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# OnaSubspaceofTwo-Forms.AQuestiononExteriorForms

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For the last few days, I have been trying to answer the following algebraic question in exterior algebra. The following question appears as an algebraic step in the context of existence of solutions of a certain system of PDE. I have asked a special case of the problem in

with the hope that this particular case will do what I have in mind, but this did not turn out to be the case. Any help in this direction is welcome.

QUESTION:

Let $N$ be a linear subspace of $\Lambda^2(\mathbb{R}^n)$ satisfying $$\omega\wedge\omega\neq 0,\text{ for all }\omega\in N,\omega\neq 0.$$ Is it true that there exists a $a\in\Lambda^4(\mathbb{R}^n)$, $a\neq 0$ such that $$\langle a;\omega\wedge\omega\rangle>0,\text{ for all }\omega\in N,\omega\neq 0?$$ Comment: My guess is that there will be such an $a$. But I could not prove it.

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# On a Subspace of Two-Forms.

For the last few days, I have been trying to answer the following algebraic question in exterior algebra. The following question appears as an algebraic step in the context of existence of solutions of a certain system of PDE. I have asked a special case of the problem in

with the hope that this particular case will do what I have in mind, but this did not turn out to be the case. Any help in this direction is welcome.

QUESTION:

Let $N$ be a subspace of $\Lambda^2(\mathbb{R}^n)$ satisfying $$\omega\wedge\omega\neq 0,\text{ for all }\omega\in N,\omega\neq 0.$$ Is it true that there exists a $a\in\Lambda^4(\mathbb{R}^n)$, $a\neq 0$ such that $$\langle a;\omega\wedge\omega\rangle>0,\text{ for all }\omega\in N,\omega\neq 0?$$ Comment: My guess is that there will be such an $a$. But I could not prove it.