Pullback of differential form is zero

Question

Let $ f:\Bbb R^m \to \Bbb R^n $ be differentiable map. Assume $ m<n$ and let $ w $ be a differential $k$-form in $\Bbb R^n $ , with $ k>m $. Show $ f^*w $ =0

Here $ f^* $ is the pullback of the $k$-form.

Answer 1

I don't think there's actually anything to do here, because $f^\ast w$ is a $k$-form in $\Bbb R^m$. Since $k> m$, every $k$-form in $\Bbb R^m$ is zero. In particular, $f^\ast w = 0$.