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

Proof

The mapping

$V^{n}{\stackrel {\varphi \times \cdots \times \varphi }{\longrightarrow }}W^{n}{\stackrel {\delta }{\longrightarrow }}\bigwedge ^{n}W$

is, due to exercise, multilinear and alternating. Due to fact, there exists a uniquely determined linear mapping

$\bigwedge ^{n}V\longrightarrow \bigwedge ^{n}W,$

with ${}v_{1}\wedge \ldots \wedge v_{n}\mapsto \varphi (v_{1})\wedge \ldots \wedge \varphi (v_{n})$.

To fact