Let $V$ be an $n$-dimensional vector space over $F$, with basis $\mathcal{B} = \{\mathbf{v_1, \cdots, v_n}\}$. Let $\mathcal{B}^{*} = \{\phi_1, \cdots, \phi_n\}$ be the dual basis for $V^{*}$. Let $\psi : V \to V$ and $A$ be the matrix representing $\psi$ w.r.t. $\mathcal{B}$. Let $\psi^t : V^{*} \to V^{*}$ and $B$ be the matrix representing $\psi^t$ w.r.t. $\mathcal{B}^{*}$. How are $A$ and $B$ related?
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Best Answer
By definition of the matrices $A,B$, we have $\psi(v_j) = \sum_k [A]_{kj} v_k$, and $\psi^t(\phi_j) = \sum_k[B]_{kj} \phi_k$.
Then we have $\phi_i(\psi(v_j)) = [A]_{ij}$, and $(\psi^t(\phi_j))(v_i) =[B]_{ij} $.
Since $(\psi^t(\phi_j))(v_i) = \phi_j(\psi(v_i))$, we get $A=B^T$.