Graphics Reference
In-Depth Information
Notation.
Let (
A
,a) be the tensor product of vector spaces
V
i
, 1 £ i £ k, constructed
as in Theorem C.6.1. The space
A
will be denoted by
V
1
ƒ
V
2
ƒ ...ƒ
V
k
and the map
a by ƒ. If
v
i
Œ
V
i
, then the element ƒ(
v
1
,
v
2
,...,
v
k
) in
A
will be denoted by
v
1
ƒ
v
2
ƒ
...ƒ
v
k
and is called the
tensor product
of the vectors
v
i
.
We summarize some basic properties of the tensor product. Part of what we
have accomplished is that we have formalized a tensor product notation for
vectors.
C.6.2. Theorem.
Let
U
,
V
, and
W
be vector spaces.
(1) Let
u
,
u
i
Œ
U
,
v
,
v
i
Œ
V
, and c
i
Œ
R
. Then
(
)
ƒ=
(
)
+
(
)
c
uuv
+
c
c
uv
ƒ
c
uv
ƒ
11
22
1 1
2 2
(
)
=
(
)
+
(
)
u
ƒ+
c
v
c
v
c
uv
ƒ
c
uv
ƒ
11
22
1
1
2
2
(2) The map
c
ƒ
vv
c
induces a natural isomorphism
R ƒ
V
.
Using this isomorphism, we shall always identify
R
ƒ
V
with
V
.
(3) (Associativity) The maps
(
)
ƒÆƒƒ
ƒƒ
uv w uvw
uvw uvw
ƒ
(
)
ƃƒ
induce natural isomorphisms
(
)
ƒÆƒƒ
ƒƒ
UV W UVW
UVW UVW
ƒ
(
)
ƃƒ,
respectively. As a result one does not have to worry about parenthesizing
tensor products.
(4) If
u
1
,
u
2
,...,
u
n
and
v
1
,
v
2
,...,
v
m
are bases for
U
and
V
, respectively, then
the
u
i
ƒ
v
j
for 1 £ i £ n and 1 £ j £ m form a basis for
U
ƒ
V
. In particular,
(
)
=
(
)(
)
dim
UV
ƒ
dim
U
dim
V
.
Proof.
The proofs are straightforward and easy. See [AusM63].
Note another property of the tensor product: there is a one-to-one correspondence
between multilinear maps from
V
1
¥
V
2
¥ ...¥
V
k
to a vector space
W
and
linear
maps
from
V
1
ƒ
V
2
ƒ ...ƒ
V
k
to
W
. In other words, rather than talking about multilinear
