Wedge product/Natural duality/Finite-dimensional/Fact

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

Then there exists a natural

\psi \colon \bigwedge ^{k}V^{*}\longrightarrow {\left(\bigwedge ^{k}V\right)}^{*},

given by

{}(\psi (f_{1}\wedge \ldots \wedge f_{k}))(v_{1}\wedge \ldots \wedge v_{k})=\det(f_{i}(v_{j}))_{ij}\,

(with ${}f_{i}\in V^{*}$ and

{}v_{j}\in V

).