Hi everyone,

could you please help me with the following?

Let

and let

be the subspace of

of dimension n+1 consisting of homogeneous polynomials of degree n, that is, the subspace spanned by

. Let P and Q be linear transformations on

defined for f in

by

and

Find the minimal polynomial of (PQ-QP).

Now, if my calculations are correct, for an arbitrary basis vector of the form

,

.

Now, the minimal polynomial (denote m) is the unique monic polynomial of least degree s.t. m(PQ-QP)=0 (0 transformation). So in particular, I want

, and moreover any linear combination of

to be equal to zero. How am I supposed to achieve this?

Thanks a lot for all the help.