Graphics Reference
In-Depth Information
()
≈
()
V
=
im T
ker
T
.
Proof.
This is an easy consequence of the fact that every
v
Œ
V
can be written in the
form
()
+-
()
(
)
v
=
T
v
v
T
v
and
v
- T(
v
) clearly belongs to ker(T).
If
V
=
X
≈
Y
, then the map
V
Æ
+Æ,
X
xy
x
for
x
Œ
X
and
y
Œ
Y
, is clearly a projection operator. Therefore, there is a one-to-one
correspondence between projection operators on a vector space and direct summands
of it.
C.2
Inner Products
Definition.
Let
V
i
, 1 £ i £ n, and
W
be vector spaces over a field k. A map
f
:
VV
¥
¥◊◊◊¥
V W
Æ
1
2
n
is called a
multilinear map
if, for each i, 1 £ i £ n,
f
(
v
,
v
,...,
v
+¢
v
,...,
v
)
=
f
(
v
,
v
,...,
v
,...,
v
)
+
f
(
v
,
v
,...,
v
¢
,...,
v
)
12
i
i
n
12
i
n
12
i
n
f
(
vv
,
,...,
c
v
,...,
v
)
=
cf
(
vv
,
,...,
v
,...,
v
)
,
12
i
n
12
i
n
for all
v
j
Œ
V
j
and c Πk. Equivalently, the map f is multilinear if, for each i and any
elements
v
1
,
v
2
,...,
v
i-1
,
v
i+1
,...,
v
n
with
v
j
Œ
V
j
, the map from
V
i
to
W
defined by
(
)
v
Æ
f
v
,
v
,...,
v
, ,
v v
,...,
v
12
i
-
1
i
+
1
n
is a linear transformation. If n = 2, then f is also called a
bilinear map
.
Definition.
Let
V
be a vector space over the field k =
R
or
C
. A bilinear map
<>
,:
VY
uv
¥
Æ
k
(
)
Æ<
,
uv
,
>
,
is called an
inner
or
scalar product
on
V
if it satisfies the following two additional
properties for all
u
,
v
Œ
V
:
