Linear Mapping Practice Test

Let f be a mapping such that $f:R^2 \rightarrow P^2$ where is the vector space of polynomials up to the term. (ie polynomials of the form )

Let f be defined such that

Is f a homomorphism?

The previous two problems showed how the dimension of the domain and codomain can be used to predict if it is possible for the mapping to be 1-to-1 or onto. Now we'll apply that knowledge to isomorphism.

Let f be a mapping such that $f:V\rightarrow W$. Also the vector space V has dimension 4 and the vector space W has dimension 8. What property of isomorphism can f NOT satisify.