I am stuck here please should be detail work!

Q1

Let V be a real vector space of dimension at least 3 and let Te EndR(V). Prove that there is a non-zero subspace W of V, W # V, such that T(W) C W.Let V be a finite dimensional vector space over a field K. Let S be a linear transformation of V into itself. Let W be an invariant subspace of V (that is, SW C W). Let m(t), mi(t), and my(t) be the minimal polynomial of S as linear transformation of V, W and V/W respectively. (a) Prove that m(t) divides mi(t) . my(t).Let V be a finite dimensional vector space over MR and T : V - V be a linear transformation such that (a) the minimal polynomial of T is irreducible and (b) there exists a vector v E V such that {T"v | i > 0} spans V. Show that V doesn't have proper T-invariant subspace.(b) Prove that if mi(t) and my(t) are relatively prime, then m(t) = mi(t) . my(t). (c) Give an example of a case in which m(t) * my(t) . ma(t)