degree of proper mapping for linear case

Question

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 $\deg(A)=sign(\det (A))$?

Answer 1

The degree of a linear map is the signum of $\det$, not $\det$ (the degree is integer valued, while $\det$ is real valued -- already this should make you suspicious). (And the answer to your question is, basically, the change of variables formula for the integral, once you replace $\det$ by it's sign).