In this proof https://math.stackexchange.com/a/661196/205193
I do not understand this part:
then $B$ and $B^T$ descend to invertible linear transformations $$ \tilde{B}, \; \tilde{B^T} : V/N \to R \subset V^\ast, $$ and hence, $B^T = CB$ for $C := \tilde{B^T}\tilde{B}^{-1} \in L(R)$
I see why they descend into invertible linear transformations on these domains but how can we affirm the last equality, $B^T = CB$ for $C := \tilde{B^T}\tilde{B}^{-1} \in L(R)$
I think what the author meant to write was $\widetilde{B^T} = C \tilde{B}$ for $C := \widetilde{B^T} \tilde{B}^{-1}$ which makes sense because both $\tilde{B},\widetilde{B^T}$ are invertible. Then the author proceeds to show that $C = cI_{R}$ for some $c \in \mathbb{F}$ which means that $\widetilde{B^T} = c \tilde{B}$.
However, this also implies that $B^T = cB$ for some $c \in \mathbb{F}$ because
$$ B^T(x) = \widetilde{B^T}([x]) = c\tilde{B}([x]) = cB(x) $$
for all $x \in V$ and the proof goes through.