Let $U,V$ be two open sets in $R^n$ and $f:U\to V$ proper $C^{\infty}$ map (proper = preimage of compact set is compact). Then we have
$\int f^{*}\omega=\deg(f)\int \omega,$
for $\omega \in \Omega_c^{n}(V)$. How to prove that if $f$ is linear mapping i.e. $f(x)=Ax$ for nonsignular $n\times n$ matrix we have $\deg(A)=sign(\det (A))$?