Okay, I see. You're right, I was using a lot of my intuition to prove the axioms (kinda funny how you knew my exact thought process). I guess I need to revisit the vector space chapter in the text. I'm using the anton 10th edition linear algebra book which I find not to be very concise. I need to see problems worked out and explained, perhaps I should look into finding a solutions manual.
I started problem 2 but I am not sure how to proceed. I tried writing everything down into words so it'll make more sense for me but I am still kind of confused as to how to prove W is not a subspace of ℝ^2. Perhaps you can? Btw, thanks for taking all this time in helping me.