Eigenvalue of vector space of polynomials

[specialnlovin]

Let V=C[x]₁₀ be the fector space of polynomials over C of degree less than 10 and let D:V\rightarrowV be the linear map defined by D(f)=f' where f' denotes the derivatige. Show that D¹¹=0 and deduce that 0 is the only eigenvalue of D. find a basis for the generalized eigenspaces V₁(0), V₂(0), V₃(0)
I get that D¹¹=0 because a polynomial with degree 10 has an 11th derivative equal to zero since. However I am not sure how exactly to write that in a proof, also how to then deduce that the eigenvalue must equal zero.
I also don't know exactly how to start the problem of finding a basis for the eigenspaces.

 

[Dick]

If f is in V, so it's a polynomial of degree n<=10, what's the degree of D(f)? What does that tell you about the possibility of a nonzero eigenvalue?