Bilinear form/Gram matrix under base change/Fact

Let ${}K$ be a field, ${}V$ a finite-dimensional ${}K$-vector space, and let ${}\left\langle -,-\right\rangle$ denote a bilinear form on ${}V$. Let ${}{\mathfrak {v}}=v_{1},\ldots ,v_{n}$ and ${}{\mathfrak {w}}=w_{1},\ldots ,w_{n}$ be two bases of ${}V$, and let ${}G$ and ${}H$ be the Gram matrices of ${}\left\langle -,-\right\rangle$ with respect to these bases. Suppose that we have the relations

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

between the basis elements, which we encode in the transformation matrix ${}A={\left(a_{ij}\right)}_{i,j}$.

Then, we have the relation

${}H={A^{\text{tr}}}GA\,$

among the Gram matrices.

Proof, Write another proof