Page 177 - A Course in Linear Algebra with Applications

6.2: Linear Transformations and Matrices 161

/ in Doo[a,b], define T(f) to be
a nf^ + a n - i / ( n _ 1 ) + • • • + a i / ' + «o/.

Then T is a linear operator on Doo[a,b], once again by ele-
mentary results from calculus. Here one can think of T as a
sort of generalized differential operator that can be applied to
functions in -Doo[a, 6].
Our next example of a linear transformation involves quo-
tient spaces, which were defined in 5.3.

Example 6.2.4
Let U be a subspace of a vector space V and define a function
T : V -» V/U by the rule T(v) = v + U. It is simple to verify
that T is a linear transformation: indeed,

T(vx + v 2 ) = (vi + v 2 ) + U = (vi + U) + (v 2 + *7)
= T( V l )+T(v 2 )

by definition of the sum of two vectors in a quotient space. In
a similar way one can show that T(cv) = c(T(v)).
The function just defined is often called the canonical
linear transformation associated with the subspace U.
Finally, we record two very simple examples of linear
transformations.

Example 6.2.5
(a) Let V and W be two vector spaces over the same field. The
function which sends every vector in V to the zero vector of W
is a linear transformation called the zero linear transformation
from V to W; it is written

Ov,w or simply 0.

(b) The identity function \y : V —> V is a linear operator on
V.
After these examples it is time to present some elemen-
tary properties of linear transformations.