Let $\mathbb{F}$ be a field, and $A\in M_{n\times n}(\mathbb{F})$.
Define $M,L=\mathbb F^n$ to be $\mathbb {F}[x]$-modules, s.t. for every $m\in M,l\in L$ :
$f(x)m=f(A)m$
$f(x)l=f(A^t)l$
Prove that both modules are isomorphic.
I proved that $xI-A$ and $xI-A^t$ have the same Smith Normal Form.
My question is how to prove that $M\cong L $ by that $SNF(xI-A)=SNF(xI-A^t)$?