the proof of this is in the book and it's quite easy! see Theorem 3, page 94.

no. it's easier to work with matrices rather than transformations here. they're the same anyway: (here has 1 as its (i,j)-entry and 0 everywhere else.)

Problem (2)

If A,B,C are linear transformations on a finite dimensional vector space, then does (AB - BA)^2 commutes with C always?

in this case, the claim is true: let clearly thus by Cayley Hamilton: for some scalar now it's obvious that for all

What happens when the dimension of the vector space is 2?