1. ## Matrix mapping

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

(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

2. ## Re: Matrix mapping

Originally Posted by bejscs
(i)Show that the map sending A to TA determines a linear isomorphism f: Mnxn(F) -> Mnxn(F)*
For every $A\in \mathcal{M}_{n\times n}(F)$ we have the map $T_A:\mathcal{M}_{n\times n}(F)\to F$ defined by $T_A(B)=\textrm{tr}(AB)$ . First of all you have to prove that $T_A\in \mathcal{M}_{n\times n}(F)^*$ that is, $T_A$ is a linear form. Write $T_A(\lambda_1B_1+\lambda_2B_2)=\ldots$ and apply well known trace properties. Show some work for (i) and I'll give you hints for (ii) if you need them.