Page 204 - A Course in Linear Algebra with Applications

188 Chapter Six: Linear Transformations

and

TiT 2 (/) = T 1(T 2(f))=T 1(xf" - 2/')
= ( * / " - 2 / ' ) ' - ( * / " - 2 / ' ) ,

which reduces to TiT 2 (/) = 2/' - (x + 1)/" + xf^\ after
evaluation of the derivatives.

At this point one can sit down and check that those
properties of matrices listed in 1.2.1 which relate to sums,
scalar multiples and products are also valid for linear oper-
ators. Thus there is a similarity between the set of linear
operators L(V) and M n(F), the set o f n x n matrices over F
where n = dim(V). This similarity should come as no sur-
prise since the action of a linear operator can be represented
by multiplication by a suitable matrix.
The relation between L(V) and M n(F) can be formalized
by defining a new type of algebraic structure. This involves
the concept of a ring, which was was described in 1.3, and
that of a vector space.
An algebra A over a field F is a set which is simultane-
ously a ring with identity and a vector space over F, with
the same rule of addition and zero element, which satisfies the
additional axiom

c(xy) = (cx)y = x(cy)

for all x and y in A and all c in the field F. Notice that this
axiom holds for the vector space M n(F) because of property
(j) in 1.2.1. Hence M n{F) is an algebra over F. Now the
additional axiom is also valid in L(V), that is,

c(T lT 2) = (cT 1)T 2 = T l(cT 2).

This is true because each of the three linear operators men-
tioned sends the vector v to c(Ti(T 2 (v))). It follows that