# wedge complements in finite dimension

in the next counting experiment we are going  to calculate the dimensions of some linear subspaces of the Grassmann algebra of a vector space of dimension $n$.

We should  be using the intuition granted by $\Lambda(\mathbb{R}^n)$

The definitions are:

• $C_{dx}=\{\alpha\mid \alpha\wedge dx=0\}$
• $M_{dx}={C_{dx}}^{\top}$
• $C_{dx\wedge dy}=\{\alpha\mid \alpha\wedge dx\wedge dy=0\}$
• $M_{dx\wedge dy}={C_{dx\wedge dy}}^{\top}$
• $C_{dx\wedge dy\wedge dz}$
• $M_{dx\wedge dy\wedge dz}$

doesn’t anybody know the name of the result?…  ‘cuz if it hasn’t, I will claim mine : )

Meanwhile, let me refrain the definition  that says:

$\Lambda(\Omega)$

is the $C^{\infty}(\Omega)$module over the symbols $dx^1,dx^2,...,dx^n$ and over an open set $\Omega\subseteq\mathbb{R}^n$

stay tune…

i think that C_{dx} could be called the Ann(dx) in $\Lambda(R^n)$: the annihilator