Let be an -dimensional vector space over a field with basis , let be a linear transformation on with matrix , and make into an -module in the usual way. That is, we have for each and . Let be the free module of rank over and let be a basis. Let be the (surjective) -module homomorphism given by defining and extending linearly.
As demonstrated in the series of exercises beginning here, once we have a generating set for , we can compute the invariant factors of . In the next few exercises, we will find such a generating set.
Show that the elements are in , where is the Kronecker delta.
Note that as desired.