Is Composition by a Mapping a Linear Isomorphism in Vector Spaces?

[fluxions]

Homework Statement

Sorry for the vague title!

Let R denote the set of real numbers, and F(S,R) denote the set of all functions from a set S to R.

Part 1: Let \phi be any mapping from a set A to a set B. Show that composition by \phi is a linear mapping from F(B,R) to F(A,R). That is, show that T : F(B,R) \rightarrow F(A,R) : f \mapsto f \circ \phi is linear.

Part 2: In the situation given in part 1, show that T is an isomorphism if \phi is bijective by show that:
(a) \phi injective implies T surjective;
(b) \phi surjective implies T injective.

The Attempt at a Solution

Well, I got part 1.
As for part 2... I have no clue. Any ideas?

 

[LCKurtz]

OK, I will get you started on (a.) So we are supposing \phi is 1-1. So for each a in A there is a unique b in B such that b = \phi(a) and a =\phi^{-1}(b). Now suppose you are given g in F(A,R). To show T is onto, you need to build an f in F(B,R) such that T(f) = g, that is f\circ\phi = g.

For b \in \phi(A), try defining f(b) = g(\phi^{-1}(b)). Now see if you can show that T(f) = g, which is the same as showing f\circ\phi = g.

Also note, if b is exterior to \phi(A), it doesn't matter what you define f(b).