i am considering a number of pairs of vector spaces like the following.
$F^3$ and $P_3(F)$
i think this pair of vector spaces is not isomorphic because $dim(F^3) = 3 \neq dim(P_3(F)) = 4$
i also think that in general whenever two different vector spaces have different dimension, they cannot be isomorphic.
am i correct? thank you for your time.