Let T : V V be a linear operator on an n-dimensional F-vector space V. For any n N+, we define Tn.=TTT (n times) and T0.=IV,where IV denotes the identity operator on V. Let v V be any non-zero vector.

i. Prove that there is a positive kn such that T k(v) = a0T 0(v)+a1T 1(v)+ +ak1Tk1(v) for some a0,a1,...,ak1 F.

ii. Let kv denote the smallest number satisfying part (i), i.e. kv is the smallest positive natural number n such that T kv (v) = a0T 0(v) + a1T 1(v) + + akv1T kv1(v) for some a0,a1,...,ak1 F. Prove that .= T 0(v),T 1(v),...,T kv1(v) is linearly independent.