Matrix representation of the dual space

Question

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?

Are there any suggestions for this problem?

Answer 1

Write the functions $\psi^t(\phi_i)$ as linear combination of the functions $\phi_i$, using the definition of $B$. Then use that if $\phi\in V^*$ with $\displaystyle\phi=\sum_{i=1}^{n}\lambda_i\phi_i$ then $\lambda_j=\phi(\mathbf{v}_j) \ (\star)$.

By the definition of the matrix $B=(B_{i,j})_{1\leq i,j\leq n}$ we have $$\psi^t(\phi_i)=\sum_{j=1}^{n}B_{j,i}\phi_j .$$ Therefore $B_{k,i}\stackrel{(\star)}{=}\psi^t(\phi_i)(\mathbf{v}_k)\stackrel{\text{Definition of }\psi^t}{=}\phi_i(\psi(\mathbf{v}_k))\stackrel{\text{Definition of }A}{=}\phi_i\left(\sum_? A_{?}\mathbf{v}_?\right)\overset{\phi_i \text{ is linear}}{=}\sum_? A_{?}\phi_i(\mathbf{v}_?)\ldots$

Answer 2

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 $A=B^T$.