Eigenvector and Its Span Let $V$ be a vector space over the field $F$ and let $T$ be a linear transformation from $V$ to $V$. Let $v\in V$ such that $v\neq 0$, let $W=span\{v\}$. Prove that if $T(W) \subset W$, then $v$ is an eigenvector for $T$. 
So, my idea for this problem was since $v\in V, v=1.v\in span\{v\}=W$. So $v\in W$.  
Then, since $W=span\{v\}$, can this be inferred that $T(v)\in T(W)$? If this is true, then I know how to do the rest, but I just want to know that if I can say that? 
Thank you for any input!

Yes, $T(v)\in T(W)$. You can convince yourself of this by looking at the definition of $T(W)$ which is $T(W):=\{T(w)\colon w\in W\}$.
