Wedge product/Universal property/Fact/Proof

Proof

We use the notation from fact. The assignment

$e_{(v_{1},\ldots ,v_{n})}\longmapsto \psi (v_{1},\ldots ,v_{n})$

defines, due to fact, a ${}K$-linear mapping

${\bar {\psi }}\colon H\longrightarrow W.$

As ${}\psi$ is multilinear and alternating, ${}{\bar {\psi }}$ sends the linear subspace ${}U\subseteq H$ to ${}0$. According to fact, there exists a ${}K$-linear mapping

${\tilde {\psi }}\colon H/U\longrightarrow W,$

which is compatible with ${}{\bar {\psi }}$.
The uniqueness follows from the fact that the ${}v_{1}\wedge \ldots \wedge v_{n}$ form a generating system of ${}\bigwedge ^{n}V$, and these have to be sent to ${}\psi (v_{1},\ldots ,v_{n})$.

To fact