Originally Posted by

**SlipEternal** I will give you a bigger hint. The standard basis for $\displaystyle V$ is $\displaystyle \{1,x,x^2,\ldots, x^n\}$. Through linear combinations of those elements, you can get any polynomial of degree less than or equal to n. The dimension of $\displaystyle V$ is the number of elements in any of its bases (obviously $\displaystyle V$ is an $\displaystyle n+1$-dimensional vector space). There is only one $\displaystyle n+1$-dimensional subspace of $\displaystyle V$. How many $\displaystyle n$-dimensional subspaces are there? How many of them have polynomials with maximum degree $\displaystyle n-1$? If that claim is true, it tells you the dimension of $\displaystyle W$. It also gives you a very good idea of which subspaces of $\displaystyle V$ are invariant under $\displaystyle \phi$.