Graphics Reference
In-Depth Information
F
()
ææ
()
T
V
*
L
V
i
»
»
i
F
()
ææ
()
E
V
*
L
V
of inclusion maps i and algebra isomorphisms F.
We finish this section with a few more facts about L(
V
).
C.6.16. Theorem.
Let
V
be an n-dimensional vector space and
V
* its dual. If a
i
Œ
V
*, then the element a
1
Ÿa
2
Ÿ ...Ÿa
k
ŒL
k
(
V
) satisfies
(
)(
)
=
(
()
)
aa
Ÿ
Ÿ◊◊◊Ÿ
a
vv
,
,...,
v
det
a
v
1
2
k
1
2
k
i
j
for all
v
j
Œ
V
.
Proof.
This is a corollary of Theorem C.6.11.
Finally, if T :
V
Æ
W
is a linear transformation, then the induced map
T* :
WV
*
Æ
*
on dual spaces defines
k
()
k
( )
Æ
k
()
ET
*:
E
WV
*
E
*.
Under our identifications, the map E
k
(T*) induces the map
C.6.17. Theorem.
k
()
Æ
k
()
T
*: L
WV
defined by
( (
)
=
(
()( )
( )
)
k
()
T
*
a
vv
,
,...,
v
a
T
v
,
T
v
,...,
T
v
,
a
Œ
L
W v V
,
Œ
.
12
k
1
2
k
i
Proof.
One simply has to carefully work through all the appropriate identifications.
It is the tensor algebra T
V
* and the exterior algebra E
V
* that are the most inter-
esting because they formalize the algebra of multilinear maps and the algebra of alter-
nating multilinear maps, respectively, which are needed for defining differential forms.
We finish with one final observation. One can use the exterior algebra to define
determinants. For example, let
V
be an n-dimensional vector space and
T:
VV
Æ
a linear transformation. We know that the vector space E
n
V
has dimension 1. There-
fore, the linear transformation
n
n
n
ET E
:
VV
Æ
E