Here is the relation I have seen a lot in the books, but I am not sure why it is always true:
If $T \subseteq E(M)$ then $$M \setminus T = (M^* / T)^* \quad\quad (*)$$
I know that contraction is defined as $$M/T = (M^* \setminus T)^*$$ I tried taking the dual of this definition but I do not think that this is the correct way of thinking. I also know the following proposition:
$M \setminus T = M/T$ iff $r(T) + r(E - T) = r(M)$ but I do not know hoe can this help.
Is there a proof for why is this relation $(*)$ true?
Take your definition of contraction and apply it to the matroid $M^*$ to get $M^*/T=(M^{**}\setminus T)^*$. Therefore, in your desired equation $(*)$, the right side is $(M^{**}\setminus T)^{**}$. Since every matroid is equal to its double-dual, this reduces to $M\setminus T$, i.e., the left side of (*).