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 \colon V^{n}\longrightarrow W

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

Then there exists a uniquely determined

{\tilde {\psi }}\colon \bigwedge ^{n}V\longrightarrow W

such that the diagram

{\begin{matrix}V^{n}&{\stackrel {}{\longrightarrow }}&\bigwedge ^{n}V&\\&\searrow &\downarrow &\\&&W&\!\!\!\!\!\!\!\!\\\end{matrix}}

commutes.