ERG 2018 Tutorial 5
October 9, 2008
1. Projection, Orthogonal Vectors
We can express a given vector
u
in the form
u
v
w
=
+
where
•
The vector
v
is parallel to a given nonzero vector
q
.
•
The vector
w
is orthogonal to
q
.
q
v
proj u
=
v
v
=
q
q
We have
cos
q
v
u
q
q
=
Recall the definition of dot product, we can get the form
u q
v
q
=
g
q
q
(
)
u e
=
g
e
, where
q
e
q
=
1.1 Example:
Let
4
2
3
u
=

and
3
0
1
q
=
. Express
u
in the form
u
v
w
=
+
where
v
is parallel to
q
and
w
is
orthogonal to
q
.
2. GramSchmidt process
Given a set of n LI vectors
1
2
{
,
,
,
}
n
V
v v
v
=
v
v
v
K
, GramSchmidt process generate a set of n orthonormal
vectors
1
2
{ ,
,
,
}
n
E
e e
e
=
v
v
v
K
, such that
Span
V
=
Span E
. It works as follows:
1
1
u
v
=
,
1
1
1
u
e
u
=
2
2
2
1
1
(
)
,
u
v
v
e e
=

g
2
2
2
u
e
u
=
3
3
3
1
1
3
2
2
(
)
(
)
u
v
v
e e
v
e
e
=


g
g
3
3
3
u
e
u
=
M
M
1
1
(
)
n
n
n
n
j
j
j
u
v
v
e
e

=
=

∑
g
n
n
n
u
e
u
=
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