Alternating iff anticommutative away from 2

Let $𝑉, 𝑊$ be vector spaces away from 2. Then a bilinear map $𝐵 : 𝑉 \times 𝑉 \to 𝑊$ is alternating iff it is anticommutative (i.e. antisymmetric). #m/thm/linalg

Proof

Let $𝐵$ be alternating

𝐵 (𝑥, 𝑦) + 𝐵 (𝑦, 𝑥) = 𝐵 (𝑥, 𝑥) + 𝐵 (𝑥, 𝑦) + 𝐵 (𝑦, 𝑥) + 𝐵 (𝑦, 𝑦) = 𝐵 (𝑥, 𝑥 + 𝑦) + 𝐵 (𝑦, 𝑥 + 𝑦) = 𝐵 (𝑥 + 𝑦, 𝑥 + 𝑦) = 0

hence $𝐵$ is anticommutative.

Let $𝐵$ be anticommutative. Then

2 𝐵 (𝑥, 𝑥) = 𝐵 (𝑥, 𝑥) + 𝐵 (𝑥, 𝑥) = 0

and since $2$ is a multiplicative unit it follows $𝐵 (𝑥, 𝑥) = 0$ .

From the proof, it is clear that only the forward implication holds for $c h a r (𝕂) = 2$ .

#state/tidy | #lang/en | #SemBr