I'm reading about matrix representations. The book explicitly defines those for linear maps between vector spaces (using Einstein summation):
Matrix representation of a linear map: Let $V$ be $m$-dim with basis $H=\{h_i\}$, and let $W$ be $n$-dim with basis $F=\{f_i\}$. Let $A:V\to W$ be a linear map and $A(h_j)=a^i_jf_i$. $[A]^F_H$ is the matrix whose columns are $[A(h_j)]_F$. Thus $[A(h_j)]_F$ or $j$-th column of $[A]^F_H$ is the representation of the $j$-th old basis vector w.r.t. the new basis. With superscript denoting row no. and subscript the column no., we can thus say that $([A]_H^F)^i_j=a^i_j$.
But it doesn't define matrix representations of linear maps between dual vector spaces. So I analogously defined them as follows:
Let $V^*,W^*$ be dual spaces with bases $\tilde H=\{h^i\}$ and $\tilde F=\{f^i\}$ respectively. For a map $\tilde A:V^*\to W^*$ we have $\tilde A(h^i)=a^i_jf^j$. $[\tilde A]^{\tilde F}_{\tilde H}$ is the matrix whose rows are $[\tilde A(h^i)]_{\tilde F}$. Thus $[\tilde A(h^i)]_{\tilde F}$ or $i$-th row of $[\tilde A]^{\tilde F}_{\tilde H}$ is the representation of the $i$-th old dual basis vector w.r.t. the new dual basis.
I'm trying to solve the following exercise:
Let $H,F$ be bases for a vector space $V$, and let $\tilde H$ and $\tilde F$ be the corresponding dual bases. If $A:V\to V$ is linear, then prove that $$[A^*]^{\tilde H}_{\tilde F}=([A]^F_H)^T$$
My attempt:
For notational simplicity let $P=[A^*]^{\tilde H}_{\tilde F}$ and $Q=[A]^F_H$ respectively. Then the $i$-th row of $P$ is $[A^*(f^i)]_{\tilde H}$ and $$A^*(f^i)(h_k)=P^i_jh^j(h_k)=P^i_j\delta^j_k=P^i_k$$ $$A^*(f^i)(h_k)=f^i(A(h_k))=([A(h_k)]_F)^i=Q^i_k$$
I'm not getting the transpose bit right. Is it that the book assumes that $[\tilde A(h^i)]_{\tilde F}$ is the $i$-th column and not the row of $[\tilde A]^{\tilde F}_{\tilde H}$? I want to ask the people who have exposure to physics/mathematical physics about what's the more widespread notation/definition of dual representations. Should I use the book (column) convention or the row convention for representations of maps between dual spaces?