Exercise 5 Let E be a K-finite dimensional vector space n. Definition 0.1 An endomorphism fEL(E) is said to be nilpotent if there exists an integer p > O such that fk is the zero endomorphism for all k 2 p. Let fel (E) be a nonzero nilpotent endomorphism. 1. Show that 0 is an eigenvalue of f. 2. We say that a basis B = {e1, e2,..., en} of E is triangular for fif f(ei) e Vector(e1,e2,..., en} for i = 1, 2,..., not show that there is a triangular basis of E for f. 3. Let B = {e1, e2, ..., en} a triangular basis for f. We set e0 = 0, we call Jordan index of B, denoted iB, the largest integer is n such that f(ej ) = ei ej-1, ej {0; 1} for j = 1,2,..., i Show that iB is definite and that iB 1. 4. In the following, for x E E: we set e(x) = inf{k E NOT* / fk(x) = OE). (a) Let B = {e1,e2, ..., en} a triangular basis for f, show that for any integer j. with 1