Vector space/Dual basis/Base change/Fact

Let ${}V$ be a finite-dimensional ${}K$-vector space. Let ${}v_{1},\ldots ,v_{n}$ be a basis of ${}V$ with the dual basis ${}v_{1}^{*},\ldots ,v_{n}^{*}$, and let ${}w_{1},\ldots ,w_{n}$ be another basis with the dual basis ${}w_{1}^{*},\ldots ,w_{n}^{*}$, and with

${}w_{r}=\sum _{k=1}^{n}a_{kr}v_{k}\,.$

Then

${}w_{j}^{*}=\sum _{i=1}^{n}b_{ij}v_{i}^{*}\,,$

where ${}{\left(b_{ij}\right)}_{ij}={{\left(A^{-1}\right)}^{\text{tr}}}$ is the transposed matrix of the inverse matrix of

${}A={\left(a_{kr}\right)}_{kr}$.

Proof, Write another proof