Page 42 - A Course in Linear Algebra with Applications

26 Chapter One: Matrix Algebra

C and R,

the fields of complex numbers and real numbers respectively,
where the addition and multiplication used are those of arith-
metic. These are the examples that motivated the definition
of a field in the first place. Another example is the field of
rational numbers
Q

(Recall that a rational number is a number of the form a/b
where a and b are integers). On the other hand, the set of all
integers Z, (with the usual sum and product), is a ring with
identity, but it is not a field since 2 has no inverse in this ring.
All the examples given so far are infinite fields. But there
are also finite fields, the most familiar being the field of two
elements. This field has the two elements 0 and 1, sums and
products being calculated according to the tables

+ 0 1
0 0 1
1 1 0

and

X 0 1
0 0 0
1 0 1

respectively. For example, we read off from the tables that
1 + 1 = 0 and 1 x 1 = 1. In recent years finite fields have be-
come of importance in computer science and in coding theory.
Thus the significance of fields extends beyond the domain of
pure mathematics.
Suppose now that R is an arbitrary ring with identity.
An m x n matrix over R is a rectangular m x n array of
elements belonging to the ring R. It is possible to form sums