Graphics Reference
In-Depth Information
2
()
=+
(
)
=
(
)
=
()
aa
v
x
c
u
a
c
u
c
a
u
0
and
2
.
(
)
=∑=
()
uv u x u
∑= ∑ +
c
c
uu
c a
u
0
The existence part of the theorem is proved. To prove uniqueness, assume that there
is another vector
u
¢ in
V
with a(
v
) =
u
¢•
v
. Then (
u
-
u
¢)•
v
= 0 for all
v
in
V
. In par-
ticular, letting
v
=
u
-
u
¢, we get that
(
)
∑-¢
(
)
= 0,
uu uu
-¢
which implies that
u
=
u
¢ and we are done.
Next, assume that
V
is a vector space and T :
V
Æ
V
is a linear transformation.
Given
v
Œ
V
, define a linear functional T
v
by
()
=
()
∑
T
v
wwv
T
.
By Theorem 1.8.2, there is a unique vector
v
*, so that
()
=*.
T
v
wvw
Definition.
The map
T*:
VV
Æ
defined by
()
=
T*
vv
*
is called the
adjoint
of T.
1.8.3. Lemma.
The adjoint map T* satisfies
()
∑=∑
()
T
vwv
T
*
w
for all
v
,
w
Œ
V
.
Proof.
By definition, T(
v
)•
w
= T
w
(
v
) =
w
*•
v
= T*(
w
)•
v
.
1.8.4. Lemma.
The adjoint map T* is a linear transformation.
Proof.
Using Lemma 1.8.3 and the linearity of the dot product, we have that
(
)
=
()
∑
(
)
u
∑
Ta
*
v
+
b
wu
T
a
v
+
b
w
()
∑+
()
∑
=
aT
uv
bT
uw
()
+∑
()
=∑
aT
uv
*
bT
uw
*
(
()
+
()
)
=∑
u
aT
*
v
bT
*
w
.
