Alternating 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 alternating.


Then $b$ is reflexive.


Proof

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

We have:

\(\ds 0\) \(=\) \(\ds \map b {v + w, v + w}\) $b$ is alternating
\(\ds \) \(=\) \(\ds \map b {v, v} + \map b {v, w} + \map b {w, v} + \map b {w, w}\) Definition of Bilinear Form (Linear Algebra)
\(\ds \) \(=\) \(\ds \map b {v, w} + \map b {w, v}\) $b$ is alternating
\(\ds \) \(=\) \(\ds \map b {w, v}\) $\map b {v, w} = 0$

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

$\blacksquare$


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