i)let v be a vector in V such that but . suppose that .

taking of both sides, every term but vanishes. since , we must have .

thus we have , and now we take of both sides to show that .

continuing in this way, we eventually find that , which shows that is linearly independent.

ii) if , then

so that

(since ,

which is certainly in , hence this span is a T-invariant subspace (if we call it W, T(W) is contained in W).