Bilinear form/Gram matrix/Definition

Gram matrix (bilinear form)

Let $K$ be a field, $V$ a finite-dimensional $K$-vector space, and let $⟨-,-⟩$ denote a bilinear form on $V$. Let $v_1,\dots ,v_n$ be a basis of $V$. The the $n\times n$-matrix

${⟨v_i,v_j⟩}_{1\le i,j\le n}$

is called the Gram matrix of $⟨-,-⟩$ with respect to this basis.