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
=
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
u
i
= 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
u
1
u
1
,
1
u
2
u
2
, . . . ,
1
u
k
u
k
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
R
n
If
{
u
1
, u
2
, . . . , u
n
}
is orthogonal, then
u
1
+
u
2
+
. . .
+
u
k
2
=
u
1
2
+
u
2
2
+
. . .
+
u
k
2
2