I will give you a bigger hint. The standard basis for

is

. Through linear combinations of those elements, you can get any polynomial of degree less than or equal to n. The dimension of

is the number of elements in any of its bases (obviously

is an

-dimensional vector space). There is only one

-dimensional subspace of

. How many

-dimensional subspaces are there? How many of them have polynomials with maximum degree

? If that claim is true, it tells you the dimension of

. It also gives you a very good idea of which subspaces of

are invariant under

.