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ICMC 2018/19 Round 1, Problem 4

Source: Imperial College Mathematics Competition 2018/19 - Round 1

August 7, 2020
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Problem Statement

For u,v∈R4u,v \in\mathbb{R}^4, let <u,v><u,v> denote the usual dot product. Define a vector field to be a map ω:R→R\omega:\mathbb{R}\to\mathbb{R} such that <ω(z),z>=0, ∀z∈R4.<\omega(z),z>=0,\ \forall z\in\mathbb{R}^4.
Find a maximal collection of vector fields {ω1,...,ωk}\left\{\omega_1,...,\omega_k\right\} such that the map Ω\Omega sending zz to λ1ω1(z)+⋯+λkωk(z)\lambda_1\omega_1(z)+\cdots+\lambda_k \omega_k(z), with λ1,…,λk∈R\lambda_1,\ldots,\lambda_k\in\mathbb{R}, is nonzero on R4\{0}\mathbb{R}^4\backslash\{0\} unless λ1=⋯=λk=0\lambda_1=\cdots=\lambda_k=0
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