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 $(v,w)\in V\times V$ with $b(v,w) = 0$.

Because $b$ is symmetric, $b(w,v) = 0$.

Because $(v,w)$ was arbitrary, $b$ is reflexive.

Also see

 * Bilinear Form is Reflexive iff Symmetric or Alternating