T-invariant basis problem

Hello all, I am having trouble showing linear independence at the indicated step.

The setup:

Let $\displaystyle v\in V$ be a fixed vector in a finite-dimensional vector space $\displaystyle V$. Let $\displaystyle W=\{v,Tv,T^2v,...\}$.

Prove $\displaystyle W$ is $\displaystyle T$-invariant. (Easy)

Suppose that $\displaystyle \dim(W)=k$ and show that $\displaystyle B=\{v,Tv,T^2v,\ldots,T^{k-1}v\}$ is a basis for $\displaystyle W$.

It seems obvious that checking linearly independent is easier than span, but I am getting stuck trying to get all of the coefficients to be zero. Thanks to any and all help ahead of time!