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Proof. This is an easy exercise working through the definitions. Note that the left side
of equation (C.12b) has a tensor product of simply “elements” or symbols a i that happen
to belong to V * whereas the right side is a tensor product of maps as defined by (C.7).
Because multilinear maps are more intuitive than abstract tensor products,
Theorem C.6.6 justifies our always treating T( V *) as if it were L( V ). Notice, however,
that we had to use the dual space V * to make this identification.
Now we move on to a definition of exterior algebras. Let S k be the group of per-
mutations of {1,2, . . . ,k}. Any element s of S k induces a unique isomorphism
k
k
s :T
VV
Æ
T
(C.13a)
satisfying
(
) =
s
vv
ƒ
ƒ◊◊◊ƒ
v v
ƒ
v
ƒ◊◊◊ƒ
v vV
,
Œ
.
(C.13b)
( )
( )
(
)
1
2
k
s
1
s
2
s
k
i
Definition. A tensor a in T k V is said to be alternating if s(a) = sign(s)a, for all sŒ
S k . A linear transformation
TT k
:
VW
Æ
is said to be alternating if T
s=sign(s)T, for all sŒS k .
Now, by Theorem C.6.6 we can identify T k ( V *) with (T k V )*.
C.6.7. Theorem. A tensor in T k ( V *) is alternating if and only if it corresponds to an
alternating linear transformation in (T k V )* under the natural isomorphism j defined
by Theorem C.6.5(2).
Proof.
See [AusM63].
The alternation map Alt : T k V Æ T k V is defined by
Definition.
1
Â
(
)
Alt
=
sign
ss
,
if k
1
,
k
!
s
Œ
S k
=
the identity map
,
if k
=
0
.
C.6.8. Theorem.
(1) The alternation map Alt : T k V Æ T k V is a linear transformation.
(2) If aŒT k V , then Alt(a) is an alternating tensor.
(3) If aŒT k V is an alternating tensor, then Alt(a) =a.
Proof.
This is an easy exercise.
Theorem C.6.8 shows that Alt 2
= Alt, so that Alt is a projection of T k V onto the
subspace of alternating tensors.
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