Symmetric Bilinear Form is Reflexive
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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.
$\blacksquare$