practice question not for marked

2. Let T be the linear operator on a nite dimensional vector space V whose Jordan canon- ical form is: 4 1 I] I] 0 I] I] 0 I] 0 I] I] 4 1 I] I] I] I] 0 I] 0 I] I] I] 4 I] 0 I] I] 0 I] I] I] I] D I] 4 1 I] I] I] I] I] I] I] D I] I] 4 I] I] I] I] I] I] I] D I] I] 0 5 1 I] I] I] I] I] D I] I] 0 I] 5 1 I] I] I] I] D I] I] I] I] I] 5 1 I] I] I] D I] I] I] I] I] I] 5 I] I] I] D I] I] I] I] I] I] I] 2 I] I] I] I] I] I] I] I] I] I] I] 2 (a) Find the characteristic polynomial of T. (in) Find the dot diagram for each eigenvalue of T, (c) For each eigenvalue )1.- nd E,' and a basis for M. For which eigenvalue Ag, if any, [1083 EA.- = K111? (d) Find dim N((T MP}, for every positive integer j and ever).r eigenvalue A of T [Use Theorem 7.9]. (e) Compute rank U and rank U2, where U denotes the restriction of T A1! = T 4! t0 Kay