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In-Depth Information
and
4
1
1
1
0
a
=
+
(
−
1
)
so the representation of
a
with respect to the first basis set is (3, 4), and the representation of
a
with respect to the second basis set is (4, -1).
In the beginning of this section we had described a mathematical machinery for finding the
components of a vector that involved taking the dot product or inner product of the vector to
be decomposed with basis vectors. In order to use the same machinery in more abstract vector
spaces we need to generalize the notion of inner product.
12.3.5 Inner Product—Formal Definition
An inner product between two vectors
x
and
y
, denoted by
, associates a scalar value
with each pair of vectors. The inner product satisfies the following axioms:
x
,
y
∗
, where
∗
denotes complex conjugate.
x
,
y
=
y
,
x
1.
x
+
y
,
z
=
x
,
z
+
y
,
z
.
2.
α
x
,
y
=
α
x
,
y
, where
α
can be a real or complex number.
3.
. The quantity
√
x
,
x
0, with equality if and only if
x
=
θ
x
,
x
denoted by
x
is
4.
called the
norm
of
x
and is analogous to our usual concept of distance.
12.3.6 Orthogonal and Orthonormal Sets
As in the case of Euclidean space, two vectors are said to be
orthogonal
if their inner product
is zero. If we select our basis set to be orthogonal (that is, each vector is orthogonal to every
other vector in the set) and further require that the norm of each vector be one (that is, the
basis vectors are unit vectors), such a basis set is called an
orthonormal basis set
. Given an
orthonormal basis, it is easy to find the representation of any vector in the space in terms of the
basis vectors using the inner product. Suppose we have a vector space
S
N
with an orthonormal
basis set
N
i
1
. Given a vector
y
in the space
S
N
, by definition of the basis set we can write
y
as a linear combination of the vectors
x
i
:
{
x
i
}
=
N
y
=
1
α
i
x
i
i
=
To find the coefficient
α
k
, we find the inner product of both sides of this equation with
x
k
:
N
y
,
x
k
=
1
α
i
x
i
,
x
k
i
=
