Vector space/Orientation/Wedge product/Fact

Let ${}V\neq 0$ be a finite-dimensional real vector space of dimension ${}n$.

Then we get, via the

assignment

$[v_{1},\ldots ,v_{n}]\longmapsto [v_{1}\wedge \ldots \wedge v_{n}],$

a correspondence between the orientations

on ${}V$ and the orientations on ${}\bigwedge ^{n}V$.

Proof, Write another proof