Matrix of a bilinear form

Definition

Let $F$ be a field and let $V$ be a vector space over $F$ with a basis $(e_1, \dots, e_n)$. Let $B$ be a bilinear form on $V$. The matrix of $B$ in this basis is the $n \times n$ matrix $M$ with entries $$M_{ij} = B(e_i, e_j) \qquad (1 \le i, j \le n).$$

Remarks

If $x$ and $y$ are the coordinate vectors of $u$ and $v$, then $B(u, v) = x^T M y$. The map $B \mapsto M$ is a linear bijection from the bilinear forms on $V$ to the $n \times n$ matrices over $F$. For the dot product on $\R^n$ and the standard basis, $M$ is the identity matrix.

Explanations

L1 human

A bilinear form is fixed by its values on pairs of basis vectors. We store these $n^2$ values in a matrix. Then the form becomes a product of a row vector, the matrix and a column vector.