Kunze Linear Algebra · Problem 10.3.12

Bilinear Forms

Mathematics — Methods · equivalence-skew-symmetric-forms

Problem

Let VV be a finite-dimensional vector space over a subfield of the complex numbers, and let ff, gg be skew-symmetric bilinear forms on VV. Show that there is an invertible linear operator TT on VV such that f(Tα,Tβ)=g(α,β)f(T\alpha, T\beta) = g(\alpha, \beta) for all α\alpha, β\beta if and only if ff and gg have the same rank.

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