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on this later), decomposition means finding the coefficients with which to weight the unit
vectors of the basis set. In our simple example it is easy to see what these coefficients should
be. However, we will encounter situations where it is not a trivial task to find the coefficients
that constitute the decomposition of the vector. We therefore need some machinery to extract
these coefficients. The particular machinery we will use here is called the
dot product
or the
inner product
.
12.3.1 Dot or Inner Product
Given two vectors
a
and
b
such that
a
1
a
2
b
1
b
2
a
=
,
b
=
the inner product between
a
and
b
is defined as
a
·
b
=
a
1
b
1
+
a
2
b
2
Two vectors are said to be
orthogonal
if their inner product is zero. A set of vectors is said
to be orthogonal if each vector in the set is orthogonal to every other vector in the set. The
inner product between a vector and a unit vector from an orthogonal basis set will give us the
coefficient corresponding to that unit vector. It is easy to see that this is indeed so. We can
write
u
x
and
u
y
as
1
0
0
1
u
x
=
,
u
y
=
These are obviously orthogonal. Therefore, the coefficient
a
1
can be obtained by
a
·
u
x
=
a
1
×
1
+
a
2
×
0
=
a
1
and the coefficient of
u
y
can be obtained by
a
·
u
y
=
a
1
×
0
+
a
2
×
1
=
a
2
The inner product between two vectors is in some sense a measure of how “similar” they are,
but we have to be a bit careful in how we define “similarity.” For example, consider the vectors
in Figure
12.2
. The vector
a
is closer to
u
x
than to
u
y
. Therefore
a
·
u
x
will be greater than
·
a
u
y
. The reverse is true for
b
.
12.3.2 Vector Space
In order to handle not just two- or three-dimensional vectors but general sequences and func-
tions of interest to us, we need to generalize these concepts. Let us begin with a more general
definition of vectors and the concept of a vector space.
A
vector space
consists of a set of elements called vectors that have the operations of
vector addition and scalar multiplication defined on them. Furthermore, the results of these
operations are also elements of the vector space.
