Page 44 - A Course in Linear Algebra with Applications

28 Chapter One: Matrix Algebra

matrices over R. The validity of the ring axioms follows from
Theorem 1.2.1. An obviously important example of a ring is
M n ( R ) . Later we shall discover other places in linear algebra
where rings occur naturally.
Finally, we mention another important algebraic struc-
t
ture hat appears naturally in linear algebra, a group. Con-
sider the set of all invertible n x n matrices over a ring with
identity R; denote this by

GL n(R).

This is a set equipped with a rule of multiplication; for if A
and B are two invertible n x n matrices over R, then AB
is also invertible and so belongs to GL n(R), as the proof of
Theorem 1.2.3 shows. In addition, each element of this set
has an inverse which is also in the set. Of course the identity
nxn matrix belongs to GL n{R), and multiplication obeys the
associative law.
All of this means hat GL n(R) is a group. The formal
t
definition is as follows. A group is a set G with a rule of
multiplication; thus if g\ and gi are elements of G, there is
a unique product gig2 in G. The following axioms must be
satisfied:

(a) (0102)03 = (0102)03, {associative law):
(b) there is an identity element 1Q with the property
1 G 0 = 0 = 0 1 G :
(c) each element g of G has an inverse element 0 _ 1 in G
1
such hat gg~ l = 1Q = 9' g-
t
These statements must hold for all elements g, gi, 02, 03 of G.
Thus the set GL n (R) of all invertible matrices over R, a
ring with identity, is a group; this important group is known
as the general linear group of degree n over R. Groups oc-
cur in many areas of science, particularly in situations where
symmetry is important.