrices to solve problems.

The first three chapters present the basics of matrices, including addition, multiplication, and division, and give solid practice in the areas of matrix manipulation where the laws of algebra do not apply. In later chapters the author introduces vectors and shows how to use vectors and matrices to solve systems of linear equations. He also covers special matrices — including complex numbers, quaternion matrices, and matrices with complex entries — and transpose matrices; the trace of a matrix; the cross product of matrices; eigenvalues and eigenvectors; and infinite series of matrices. Exercises at the end of each section give students further practice in problem solving.
Prerequisites include a background in algebra, and in the later chapters, a knowledge of solid geometry. The book was designed as an introductory text for college freshmen and sophomores, but selected chapters can also be used to supplement advanced high school classes. Professionals who need a better understanding or review of the subject will also benefit from this concise guide.

Chapter 1

DEFINITION, EQUALITY, AND ADDITION OF MATRICES

1-1. Introduction

As we have done more and more sophisticated mathematics in our previous studies, we have had occasion to use more and more sophisticated kinds of “numbers.” We began with positive whole numbers, like 1, 2, 3, . . . . Then, in order to make subtractions like 3 − 7 possible, the negative whole numbers, like −1, −2, −3, . . . , had to be introduced. Next, in order to make it possible to divide any two numbers, fractions were invented:

, etc. This still did not bring us to the end of our story; for, in order that every number have a square root, a cube root, a logarithm, a sine, and so forth, it was necessary to invent still more numbers: the infinite decimals, or real numbers, like 1.4142 . . . , 3.1415928 . . . , 0.13131313 . . . . Finally, in order that even negative numbers have square roots, it was necessary to invent complex numbers, like

Whenever there seemed to be good reason to do so, we have invented new kinds of numbers. For instance, in inventing complex numbers, we began not with the numbers but with a purpose: to find a system of numbers, including all the real numbers, in which every number had a square root. Once we have made such an invention, it should not be hard for us to realize that there is no reason to stop inventing; that is, there is no reason why other kinds of numbers should not be useful for other purposes. There is no reason why we should not hope to invent many different kinds of new numbers.

Of course, as an inventor, it is easy to invent things that do not work, but harder to invent things that do work; easy to invent things that are useless, but hard to invent things that are useful. The same is true about the invention of new kinds of numbers. The hard thing is to invent useful kinds of numbers, and kinds of numbers that work. It is easier to make inventions than to make successful inventions. Nevertheless, a large variety of more or less successful new kinds of numbers have been invented by mathematicians. In this course you are going to study one of the most successful of these new kinds of numbers: the matrices.

Before you are told what matrices are, it is well to emphasize their importance. Matrices are useful in almost every branch of science and engineering. A great number of the computations made on the giant “electronic brains” are computations with matrices. Many problems in statistics are expressed in terms of matrices. Matrices come up in the mathematical problems of economics. Matrices are extremely important in the study of atomic physics; indeed, atomic physicists express almost all their problems in terms of matrices. It would not be an exaggeration to say that the algebra of matrices is the language of atomic physics. Many other kinds of algebra, like complex-number algebra (and like vector algebra, which...

Table of contents

• –DOVER BOOKS ON MATHEMATICS
• Title Page
• Copyright Page
• PREFACE
• Table of Contents
• Chapter 1 - DEFINITION, EQUALITY, AND ADDITION OF MATRICES
• Chapter 2 - MULTIPLICATION OF MATRICES
• Chapter 3 - DIVISION OF MATRICES
• Chapter 4 - VECTORS AND LINEAR EQUATIONS
• Chapter 5 - SPECIAL MATRICES OF PARTICULAR INTEREST
• Chapter 6 - MORE ALGEBRA OF MATRICES AND VECTORS
• Chapter 7 - EIGENVALUES AND EIGENVECTORS
• Chapter 8 - INFINITE SERIES OF MATRICES
• INDEX
• A CATALOG OF SELECTED DOVER BOOKS IN ALL FIELDS OF INTEREST