Let I be a vector space, and let F : V - V be linear. (a) Fix n E N and denote by F" the n-fold composition of F with itself, e.g. Fl = F, F2 = F . F, F3 = F . F . F, etc. Suppose that for some v E V, we have F"(v) # 0, but Fnti(v) = 0. Show that the set { v, F(v), ... , F"(v) } is linearly independent. (b) Assume F2 = F. Prove the existence of subspaces U, W C V such that V = U O W, F(W) = {0} , and F(u) = u for all u E U. Recall that End(V) denotes the set of all linear maps F : V - V. In the remainder of this problem, we examine the algebraic properties of this set. (c) Show that F, G E End(V) implies F . G E End(V). This implies the composition . : End(V) X End(V) - End(V) is well-defined. In fact, it can be interpreted as some kind of {\\it multiplication} of elements in End(V). (d) Show that End(V) naturally becomes a monoid. That is, the following properties hold. 1. For every F, G, H E End(V), we have (F . G) . H = F . (G . H). 2. There exists E E End(V) such that for every F E End(V) , we have F . E = E . F = F. (e) Show that, in general, given some F E End(V) there does not exist G E End(V) such that F . G = G . F = E, where E is as in (d.2)