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

Link: http://mathoverflow.net/questions/112659/inequalities-involving-wedge-product-reference-request

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.

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

Link: http://mathoverflow.net/questions/112659/inequalities-involving-wedge-product-reference-request

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.