Q:

Help with Linear Transformations Proof

So here is the (probably simple) proof:
Assume that T:V $\rightarrow$ V is a linear operator on the vector space V, that v $\neq$0 and that the set {v, T(v), T$^2$(v),...,T$^{k-1}$(v)} is independent but the set {v, T(v), T$^2$(v),...,T$^{k-1}$(v), T$^k$(v)} is dependent. Let U=span(v, T(v), T$^2$(v),...,T$^{k-1}$(v)). Show that for all w $\in$ U, T(w) $\in$ U.
I'm not really sure how I should approach this proof. My first thought is to extend the basis of V with another basis, but the proof isn't coming to me.  I would greatly appreciate your help. Thank you!

A:

First of all, when you're told that $\left\{v, T(v), T^{2}(v),...,T^{k−1}(v)\right\}$ is linearly independent, but that $\left\{v, T(v), T^{2}(v),...,T^{k}(v)\right\}$ is not, what does that tell you about what $T^{k}\left(v\right)$ looks like? Think about the definition of linear independence.
Now, think about what $w\in U$ looks like. We know that $w$ has to be something of the form $w=c_{0}v+c_{1}T\left(v\right)+\cdots+c_{k-1}T^{k-1}\left(v\right)$, where the $c_{i}$ are scalars in your field. Then, what does $T\left(w\right)$ look like? Think about the linearity of $T$.
When you look at your last term, you'll notice that you already have something to put there in terms of the basis vectors of $U$. So, you have shown that $T\left(w\right)\in U$, as required.

