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ORTHOGONALITY TUTORIAL
Orthogonal Sets
Vectors
v
,
u
are
orthogonal
or
perpendicular
to each other if
v
u
= 0 whenever
vu
. 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