Editing Exterior algebra (section)

==== Rank of a ''k''-vector ====
If {{tmath|\alpha \in {\textstyle\bigwedge}^{\!k}(V)}}, then it is possible to express <math>\alpha</math> as a linear combination of decomposable [[p-vector|{{math|''k''}}-vector]]s:
: <math> \alpha = \alpha^{(1)} + \alpha^{(2)} + \cdots + \alpha^{(s)} </math>
where each <math>\alpha^{(i)}</math> is decomposable, say
: <math>
\alpha^{(i)}
= \alpha^{(i)}_1 \wedge \cdots \wedge \alpha^{(i)}_k,\quad i 
= 1,2,\ldots, s.
</math>

The '''rank''' of the {{math|''k''}}-vector <math>\alpha</math> is the minimal number of decomposable {{math|''k''}}-vectors in such an expansion of {{tmath|\alpha}}.  This is similar to the notion of [[tensor rank]].

Rank is particularly important in the study of 2-vectors {{harv|Sternberg|1964|loc=§III.6}} {{harv|Bryant|Chern|Gardner|Goldschmidt|1991}}.  The rank of a 2-vector <math>\alpha</math> can be identified with half the [[rank of a matrix|rank of the matrix]] of coefficients of <math>\alpha</math> in a basis.  Thus if <math>e_i</math> is a basis for {{tmath|V}}, then <math>\alpha</math> can be expressed uniquely as
: <math> \alpha = \sum_{i,j}a_{ij}e_i \wedge e_j </math>
where <math>a_{ij} = -a_{ji}</math> (the matrix of coefficients is [[skew-symmetric matrix|skew-symmetric]]).  The rank of the matrix <math>a_{ij}</math> is therefore even, and is twice the rank of the form <math>\alpha</math>.

In characteristic 0, the 2-vector <math>\alpha</math> has rank <math>p</math> if and only if
: <math>
\underset{p}{\underbrace{\alpha \wedge \cdots \wedge \alpha}} \neq 0 \ 
</math> and <math>
\ \underset{p+1}{\underbrace{\alpha \wedge \cdots \wedge \alpha}} = 0.
</math>