Alternating bilinear form

Definition

Let $F$ be a field and let $V$ be a vector space over $F$. A bilinear form $B$ on $V$ is alternating when $$B(v, v) = 0 \qquad \text{for all } v \in V.$$

Remarks

An alternating form is skew-symmetric: $B(u, v) = -B(v, u)$ for all $u, v$. If the characteristic of $F$ is not $2$, each skew-symmetric form is alternating. On $F^2$, the form $B(x, y) = x_1 y_2 - x_2 y_1$ is alternating; it is the determinant of the matrix with the columns $x$ and $y$. A nondegenerate alternating form is a symplectic form.

Explanations

L1 human

An alternating form gives zero when both vectors are equal. Then swapping the two vectors changes the sign of the value. The signed area of the parallelogram spanned by two plane vectors is an example.