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## 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 [tex]\phi[/tex] be any mapping from a set A to a set B. Show that composition by [tex]\phi[/tex] is a linear mapping from F(B,R) to F(A,R). That is, show that [tex] T : F(B,R) \rightarrow F(A,R) : f \mapsto f \circ \phi[/tex] is linear.

Part 2: In the situation given in part 1, show that T is an isomorphism if [tex]\phi[/tex] is bijective by show that:

(a) [tex]\phi[/tex] injective implies T surjective;

(b) [tex]\phi[/tex] surjective implies T injective.

## The Attempt at a Solution

Well, I got part 1.

As for part 2... I have no clue. Any ideas?