Wedge product/Canonical mapping for linear mapping/Existence/Fact/Proof

Proof

The mapping

$V^n\stackrel{\phi \times \cdots \times \phi }{⟶}{W}^n\stackrel{\delta }{⟶}\stackrel{n}{\bigwedge }W$

is, due to exercise, multilinear and alternating. Due to fact, there exists a uniquely determined linear mapping

$\stackrel{n}{\bigwedge }V⟶\stackrel{n}{\bigwedge }W,$

with $v_1\wedge \dots \wedge v_n\mapsto \phi (v_1)\wedge \dots \wedge \phi (v_n)$.

To fact