Assume $\mathcal U$ is an $n$-dimensional vector space, and $\mathcal V$ and $\mathcal W$ are subspaces of $\mathcal U$ such that $\mathcal V\oplus \mathcal W= \mathcal U$. Let us set $\operatorname{dim} \mathcal V=q$, and therefore $\operatorname{dim} \mathcal W=n-q$.
I was wondering if the decomposition $\mathcal U=\mathcal V\oplus \mathcal W$ naturally induces a similar decomposition on the space of $k$-forms over $\mathcal U$. Maybe considering $k$-forms with some kind of vanishing property over the subspaces $\mathcal V$ and $\mathcal W$... but I am clueless.
Thanks!