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!

Intuitively, dimension is the number of degrees of freedom.  The elements of $\mathcal{P}_n(\mathbb R)$ are polynomials of degree $n$ (more precisely, at most $n$), so they look like
$$ a_0 + a_1 x + a_2 x^2 + \dotsb + a_{n-1} x^{n-1} + a_n x^n $$
To specify such a polynomial, you have to specify $n+1$ numbers, the coefficients $a_0,a_1,\dotsc,a_n$.  So there are $n+1$ degrees of freedom in this "space" of polynomials.
To prove that formally, you'd want to think of polynomials $a_0+\dotsb+a_nx^n$ as being linear combinations of the polynomials $1,x,x^2,\dotsc,x^n$, and show that these latter polynomials form a basis.  This is done in chapter 2 of Axler.
Again intuitively, a constraint that specifies a single number reduces the number of degrees of freedom by 1.  Thus imposing the constraint that we will only work with polynomials $f(x)$ satisfying $f(1)=0$ should, we expect, reduce the dimension from $n+1$ to $n$.
The formal version of this is the rank-nullity theorem (Axler's theorem 3.4), which is why everybody's giving answers involving it.  I see Axler doesn't do that until chapter 3, though.
So I think the only thing you can do at this point is to produce an explicit basis for the subspace in question.  Exercise 8 in chapter 2 is similar; have you tried that?  (And for playing with polynomials, exercises 9 and 12 in the same chapter are good.)
(I have the 2nd edition of Axler's text; hopefully it matches yours.)
