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.
$\blacksquare$