Kunze Linear Algebra · Problem 10.3.10

Bilinear Forms

Mathematics — Methods · rank-two-skew-symmetric-form

Problem

Let

V

be a finite-dimensional vector space over a subfield of the complex numbers and

f

a skew-symmetric bilinear form on

V

. Show that

f

has rank

2

if and only if there exist linearly independent linear functionals

L_1

L_2

V

such that

f(\alpha, \beta) = L_1(\alpha)L_2(\beta) - L_1(\beta)L_2(\alpha).