Editing Multilinear form (section)

==== Exterior product ====
The tensor product of alternating multilinear forms is, in general, no longer alternating.  However, by summing over all permutations of the tensor product, taking into account the parity of each term, the ''[[exterior product]]'' (<math>\wedge</math>, also known as the ''wedge product'') of multicovectors can be defined, so that if <math>f\in\mathcal{A}^k(V)</math> and <math>g\in\mathcal{A}^\ell(V)</math>, then <math>f\wedge g\in\mathcal{A}^{k+\ell}(V)</math>:

: <math>(f\wedge g)(v_1,\ldots, v_{k+\ell})=\frac{1}{k!\ell!}\sum_{\sigma\in S_{k+\ell}} (\sgn(\sigma)) f(v_{\sigma(1)}, \ldots, v_{\sigma(k)})g(v_{\sigma(k+1)}
,\ldots,v_{\sigma(k+\ell)}),</math>

where the sum is taken over the set of all permutations over <math>k+\ell</math> elements, <math>S_{k+\ell}</math>.  The exterior product is bilinear, associative, and graded-alternating: if <math>f\in\mathcal{A}^k(V)</math> and <math>g\in\mathcal{A}^\ell(V)</math> then <math>f\wedge g=(-1)^{k\ell}g\wedge f</math>. 

Given a basis <math>(v_1,\ldots, v_n)</math> for <math>V</math> and dual basis <math>(\phi^1,\ldots,\phi^n)</math> for <math>V^*=\mathcal{A}^1(V)</math>, the exterior products <math>\phi^{i_1}\wedge\cdots\wedge\phi^{i_k}</math>, with <math>1\leq i_1<\cdots<i_k\leq n</math> form a basis for <math>\mathcal{A}^k(V)</math>.  Hence, the dimension of <math>\mathcal{A}^k(V)</math> for ''n''-dimensional <math>V</math> is <math display="inline">\tbinom{n}{k}=\frac{n!}{(n-k)!\,k!}</math>.