Diagonalizable Proof Homework: True or False?

[metder]

Homework Statement

Let V be the space of n X n matrices over F. Let A be a fixed n X n matrix
over F. Let T and U be the linear operators on V defined by
T(B) = AB
U(B) = AB - BA.
1. True or false? If A is diagonalizable (over F), then T is diagonalizable.
2. True or false? If A is diagonalizable, then U is diagonalizable
Thanks for the help.

The Attempt at a Solution

I'm guessing that 1 is true and 2 is false. I'm not sure, since these are linear operators rather than simple matrices.

 

[HallsofIvy]

Consider, first, the case in which A is diagonal. What do T and U do to the "basis" matrices?