Graphics Reference
In-Depth Information
k
(
)
=
()
Ÿ
()
Ÿ◊◊◊Ÿ
()
ET
vv
Ÿ
Ÿ◊◊◊Ÿ
v
T
v
T
v
T
v
1
2
k
1
2
k
for
v
i
Œ
V
.
Proof.
See [AusM63].
The map E
k
T in Theorem C.6.12 is called the
k-fold exterior product
of T.
Definition.
Theorem C.6.6 showed the relationship between tensor algebras and multilinear
map algebras. Now we want to show how alternating tensors are related to special
types of multilinear maps in a similar way.
Given a permutation sŒS
k
, define
k
k
s :
VV
Æ
by
(
)
=
(
)
s
vv
,
,...,
v
v
,
v
,...,
v
.
12
k
s
()
1
s
()
2
s
( )
k
Definition.
Let
V
and
W
be vector spaces. A multilinear map
k
:
VW
T
Æ
is said to be
alternating
if T
s=(sign s)T for all sŒS
k
. The set of such alternating
maps will be denoted by L
k
(
V
;
W
). If
W
=
R
, then L
k
(
V
;
W
) will be abbreviated to L
k
(
V
).
It is easy to show that L
k
(
V
;
W
) is a vector space. By definition,
k
(
)
Õ
k
(
)
k
(
)
Õ
k
()
L
VW
;
L
VW
;
and
L
V
L
V
.
The well-known properties of the determinant function lead to the standard example
of an alternating mulilinear map.
Definition.
The map
n
det :
RR
Æ
defined by
(
(
) (
)
(
)
)
=
()
det
aa
,
,...,
a aa
,
,
,...,
a
,...,
aa
,
,...,
a
det
a
11
12
1
n
21
22
2
n
nl
n
2
nn
ij
is called the
determinant map
of
R
n
.
Clearly, det ŒL
n
(
R
n
).
The next theorem shows that, like the tensor product, the k-fold exterior product
could have been defined in terms of a
universal factorization property
with respect to
alternating multilinear maps. Let
