Graphics Reference
In-Depth Information
(
)
aa
Ÿ
Ÿ◊◊◊
a
=
k Alt
!
a a
ƒ
ƒ◊◊◊ƒ
a
1
2
k
1
2
k
for a
i
Œ E
1
(
V
).
(3) If wŒE
r
(
V
) and hŒE
s
(
V
), then
1
rs
Ÿ=
()
Ÿ
wh
hw
.
In particular, if a, bŒE
1
(
V
), then aŸb=- bŸaand aŸa=0.
(4) If sŒS
k
and a
i
Œ E
1
(
V
), then
=
(
)
aa
Ÿ
Ÿ◊◊◊Ÿ
a
sign
s aa
Ÿ
Ÿ◊◊◊Ÿ
a
.
(
)
(
)
(
)
s
1
s
2
s
k
1
2
k
(5) Let a
1
, a
2
,..., anda
n
form a basis for E
1
(
V
). If 1 £ k £ n, then the set of all
aa
Ÿ
Ÿ◊◊◊Ÿ
a
,
1
£
i
<
i
<◊◊◊<
i
£
n
,
i
i
i
12
k
1
2
k
n
k
Ê
Ë
ˆ
¯
is a basis for E
k
(
V
). It follows that E
k
(
V
) has dimension
for 0 £ k £ n.
(6) If k > n, then E
k
(
V
) = 0.
Proof.
The proofs in [Spiv65] for alternating multilinear maps readily translate to
our situation here.
Definition.
Let E
V
denote the direct sum of the vector spaces E
k
(
V
), k ≥ 0. The exte-
rior product Ÿ makes E
V
into an algebra called the
exterior algebra
or
Grassmann
algebra
of
V
.
C.6.11. Theorem.
Let
V
be a vector space and
V
* its dual space. For k ≥ 1, there is
a unique isomorphism
(
Æ
(
)
k
k
F :
E
V
*
E
V
*
such that
(
)
(
)
=
(
()
)
Fa
Ÿ
a
Ÿ◊◊◊Ÿ
a
vv
Ÿ
Ÿ◊◊◊Ÿ
v
det
a
v
,
1
2
k
1
2
k
i
j
for a
i
Œ
V
* and
v
j
Œ
V
= E
1
(
V
).
Proof.
See [AusM63].
C.6.12. Theorem.
Let
V
and
W
be vector spaces. A linear transformation T :
V
Æ
W
induces a unique linear transformation
k
k
k
ET E
:
VW
Æ
E
such that