# 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.
Please, help me:
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!

2. Originally Posted by maria_stoeva
I posted this before but I did typo. Now should ne correct but I still can't make it.
Please, help me:
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!
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 $T^{m-1}(v) \neq 0. \ \ \ \Box$