Mathematics;eigenvalues and eigenvectors

Be sure to show and explain all of your work.

1. Let T : P4(R) → P4(R), p(x) 7→ xp0
   (x) + p
   00(x), and define the bases
   B =
   
   1, x, x2
   , x3
   , x4

and D =

1, 1 + x, 1 + x
2
, 1 + x
3
, 1 + x
4

(a) Compute MB(T) and MD(T).
(b) Compute PDB.
(c) Find the eigenvalues and eigenvectors of T.

2. Suppose T : V → V satisfies T
   2 = 4T.
   (a) Define U = {v ∈ V : T(v) = 4v}, the 4-eigenspace of T. Show that V = U ⊕ ker T.
   (b) Suppose V is finite dimensional, and that r = rank(T). Show that there is a basis B of
   V such that
   MB(T) = 
   4Ir 0
   0 0
   .
   1