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ORTHOGONALITY TUTORIAL
Orthogonal Sets
Vectors
v
,
u
are
orthogonal
or
perpendicular
to each other if
v
u
= 0 whenever
v
u
. We say a set of vectors
{
v
1
, v
2
, ... , v
k
}
is an
orthogonal set
if for all
v
j
and
v
i
,
v
j
v
i
= 0 where i
j and i, j = 1, 2, ... , k
We can show easily that the
standard basis in
is an
orthogonal set
This is also true for any subset of
the standard basis. Example one
checks some other vectors for
orthogonality.
Next we will look at some theorems
that apply to orthogonal sets.
Theorem 1:
If we have an orthogonal set
{
v
1
, v
2
, ... , v
k
}
in
then vectors
v
1
,
v
2
, ... , v
k
are all linearly independent.
1  Proof
Theorem 2:
An orthogonal set of vectors
v
1
, v
2
, ... , v
n
in
form an orthogonal
basis of
Theorem 3:
Letting
{
v
1
, v
2
, ... , v
k
}
be an orthogonal basis for a subspace of
and let
w
be any vector in the subspace
. Then there exists scalars c
1
, c
2
, ... , c
i
,
... , c
k
such that:
w
= c
1
v
1
+c
2
v
2
+ ... + c
i
v
i
+ ... +c
k
v
k
Examples:
1  Check vectors for orthogonality
2  Find coordinates with respect to a basis
Page 1 of 7
Orthogonality Tutorial
10/10/2011
http://www.nipissingu.ca/algebra/tutorials/orthogonality.html
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Orthonormal Sets
Next we will begin to explain the difference between an orthogonal and an
orthonormal set.
The
length
or
magnitude
of
v
in
is defined as 
v
 =
A
unit
vector
u
in
is a vector that has a length or magnitude of one. In other
words: 
u
 = 1 or
u u
= 1
An
orthonormal set
is a set of vectors
{
v
1
, v
2
, ... , v
k
}
which is orthogonal and
every vector in the set is a unit vector. In other words for every vector
v
i
in the set:

v
i
 = 1 and
where i = 1, 2, ... , k
It is quick and easy to obtain an orthonormal set for vectors from an orthogonal set
of vectors, simply divide each vector by its length.
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 Spring '12
 MUNK
 Linear Algebra, WI, Netscape, orthogonal complement, orthogonality tutorial

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