Suppose V is a vector space of dimension n, that T is a linear transformation on V and that there exists $\displaystyle v \in V$ such that $\displaystyle \{v, Tv, ..., T^{n-1}(v)\}$ is a basis of V.

Then clearly $\displaystyle T^{n}(v)$ can be written as a linear combination of the basis elements, but can we find this linear combination explicitly? I suspect not...