This problem comes from our Final Review. It states:

Suppose V is a vector space over a field F, and T: V-->V is a linear map. Suppose also that, for a vector v in V, that T^(k+1) * v = 0 for some positive integer k. If (T^k) is not equal to zero, show:

i.) {v, Tv, (T^2)v .. (T^k)v} is linearly independent;

ii.) The span of {v, Tv, (T^2)v .. (T^k)v} is a T-invariant subspace of V.

Our professor hasn't been using a textbook and google doesn't understand maths .