i can solve the problem provided that the following is true (so you need to tell me if such a thing basically exists in your textbook or not):

Theorem (i just made it up!): suppose is a finite dimensional vector space over and is a subspace of then for any linear map there exists

and a linear map such that and for all

the reason that i think there must be such a theorem in linear algebra is that we have the same situation in ring theory, i.e. if is self-injective and are two ideals

of then