I've not made that much progress on this, but I'll show right now what else I've done so far. Some of it may not be correct in any case. My mind's not in the best shape right now due to sickness, so I've probably got a few things muddled.
My Prof. told me that for this, I had to show that was one-to-one, onto, and linear. Proving these is probably easier than I think it is, but I can't help but feel as if I could easily miss something if I'm not careful.
I think I have the one-to-one part (injective) down, though the onto part (surjective) is much trickier due to there being two variables. I don't remember any similar situation, so I'm kinda stuck.
Primarily, I'm having difficulty showing my proofs. Either I'm over-analyzing the proofs and I'm trying to be too complex in showing them, or I don't know what to do entirely.