Math 2.0

Linear and multilinear algebra; matrix theory Definition

Alternating bilinear form

Builds on Bilinear form

Definition

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

Remarks

An alternating form is skew-symmetric: B(u,v)=−B(v,u)B(u, v) = -B(v, u) for all u,vu, v. If the characteristic of FF is not 22, each skew-symmetric form is alternating. On F2F^2, the form B(x,y)=x1y2−x2y1B(x, y) = x_1 y_2 - x_2 y_1 is alternating; it is the determinant of the matrix with the columns xx and yy. A nondegenerate alternating form is a symplectic form.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Bilinear_form, Symmetric, skew-symmetric, and alternating forms
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