Let $V$ be a $n$ dimensional vector space spanned by $\{e_{i}\}_{i=1}^{n}$.
Let $T:V\to V$ be a linear operator with matrix transformation $A$. Is there any relationship between the dual operator $T^{*}:V^{*}\to V^{*}$, and the complex conjugate $A^{*}$ of $A$?