MATH 220 Final, Spring 2005

| LIBRARY | Math 220 OF KeIRET Final Exam June 3, 2005 Time: 120 minutes. Throughout T : Y > Vis a linear transformation of vector spaces over a field K. l. (a) Define what is meant by the statement, ‘V=V,0V.®@...@8V,is a direct sum of the subspaces V;, V2, ..., V,”. (b) Let E,, ..., Ex, E; : ¥ > VY be linear transformations such that ()E,+...+E2 + E, = idy, and (ti) ExE; = 0 if i +). Prove that V=imE,;® imE,®...@imE, 2. IF V=V, @ Va, By = {y,..., v-} is a basis of Vi and By = {wi,..., Ws} a basis of V2, prove that B = B, U B2 is a basis of V. 3. (a)LetT : V > VY be a nilpotent linear transformation.. Define what is meant by the statement, “A subspace V; of a vector space V is T-cyclic of order m”. (b) If the linear transformation T : V > V is nilpotent with index of nilpotency equal to m, prove that V has a T-cyclic subspace V; of order m. Prove also that T(V,) is T-cyclic of order m —- 1. 4. Let T: V + V bea nilpotent linear transformation of index of nilpotency nj. You already know that Vis decomposable as a direct sum V=V, 8W.®8...@V, of subspaces V;, 42, .. ., V,, where every V; is T-cyclic of order n,; and nm 22 >...2n,. Prove that the sequence of natural numbers m, .. ., m, are invariants of T. (That is, prove the following: If V= U; 8 U2@6...@® U,isa direct sum decomposition of VY into T-cyclic subspaces U), ..., U, of respective orders m| 2m2,2...2 ms, thens =r andn;=m, for alli=1,2,...,7). 5. Let m(x) = (« - &)% (x - &2)7. . . (« — &)* be the minimal polynomial of a linear transformation T: ¥> V. (a) State the triangular theorem associated with m(x). (b) State the upper triangular theorem associated with m(x). (c) Describe the Jordan blocks associated with the factor (x — €))“ of m(x) and prove the truth of your description.