Editing Multilinear form (section)

===Alternating multilinear forms===
{{main|Alternating multilinear map}}

An important class of multilinear forms are the '''alternating multilinear forms''', which have the additional property that<ref name=":0">{{Cite book|title=An Introduction to Manifolds|url=https://archive.org/details/introductiontoma00lwtu_506|url-access=limited|last=Tu|first=Loring W.|authorlink = Loring W. Tu|publisher=Springer|year=2011|isbn=978-1-4419-7399-3|edition=2nd |pages=[https://archive.org/details/introductiontoma00lwtu_506/page/n40 22]–23}}</ref>

: <math>f(x_{\sigma(1)},\ldots, x_{\sigma(k)}) = \sgn(\sigma)f(x_1,\ldots, x_k), </math>

where <math>\sigma:\mathbf{N}_k\to\mathbf{N}_k</math> is a [[permutation]] and <math>\sgn(\sigma)</math> denotes its [[Sign of a permutation|sign]] (+1 if even, –1 if odd).  As a consequence, [[Alternating multilinear map|alternating]] multilinear forms are antisymmetric with respect to swapping of any two arguments (i.e., <math>\sigma(p)=q,\sigma(q)=p </math> and <math>\sigma(i)=i, 1\le i\le k, i\neq p,q </math>):

: <math>f(x_1,\ldots, x_p,\ldots, x_q,\ldots, x_k) = -f(x_1,\ldots, x_q,\ldots, x_p,\ldots, x_k). </math>

With the additional hypothesis that the [[Characteristic (field)|characteristic of the field]] <math>K</math> is not 2, setting <math>x_p=x_q=x </math> implies as a corollary that <math>f(x_1,\ldots, x,\ldots, x,\ldots, x_k) = 0 </math>; that is, the form has a value of 0 whenever two of its arguments are equal.  Note, however, that some authors<ref>{{Cite book|title=Finite-Dimensional Vector Spaces|last=Halmos|first=Paul R.|authorlink = Paul R. Halmos|publisher=Van Nostrand|year=1958|isbn=0-387-90093-4|edition=2nd |pages=50}}</ref> use this last condition as the defining property of alternating forms.  This definition implies the property given at the beginning of the section, but as noted above, the converse implication holds only when <math>\operatorname{char}(K)\neq 2 </math>.

An alternating multilinear <math>k</math>-form on <math>V</math> over <math>\R</math> is called a  '''multicovector of degree <math>\boldsymbol{k}</math>''' or '''<math>\boldsymbol{k}</math>-covector''', and the vector space of such alternating forms, a subspace of <math>\mathcal{T}^k(V)</math>, is generally denoted <math>\mathcal{A}^k(V)</math>, or, using the notation for the isomorphic ''k''th [[exterior power]] of <math>V^*</math>(the [[dual space]] of <math>V</math>), <math display="inline">\bigwedge^k V^*</math>.<ref>Spivak uses <math>\Omega^k(V)</math> for the space of <math>k</math>-covectors on <math>V</math>.  However, this notation is more commonly reserved for the space of differential <math>k</math>-forms on <math>V</math>.  In this article, we use <math>\Omega^k(V)</math> to mean the latter.</ref>  Note that [[linear functional]]s (multilinear 1-forms over <math>\R</math>) are trivially alternating, so that <math>\mathcal{A}^1(V)=\mathcal{T}^1(V)=V^*</math>, while, by convention, 0-forms are defined to be scalars: <math>\mathcal{A}^0(V)=\mathcal{T}^0(V)=\R</math>.

The [[determinant]] on <math>n\times n</math> matrices, viewed as an <math>n</math> argument function of the column vectors, is an important example of an alternating multilinear form.

==== 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>.