1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: General vector space question

  1. Mar 9, 2010 #1
    1. The problem statement, all variables and given/known data
    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.

    3. The attempt at a solution
    Well, I got part 1.
    As for part 2... I have no clue. Any ideas?
  2. jcsd
  3. Mar 10, 2010 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    OK, I will get you started on (a.) So we are supposing [itex]\phi[/itex] is 1-1. So for each a in A there is a unique b in B such that [itex]b = \phi(a)[/itex] and [itex]a =\phi^{-1}(b)[/itex]. 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 [itex]f\circ\phi = g[/itex].

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

    Also note, if b is exterior to [itex]\phi(A)[/itex], it doesn't matter what you define f(b).
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook