Let T be a linear operator on the finite-dimensional space V. Prove that there exists a vector α in V with this property: If f is a polynomial and f(T)α=0, then f(T)=0. (Such a vector α is called a separating vector for the algebra of polynomials in T.) When T has a cyclic vector, give a direct proof that any cyclic vector is a separating vector for the algebra of polynomials in T.