Thread: 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!

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 and for some choose to be the smallest such that then which gives us:

because that contradicts our assumption that