Editing Exterior algebra (section)

=== Exterior power ===
The {{math|''k''}}th '''exterior power''' of {{tmath|V}}, denoted {{tmath|{\textstyle\bigwedge}^{\!k}(V)}}, is the [[vector subspace]] of {{tmath|{\textstyle\bigwedge}(V)}} [[linear span|spanned]] by elements of the form
: <math>x_1 \wedge x_2 \wedge \cdots \wedge x_k,\quad x_i \in V, i=1,2, \dots, k .</math>

If {{tmath|\alpha \in {\textstyle\bigwedge}^{\!k}(V)}}, then <math>\alpha</math> is said to be a '''[[p-vector|{{math|''k''}}-vector]]'''.  If, furthermore, <math>\alpha</math> can be expressed as an exterior product of <math>k</math> elements of {{tmath|V}}, then <math>\alpha</math> is said to be '''decomposable''' (or '''simple''', by some authors; or a '''blade''', by others).  Although decomposable {{tmath|k}}-vectors span {{tmath|{\textstyle\bigwedge}^{\!k}(V)}}, not every element of <math>{\textstyle\bigwedge}^{\!k}(V)</math> is decomposable.  For example, given {{tmath|\mathbf{R}^4}} with a basis {{tmath|1= \{ e_1, e_2, e_3, e_4 \} }}, the following 2-vector is not decomposable:
: <math> \alpha = e_1 \wedge e_2 + e_3 \wedge e_4. </math>

==== Basis and dimension ====
If the [[dimension (linear algebra)|dimension]] of <math>V</math> is <math>n</math> and <math>\{e_1,\dots,e_n\}</math> is a [[basis (linear algebra)|basis]] for <math>V</math>, then the set
: <math>
\{\,e_{i_1} \wedge e_{i_2} \wedge \cdots \wedge e_{i_k} ~
\big| ~~ 1 \le i_1 < i_2 < \cdots < i_k \le n \,\}
</math>
is a basis for {{tmath|{\textstyle\bigwedge}^{\!k}(V)}}.  The reason is the following: given any exterior product of the form
: <math> v_1 \wedge \cdots \wedge v_k , </math>
every vector <math>v_j</math> can be written as a [[linear combination]] of the basis vectors {{tmath|e_i}}; using the bilinearity of the exterior product, this can be expanded to a linear combination of exterior products of those basis vectors.  Any exterior product in which the same basis vector appears more than once is zero; any exterior product in which the basis vectors do not appear in the proper order can be reordered, changing the sign whenever two basis vectors change places.  In general, the resulting coefficients of the basis {{mvar|k}}-vectors can be computed as the [[minor (linear algebra)|minor]]s of the [[matrix (mathematics)|matrix]] that describes the vectors <math>v_j</math> in terms of the basis {{tmath|e_i}}.

By counting the basis elements, the dimension of <math>{\textstyle\bigwedge}^{\!k}(V)</math> is equal to a [[binomial coefficient]]:
: <math> \dim {\textstyle\bigwedge}^{\!k}(V) = \binom{n}{k} ,</math>
where {{tmath|n}} is the dimension of the ''vectors'', and {{tmath|k}} is the number of vectors in the product.  The binomial coefficient produces the correct result, even for exceptional cases; in particular, <math>{\textstyle\bigwedge}^{\!k}(V) = \{ 0 \}</math> for {{tmath|k > n}}.

Any element of the exterior algebra can be written as a sum of [[p-vector|{{math|''k''}}-vector]]s.  Hence, as a vector space the exterior algebra is a [[Direct sum of modules|direct sum]]
: <math>
{\textstyle\bigwedge}(V)
= {\textstyle\bigwedge}^{\!0}(V)
  \oplus {\textstyle\bigwedge}^{\!1}(V)
  \oplus {\textstyle\bigwedge}^{\!2}(V)
  \oplus \cdots \oplus {\textstyle\bigwedge}^{\!n}(V)
</math>
(where, by convention, {{tmath|1={\textstyle\bigwedge}^{\!0}(V) = K}}, the [[field (mathematics)|field]] underlying {{tmath|V}}, and {{tmath|1={\textstyle\bigwedge}^{\!1}(V) = V}}), and therefore its dimension is equal to the sum of the binomial coefficients, which is {{tmath|2^n}}.

==== 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>