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 ΠV .
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
Search WWH ::




Custom Search