# Math Help - linear operator

1. ## linear operator

I posted this before but I did typo. Now should ne correct but I still can't make it.
suppose $\sum_{j=0}^{m-1}c_jT^j(v)=0,$ and $c_j \neq 0,$ for some $0 \leq j \leq m-1.$ choose $0 \leq k \leq m-1$ to be the smallest such that $c_k \neq 0.$ then $0=T^{m-1-k} \left(\sum_{j=0}^{m-1}c_jT^j(v) \right)=c_kT^{m-1}(v),$ which gives us:
$T^{m-1}(v)=0$ because $c_k \neq 0.$ that contradicts our assumption that $T^{m-1}(v) \neq 0. \ \ \ \Box$