Graphics Reference
In-Depth Information
1.4.5. Theorem.
If
X
is a k-dimensional subspace of an n-dimensional inner
product space
V
, then there exist n-k orthonormal vectors
n
1
,
n
2
,...,
n
n-k
, so that
{
}
XuVnu
=Œ
∑=
0
for
1
££-
i
n
k
.
i
Proof.
Choose the vectors
n
i
to be an orthonormal basis for the orthogonal
complement of
X
.
Now let
X
be a subspace of an inner product space
V
. Let
v
Œ
V
. Since
V
=
X
≈
X
^
, we can express
v uniquely
in the form
v
=
x
≈
y
, where
x
Œ
X
and
y
Œ
X
^
.
Definition.
The vector
x
, denoted by
v
||
, is called the
orthogonal projection of
v
on
X
and the vector
y
, denoted by
v
^
, is called the
orthogonal complement of
v
with respect
to
X
.
Note that in the definition, because of the symmetry of the direct sum operator,
the orthogonal complement
v
^
of
v
with respect to
X
is also the orthogonal projec-
tion of
v
on
X
^
. The next theorem shows us how to compute orthogonal projections
and complements.
1.4.6. Theorem.
Let
u
1
,
u
2
,...,
u
k
, k ≥ 1, be
any
orthonormal basis for a subspace
X
in an inner product space
V
. Let
v
Œ
V
. If
v
||
and
v
^
are the orthogonal projection
and orthogonal complement of
v
on
X
, respectively, then
||
(
)
(
)
(
)
vvuuvuu
=∑
+∑
2 2
...
++∑
vuu
(1.14)
11
kk
and
^
(
)
(
)
(
)
v
=- ∑
v vuu
-∑
vuu
2 2
...
--∑
vu u
.
kk
(1.15)
11
Proof.
Exercise 1.4.2.
In Theorem 1.4.6 it is essential that we have an orthonormal basis, otherwise it
is easy to come up with examples that show equations (1.14) and (1.15) are false.
The next definition formalizes some common terminology.
Definition.
Let
u
π
0
and
v
be vectors in an inner product space. Then the
orthogo-
nal projection of
v
on
u
, denoted by
v
||
, and the
orthogonal complement of
v
with respect
to
u
, denoted by
v
^
, are defined by
vv
u
u
u
u
Ê
Ë
ˆ
¯
||
=∑
(1.16)
and
vvv
u
u
u
u
Ê
Ë
ˆ
¯
^
=- ∑
.
(1.17)
