Bilinear form/Gram matrix/Definition

Let ${\displaystyle {}K}$ be a field, ${\displaystyle {}V}$ a finite-dimensional ${\displaystyle {}K}$-vector space, and let ${\displaystyle {}\left\langle -,-\right\rangle }$ denote a bilinear form on ${\displaystyle {}V}$. Let ${\displaystyle {}v_{1},\ldots ,v_{n}}$ be a basis of ${\displaystyle {}V}$. The the ${\displaystyle {}n\times n}$-matrix

${\displaystyle \left\langle v_{i},v_{j}\right\rangle _{1\leq i,j\leq n}}$

is called the Gram matrix of ${\displaystyle {}\left\langle -,-\right\rangle }$ with respect to this basis.