Show Linear Independence of Set with T:V->V Operator

[mivanova]

Can you help me with this, or at least give me an idea how to proceed:
Let T:V->V be a linear operator on the vector space over the field F. Let v is in V and let m be a positive integer for which v is not equal to 0, T(v) is not equal to 0, ...,T^(m-1)(v) is not equal to 0, but T^m(v) is equal to 0. Show that {v, T(v), ... , T^(m-1)(v)} is linearly independent set.
Thank you!

 

[adriank]

Given $$a_0 v + a_1 T(v) + \cdots + a_{m-1} T^{m-1}(v) = 0$$, apply T to both sides repeatedly and see what you get.