Evaluation of quadratic form.

Question

Suppose we can have a quadratic module $(V, Q)$ for $V$ a finite dimensional vector space. Let $(e_i)_{1 \leq i \leq n}$ be the basis for $V.$ Let $x = \sum x_ie_i.$ My book says that $Q(x) = \sum_{i, j} a_{ij}x_ix_j$ where $a_{ij} = e_i . e_j$ where $e_i . e_j = B(e_i, e_j)$ for $B$ the associated symmetric bilinear form. So I understand that $Q(x) = B(x, x)$ which gives us $Q(x) = B(x, x) = \sum_{i, j} B(x_ie_i, x_je_j)$ as $B$ is bilinear. However, I can't see how $B(x_ie_i, x_je_j) = a_{ij}x_ix_j,$ which would explain the equivalence. Is this even true?

Answer 1

By definition of a bilinear form, $$B(x_ie_i, x_je_j) \overset{\text{def}}= x_i x_j B(e_i, e_j) =a_{ij}x_ix_j$$

Answer 2

$ B(x_ie_i,x_je_j)= \frac{1}{2}(Q(x_ie_i+x_je_j)- Q(x_ie_i) -Q(x_je_j)) = \frac{1}{2}(a_{ii}e_ie_i+ a_{jj}e_j.e_j + 2a_{ij}e_i.e_j- a_{ii}e_i.e_i-a_{jj}e_j.e_j) = a_{ij}e_i.e_j$