Symmetric Bilinear Form is Reflexive

Theorem
Let $\mathbb K$ be a field.

Let $V$ be a vector space over $\mathbb K$.

Let $b$ be a bilinear form on $V$.

Let $b$ be symmetric.

Then $b$ is reflexive.

Proof
Let $\tuple {v, w} \in V \times V$ with $\map b {v, w} = 0$.

Because $b$ is symmetric, $\map b {w, v} = 0$.

Because $\tuple {v, w}$ was arbitrary, $b$ is reflexive.

Also see

 * Bilinear Form is Reflexive iff Symmetric or Alternating