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C.6.14. Theorem.
(1) The alternation map Alt : L k ( V ) Æ L k ( V ) is a linear transformation.
(2) If aŒL k ( V ), then Alt(a) is an alternating multilinear map.
(3) If aŒL k ( V ) is an alternating multilinear, then Alt(a) =a.
Proof.
This is easy and proved like Theorem C.6.8.
Theorem C.6.14 implies that
(
)
L k
k
() =
()
V
Alt L
V
and is a direct summand of L k ( V ). We can define a product, called the exterior or wedge
product ,
r
() ¥
s
() Æ
r
+
s
()
Ÿ
: L
VV
L
L
V
(C.17)
by the same formula (C.15) that we used for the exterior algebra. Theorem C.6.10
holds verbatim for alternating multilinear maps. This is exactly how Spivak ([Spiv65])
develops the exterior algebra.
Definition. An element of L k ( V ) is called an exterior k-form on V . Let L V denote the
direct sum of the vector spaces L k ( V ), k ≥ 0. The exterior product Ÿ makes L V into
an algebra called the algebra of exterior forms on V .
Finally, we already know from Theorem C.6.11 and Theorem C.6.13 that there are
natural isomorphisms
( Æ (
) ¨
k
k
k
()
E
V
*
E
V
*
L
V
,
(C.18)
so that E k ( V *) and L k ( V ) are isomorphic.
C.6.15. Theorem.
The isomorphisms in (C.18) induce an isomorphism of algebras
( Æ
()
F
:
E V
*
L
V
,
(C.19a)
where
Fa
(
Ÿ
a
Ÿ◊◊◊Ÿ
a
) =
a
Ÿ
a
Ÿ◊◊◊Ÿ
a
(C.19b)
1
2
k
1
2
k
for all a i ΠV *.
Proof. This is another easy exercise working through the definitions. Note that the
left side of Equation (C.19b) has an exterior product of simply “elements” or symbols
a i that happen to belong to V * whereas the right side is an exterior product of maps
as defined by (C.17).
Theorems C.6.6 and C.6.15 can be summarized compactly by saying that we have
a commutative diagram
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