Hi looking over a few past papers and I've got stuck.

Any answers greatly appreciated

(i)Show that the map sending A to TA determines a linear isomorphism

f: Mnxn(F) -> Mnxn(F)*

(ii) Let E = {TA|A belogning to Mnxn(F) and A' = A} is a subset of Mnxn(F). Compute sol(E) which is a subset of Mnxn(F)

[You may assume standard properties of solution spaces, such as the formula

relating dim solE and dimE. You may also assume 1 + 1 =/= 0 in F.]

where TA(B) is trace(AB), Mnxn is an nxn matrix, F is the field and A' is the transpose of A

Thanks