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). 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 $c h a r (𝕂) = 2$ .

tidy | en | SemBr

Alternating iff anticommutative away from 2

Graph View

Backlinks