Page 203 - A Course in Linear Algebra with Applications

6.3: Kernel, Image and Isomorphism 187

It is equally easy to show that 7\ + T2(cv) = c(2\ + T 2 (v)).
Thus the set of all linear operators on V, which will henceforth
be written
L(V),

admits natural operations of addition and scalar multiplica-
tion.
Now there is a further natural operation that can be per-
formed on elements of L(V), namely functional composition
as defined in 6.1. Thus, if T\ and T 2 are linear operators on
V', then the composite T\ oT 2 , which will in future be written

TiT 2,

is defined by the rule

TiT 2 (v)=Ti(T 2 (v)).

One has of course to check that TiT 2 is actually a linear trans-
formation, but again this is quite routine. So one can also form
products in the set L(V).
To illustrate these definitions, we consider an explicit ex-
ample where sums, scalar multiples and products can be com-
puted.

Example 6.3.4
Let Ti and T 2 be the linear operators on -Doo[a, b] defined by
Ti(/) = f - f and T 2 (/) = xf" - 2/'. The linear opera-
tors Ti + T 2, cT\ and TiT 2 may be found directly from the
definitions as follows:

x
/
Ti + r (/) = r (/) + T (/) = ' - / + /"-2/'
2
2
1
= -f-f' + xf".
Also
cT (f) = cf'-cf
1