Can Any Bivector Be Decomposed Using a Specific Basis?

[Sudharaka]

Hi everyone, :)

I am trying to find an approach to solve this but yet could not find a meaningful one. Hope you can give me a hint to solve this problem.

Problem:

Prove that for any bivector \(\epsilon\in\wedge^2(V)\) there is a basis \(\{e_1,\,\cdots,\,e_n\}\) of \( V \) such that \(\epsilon=e_1\wedge e_2+e_3\wedge e_4+\cdots + e_{k-1}\wedge e_{k}\).

 

[jvicens]

Hello!

Thank you for bringing this problem to our attention. This is known as the decomposition theorem for bivectors and it can be proven using the Gram-Schmidt process.

First, let's define a bivector \(\epsilon\) as \(\epsilon=a\wedge b\), where \(a\) and \(b\) are vectors in \(V\). We can then apply the Gram-Schmidt process to the set \(\{a,b\}\) to obtain an orthonormal basis \(\{e_1,e_2\}\) for the plane spanned by \(a\) and \(b\).

Next, we can extend this basis to a basis for \(V\) by adding vectors \(e_3,e_4,\cdots,e_n\) that are orthogonal to both \(e_1\) and \(e_2\). This can be done using the Gram-Schmidt process again.

Finally, we can write \(\epsilon\) in terms of this new basis as \(\epsilon=e_1\wedge e_2+e_3\wedge e_4+\cdots+e_{k-1}\wedge e_{k}\). This proves the decomposition theorem for bivectors.

I hope this helps and gives you a starting point for solving the problem. Good luck!