First, do you mean to define ?
I think that your first step is to determine the kernel of . If , what can be said about and ?
You have that x,y)\mapsto \left(x+S,y+T)" alt="F:V\boxplus U\to (V/S)\boxplus (U/T)x,y)\mapsto \left(x+S,y+T)" /> (I use to differentiate between external and internal direct sum), this is evidently a surjective linear homomorphism. Then, we are trying to look for what values is ...equals what? We're trying to look for the origin in the vector spaces and but these are precisely and respectively. So, for what is and for what is ? It is fairly clear that these statements are true if and only if and . Thus, a quick proof shows that and so by the F.I.T.
Come on friend. Review your definitions. You're trying to show that which is equivalent to showing . I give the solution below "hidden". Try your damnedest, and only look at it as a last result.
[spoil]Clearly if then since since and similarly . Conversely, suppose that $latex F(x,y)=(S,T)[/tex] then . But, note that since and so . Similarly and so . Combining these we get [/spoil]
EDIT: I can't figure out how to hide it...so work on the honor code haha