M is a square martix, and $φ^*$ : $V^* \to V^*$, then why $M^T=M_{φ^*_M}$? 
Let M be a matrix, transpose $M^T$ is the matrix who's i,j component is the j,i component of M.
And  let $ \phi\in Hom(V,V)$, we define $\phi^*: V^* \to V^*$ by $\phi^*(f)(v)=f(\phi(v))$ where $V^*$ is the dual space of $V$.
$M_{φ^*_M}$ is the matrix representation of $φ^*_M$

Then how to prove $M^T=M_{φ^*_M}$
 A: Fix a basis $e_1, \ldots, e_n$ of $V$, and let $e_1^*, \ldots, e_n^*$ denote the dual basis of $V^*$, defined by
$$
e_i^*(e_j) =
\begin{cases}
1 & \text{if } i = j\\
0 & \text{otherwise.}
\end{cases}
$$
Let $A$ be the matrix of $\varphi$ with respect to this basis, so the $i,j$ entry of $A$ is $a_{ij}$ where
\begin{align} \label{A}
\varphi(e_j) = \sum_i a_{ij} e_i \tag{1}
\end{align}
for all $j$.  Similarly, let $B$ be the matrix of $\varphi^*$ with respect to the dual basis, so the $i,j$ entry of $B$ is $b_{ij}$ where
\begin{align} \label{B}
\varphi^*(e_j^*) = \sum_i b_{ij} e_i^* \, . \tag{2}
\end{align}
Applying the functional $\varphi^*(e_j^*)$ to the basis vector $e_k$, by (\ref{A}) we have
\begin{align*}
(\varphi^*(e_j^*))(e_k) &= (e_j^* \circ \varphi)(e_k) = e_j^*(\varphi(e_k)) = e_j^*\left(\sum_i a_{ik} e_i\right) = \sum_i a_{ik} e_j^*(e_i) = a_{jk} \, .
\end{align*}
On the other hand, by (\ref{B}) we also have
\begin{align*}
(\varphi^*(e_j^*))(e_k) = \left(\sum_i b_{ij} e_i^*\right)(e_k) = \sum_i b_{ij} e_i^*(e_k) = b_{kj} \, .
\end{align*}
Thus $a_{jk} = b_{kj}$ for all $j,k$, hence $B = A^T$.
