Wedge product/Universal property/Fact

Universal property of the wedge product

Let $K$ be a field, let $V$ be a $K$-vector space, and $n\in \mathbb{N}$. Let

$\psi :V^n⟶W$

be an alternating multilinear mapping in another $K$-vector space $W$.

Then there exists a uniquely determined

linear mapping

$\tilde{\psi }:\stackrel{n}{\bigwedge }V⟶W$

such that the diagram

$\begin{array}{cccc}{V}^n& \stackrel{}{⟶}& \stackrel{n}{\bigwedge }V& \\ & \searrow & \downarrow & \\ & & W& \end{array}$

commutes.

Proof, Write another proof