Page 31 - Curvature and Homology

1.4. DIFFERENTIAL FORMS 13

dccomposabb p-vectors eil A ... A eiB. Since, by assumption, A (V)
is spanned by 1 and such products, it follows that A (V) is spanned by
the elements eil A ... A ei9 where (i,, ..., 6) is a subset of the set
(1, ..., n) arranged in increasing order. But there are exactly 2n subsets
of (1, ..., n), while by assumption dim A (V) = 2n. These elements
must therefore be linearly independent. Hence, any element of A (V)
can be uniquely represented as a linear combination

where now and in the sequel (il ... 6) implies i, < ... < &. An element
of the first sum is called homogeneous of degree p.
It may be shown that any two Grassman algebras over the same
vector space are isomorphic. For a realization of A (V) in terms of the
'tensor algebra' over V the reader is referred to (I.C.2).
The elements x,, ..., x, in V are linearly independent, if and only if,
their product x, A ... A x, in A (V) is not zero. The proof is an easy
exercise in linear algebra. In particular, for the basis elements el, ..., e,
of V, el A ... A en # 0. However, any product of n + 1 elements of
V must vanish.
All the elements

for a fixed p span a linear subspace of A (V) which we denote by
Ap(V). This subspace. is evidently independent of the choice of base.
An element of AP(V) is called an exterior p-vector or, simply a p-vector.
Clearly, A1(V) = V. We define AO(V) = R. As a vector space, A (V)
is then the direct sum of the subspaces Ap(V), 0 5 p 5 n.
Let W be the subspace of V spanned by y,, ..., y, E V. This gives
rise to a p-vector q = yl A ... A yp which is unique up to a constant
factor as one sees from the theory of linear equations. Moreover, any
vector y E W has the property that y A q vanishes. The subspace W
also determines its orthogonal complement (relative to an inner product)
in V, and this subspace in turn defines a 'unique' (n - p)-vector. Note
that for eachp, the spaces Ap(V) and An-p(V) have the same dimensions.
Any p-vector 5 and any (n 2 p)-vector q determine an n-vector 5 A q
which in terms of the basis e = el A ... e, of An(V) may be expressed
as
f A 7 = (f,~) (1.4.3)
e
where (5, q) E R. It can be shown that this 'pairing' defines an iso-
morphism of AP(V) with (A~-P(~))* 1.5.1 and 1I.A).
(cf.