Symmetric bilinear form

Definition

Let $F$ be a field and let $V$ be a vector space over $F$. A bilinear form $B$ on $V$ is symmetric when $$B(u, v) = B(v, u) \qquad \text{for all } u, v \in V.$$

Remarks

In each basis, $B$ is symmetric if and only if its matrix is a symmetric matrix. The dot product on $\R^n$ is symmetric. If the characteristic of $F$ is not $2$, symmetric bilinear forms correspond one to one with quadratic forms.

Explanations

L1 human

A symmetric bilinear form gives the same value for $(u, v)$ and for $(v, u)$. Inner products on real spaces are the main examples. Their matrices are symmetric.