Editing Exterior algebra (section)

=== Alternating operators ===
Given two vector spaces ''V'' and ''X'' and a natural number ''k'', an '''alternating operator''' from ''V''<sup>''k''</sup> to ''X'' is a [[multilinear map]]
: <math> f : V^k \to X </math>
such that whenever ''v''<sub>1</sub>, ..., ''v''<sub>''k''</sub> are [[linearly dependent]] vectors in ''V'', then
: <math> f(v_1,\ldots, v_k) = 0. </math>

The map
: <math> w : V^k \to {\textstyle\bigwedge}^{\!k}(V),</math>
which associates to <math> k </math> vectors from <math> V </math> their exterior product, i.e. their corresponding <math> k </math>-vector, is also alternating.  In fact, this map is the "most general" alternating operator defined on <math> V^k; </math> given any other alternating operator <math> f : V^k \rightarrow X, </math> there exists a unique [[linear map]] <math> \phi : {\textstyle\bigwedge}^{\!k}(V) \rightarrow X </math> with <math> f = \phi \circ w. </math>  This [[universal property]] characterizes the space of alternating operators on <math> V^k </math> and can serve as its definition.