Wedge product/Universal property/Alternating forms and linear forms/Fact/Proof

Proof

The mapping is just the composition ${}\psi \mapsto \psi \circ \delta$, where ${}\delta \colon V\times \cdots \times V\rightarrow \bigwedge ^{n}V$ denotes the canonical mapping. The linearity of the assignment comes from the linear structures on the dual space, and on the space of alternating forms. The bijectivity of the mapping follows from fact, applied to ${}W=K$.

To fact