Automorphism group on a matroid

Question

I'm trying to prove that the automorphism group on a matroid is (set-theoritically) equal to the automorphism group on its dual matroid.

that is, $\ Aut(M) = Aut(M^*)$

where the automorphism group is defined by

$\ Aut(M)$ = {$f $: $E \to E$ bijective | $I \in \mathcal I$ if and only if $f(I) \in \mathcal I$}

($ E $ and $ \mathcal I $ are the ground set and the (collection of) independent sets in matroid $ M $, respectively.)

So I want to show $ Aut(M) \subset Aut(M^*)$, but I stuck here.

how can I prove that for $ f \in Aut(M), I \in \mathcal I^*$ if and only if $ f(I) \in \mathcal I^*$?

Answer 1

Hint: it is easier to prove this using bases. If $f$ is an automorphism of $M$ and $B$ is a basis of $M^*$, then does $f$ fix $B$?