Wedge product/Canonical mapping for linear mapping/Properties/Fact/Proof

Proof

(1). Let ${}w_{1},\ldots ,w_{n}\in W$ be given, and let ${}v_{1},\ldots ,v_{n}\in V$ be preimages, that is, ${}\varphi (v_{i})=w_{i}$. We have

${}{\left(\bigwedge ^{n}\varphi \right)}(v_{1}\wedge \ldots \wedge v_{n})=w_{1}\wedge \ldots \wedge w_{n}\,.$

Surjectivity follows from fact  (1).
(2). We may assume, due to the construction of the wedge product, that ${}V$ and ${}W$ have finite dimension. The statement follows from the explicit description of the bases in fact.
(3). It is enough to show the equality for the generating system ${}u_{1}\wedge \ldots \wedge u_{n}$ with ${}u_{i}\in U$; but this is clear due to the explicit description.

To fact