Deﬁnition:
A set
{
~u
1
,~u
2
, . . . ,~u
n
}
of nonzero vectors in
R
n
is called an
orthogonal set
if
~u
i
·
~u
j
= 0 for all
i
6
=
j
.
This set
{
~u
1
,~u
2
, . . . ,~u
n
}
is called
orthonormal
if it is orthogonal and each
~u
i
is a unit vector. (ie.
if
k
~u
i
k
= 1 for all
i
.)
Examples
1. The standard basis
{
~e
1
,~e
2
, . . . ,~e
n
}
is an orthonormal set in
R
n
.
2. If
{
~u
1
,~u
2
, . . . ,~u
n
}
is orthogonal, then so is
{
a
1
~u
1
, a
2
~u
2
, . . . , a
n
~u
n
}
.
We can create an orthonormal set from any orthogonal set simply by dividing each vector by its length
making it a unit vector. That is, if
{
~u
1
,~u
2
, . . . ,~u
k
}
is an orthogonal set, then
±
1
k
~u
1
k
~u
1
,
1
k
~u
2
k
~u
2
, . . . ,
1
k
~u
k
k
~
u
k
²
is an orthonormal set.
Example:
If
~
f
1
=
1
1
1

1
,
~
f
2
=
1
0
1
2
,
~
v
3
=

1
0
1
0
, and
~
f
4
=

1
3

1
1
, then show that
{
~
f
1
,
~
f
2
,
~
f
3
,
~
f
4
}
is orthogonal
then normalize this set.
Pythagorean Theorem in