Kunze Linear Algebra · Problem 10.1.10

Bilinear Forms

Mathematics — Methods · rank-bilinear-form

Problem

Let ff be any bilinear form on a finite-dimensional vector space VV. Let WW be the subspace of all β\beta such that f(α,β)=0f(\alpha, \beta) = 0 for every α\alpha. Show that rank f=dim⁡V−dim⁡W.\text{rank}\, f = \dim V - \dim W. Use this result and the result of Exercise 9 to compute the rank of the bilinear form defined in Exercise 6.

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