Isomorphic spaces and bases

Question

I know P2 and R3 are isomorphic since dim(P2) = dim(R3) = 3. Does this mean that if B is a basis for R3, it is also a basis for P2? Is there a theorem that states this?

I'm coming from https://www.youtube.com/watch?v=GdXCBG3dRTo, at 2:00 in.

Answer 1

No. No theorem can possibly say that, since $\mathbb R^3\cap P_2=\emptyset$. In particular, no non-empty subset of $\mathbb R^3$ can possibly also be a subset of $P_2$.

Asserting that $\mathbb R^3$ and $P_2$ are isomorphic simply means that there is a linear isomorphism from $\mathbb R^3$ onto $P_2$.