Graphics Reference
In-Depth Information
p
2
p
1
p
1
p
0
p
0
p
2
- p
0
p
2
p
1
- p
0
p
1
- p
0
p
2
- p
0
(a)
(b)
Figure C.1.
Linearly independent/dependent points.
Definition.
Let
X
and
Y
be subsets of a vector space
V
. The
sum
X
+
Y
of
X
and
Y
is defined by
{
}
XY xyxX
+= +
Œ
and
yY
Œ
.
C.1.2. Theorem.
If
X
and
Y
are subspaces of a vector space
V
, then
X
+
Y
is a sub-
space of
V
.
Proof.
Easy.
Definition.
A vector space
V
is said to be the
direct sum
of two subspaces
X
and
Y
,
and we write
VXY
=≈,
if
VXY
=+
and
X Y0
«=
.
C.1.3. Theorem.
If
X
and
Y
are subspaces of a vector space
V
, then
V
=
X
≈
Y
if and only if each
v
Œ
V
has a unique representation of the form
v
=
x
+
y
, where
x
Œ
X
and
y
Œ
Y
.
Proof.
See [Lips68].
Let
V
be a vector space and T :
V
Æ
V
a linear transformation. If T
2
Definition.
= T,
then T is called a
projection operator
on
V
.
C.1.4. Theorem.
Let
V
be a vector space. If T :
V
Æ
V
is a projection operator on
V
,
then
