16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 10.3.12

Bilinear Forms

Mathematics — Linear Algebra · equivalence-skew-forms-same-rank

Problem

Let

V

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

f

g

be skew-symmetric bilinear forms on

V

. Show that there is an invertible linear operator

T

V

such that

f(T\alpha, T\beta) = g(\alpha, \beta)

for all

\alpha, \beta

if and only if

f

and

g

have the same rank.