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We now need to come back and revisit the topic of basis. We are going
to be looking at a special kind of basis in this section that can arise in an
inner product space, and yes it does require an inner product space to
construct. However, before we do that
we’re going to need to get some preliminary topics out of the way first.
We’ll first need to get a set of definitions out of way.
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Definition 1 Suppose
that S is a set of vectors in an
inner product space.
(a) If
each pair of distinct vectors from S
is orthogonal then we
call S an orthogonal set.
(b) If
S is an orthogonal set and each of
the vectors in S also has a norm of
1 then we call S an orthonormal set.
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Let’s take a quick look at an example.
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Example 1 Given
the three vectors  ,
and in answer each of the following.
(a) Show
that they form an orthogonal set under the standard Euclidean inner product
for but not an orthonormal set. [Solution]
(b) Turn
them into a set of vectors that will form an orthonormal set of vectors under
the standard Euclidean inner product for . [Solution]
Solution
(a) Show that they form an
orthogonal set under the standard Euclidean inner product for but not an orthonormal set.
All we need to do here to show that they form an
orthogonal set is to compute the inner product of all the possible pairs and
show that they are all zero.
So, they do
form an orthogonal set. To show that
they don’t form an orthonormal set we just need to show that at least one of
them does not have a norm of 1. For
the practice we’ll compute all the norms.
So, one of them
has a norm of 1, but the other two don’t and so they are not an orthonormal
set of vectors.
[Return to Problems]
(b) Turn them into a set of vectors that will
form an orthonormal set of vectors under the standard Euclidean inner product
for .
We’ve actually
done most of the work here for this part of the problem already. Back when we were working in we saw that we could turn any vector v into a vector with norm 1 by
dividing by its norm as follows,
This new vector
will have a norm of 1. So, we can turn
each of the vectors above into a set of vectors with norm 1.
All that
remains is to show that this new set of vectors is still orthogonal. We’ll leave it to you to verify that,
and so we have turned the three vectors into a set of
vectors that form an orthonormal set.
[Return to Problems]
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We have the following very nice fact about orthogonal sets.
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Theorem 1 Suppose
is an orthogonal set of non-zero vectors in
an inner product space, then S is
also a set of linearly independent vectors.
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Proof : Note that
we need the vectors to be non-zero vectors because the zero vector could be in
a set of orthogonal vectors and yet we know that if a set includes the
zero vector it will be linearly dependent.
So, now that we know there is a chance that these vectors
are linearly independent (since we’ve excluded the zero vector) let’s form the
equation,
and we’ll need to show that the only scalars that work here
are ,
,
… , .
In fact, we can do this in a single step. All we need to do is take the inner product
of both sides with respect to ,
,
and then use the properties of inner products to rearrange things a little.
Now, because we know the vectors in S are orthogonal we know that if and so this reduced down to,
Next, since we know that the vectors are all non-zero we
have and so the only way that this can be zero is
if . So, we’ve shown that we must have ,
,
… , and so these vectors are linearly independent.
Okay, we are now ready to move into the main topic of this
section. Since a set of orthogonal
vectors are also linearly independent if they just happen to span the vector
space we are working on they will also form a basis for the vector space.
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Definition 2 Suppose
that is a basis for an inner product space.
(a) If
S is also an orthogonal set then we
call S an orthogonal basis.
(b) If
S is also an orthonormal set then
we call S an orthonormal basis.
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Note that we’ve been using an orthonormal basis already to
this point. The standard basis vectors for are an orthonormal basis.
The following fact gives us one of the very nice properties
about orthogonal/orthonormal basis.
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Theorem 2 Suppose
that is an orthogonal basis for an inner product
space and that u is any vector
from the inner product space then,
If in addition S is in fact an orthonormal basis
then,
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Proof : We’ll
just show that the first formula holds.
Once we have that the second will follow directly from the fact that all
the vectors in an orthonormal set have a norm of 1.
So, given u we
need to find scalars ,
,
… , so that,
To find these scalars simply take the inner product of both
sides with respect to ,
.
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