Page 610 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

It follows that

Expressing this result in horizontal notation yields

The following theorem shows that a composition of one-to-one linear transformations is one-to-one, and it relates the inverse
of the composition to the inverses of the individual linear transformations.

THEOREM 8.3.3

If and are one-to-one linear transformations, then
(a) is one-to-one.
(b)
.

Proof (a) We want to show that maps distinct vectors in U into distinct vectors in W. But if and are distinct

vectors in U, then and are distinct vectors in V since is one-to-one. This and the fact that is one-to-one

imply that

are also distinct vectors. But these expressions can also be written as
so maps and into distinct vectors in W.

Proof (b) We want to show that

for every vector in the range of . For this purpose, let

(3)

so our goal is to show that