Find the dimension of a vector subspace I'm doing a problem on finding the dimension of a linear subspace, more specifically

if $\:$ {$f \in \mathcal P_n(\mathbf F): f(1)=0, f'(2)=0$} is a subspace of $P_n$, what is this dimension of this subspace? Here $\mathcal P_n(\mathbf F)$ denotes a vector space of Polynomials of degree $n$ over the real number field.

At first glance, I thought the dimension is infinity, but I think perhaps since the degree is restricted, the dimension should be finite. Yet I find it hard to specify the number of dimensions. Being a beginner of linear algebra, I would like to hear some detailed explanation on how to solve this type of problems.
Thanks in advance!

Hint
For any linear function $T:V\to W$ we have that $\dim\ker T+\dim\text{Im} T=\dim V$.
Now consider $T(f)=(f(1),f'(2))$.
