Vector space/Finite-dimensional/Gram matrices/Fact

basis ${\displaystyle {}{\mathfrak {v}}=v_{1},\ldots ,v_{n}}$, the mapping

${\displaystyle \operatorname {Bilin} _{}{\left(V\right)}\longrightarrow \operatorname {Mat} _{n}(K),\Psi \longmapsto G_{\mathfrak {v}}(\Psi ),}$

which assigns to a bilinear form ${\displaystyle {}\Psi }$ its Gram matrix with respect to the given basis, is an isomorphism