Finding minimal polynomial
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
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.