Graphics Reference
In-Depth Information
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.