Page 9 - Matrix Analysis & Applied Linear Algebra

CHAPTER 1

Linear

Equations

1.1 INTRODUCTION

A fundamental problem that surfaces in all mathematical sciences is that of
analyzing and solving m algebraic equations in n unknowns. The study of a
system of simultaneous linear equations is in a natural and indivisible alliance
with the study of the rectangular array of numbers defined by the coefficients of
the equations. This link seems to have been made at the outset.
The earliest recorded analysis of simultaneous equations is found in the
ancient Chinese book Chiu-chang Suan-shu (Nine Chapters on Arithmetic), es-
timated to have been written some time around 200 B.C. In the beginning of
Chapter VIII, there appears a problem of the following form.
Three sheafs of a good crop, two sheafs of a mediocre crop, and
one sheaf of a bad crop are sold for 39 dou. Two sheafs of
good, three mediocre, and one bad are sold for 34 dou; and one
good, two mediocre, and three bad are sold for 26 dou. What is
the price received for each sheaf of a good crop, each sheaf of a
mediocre crop, and each sheaf of a bad crop?
Today, this problem would be formulated as three equations in three un-
knowns by writing
3x +2y + z =39,
2x +3y + z =34,
x +2y +3z =26,
where x, y, and z represent the price for one sheaf of a good, mediocre, and
bad crop, respectively. The Chinese saw right to the heart of the matter. They
placed the coefficients (represented by colored bamboo rods) of this system in