Page 43 - A Course in Linear Algebra with Applications

1.3: Matrices over Rings and Fields 27

and products of matrices over R, and the scalar multiple of
a matrix over R by an element of R, by using exactly the
same definitions as in the case of matrices with numerical
entries. That the laws of matrix algebra listed in Theorem
1.2.1 are still valid is guaranteed by the ring axioms. Thus in
the general theory the only change is that the scalars which
appear as entries of a matrix are allowed to be elements of an
arbitrary ring with identity.
Some readers may feel uncomfortable with the notion of a
matrix over an abstract ring. However, if they wish, they may
safely assume in the sequel that the field of scalars is either
R or C. Indeed there are places where we will definitely want
to assume this. Nevertheless we wish to make the point that
much of linear algebra can be done in far greater generality
than over R and C.

Example 1.3.1

Let A = I 1 and B = I n J be matrices over the
field of two elements. Using the tables above and the rules of
matrix addition and multiplication, we find that

Algebraic structures in linear algebra
There is another reason for introducing the concept of a
ring at this stage. For rings, one of the fundamental structures
of algebra, occur naturally at various points in linear algebra.
To illustrate this, let us write

M n(R)

for the set of all n x n matrices over a fixed ring with identity
R. If the standard matrix operations of addition and multipli-
cation are used, this set becomes a ring, the ring of all n x n