In order to be able to define orthogonality and the “size” of a vector, we
need the notion of an inner product. This is just like the dot product of two
vectors. This is basically a rule for assigning a (complex) number to a pair of
vectors.
For this we define a dual vector space V
†
of same dimensions. Vectors in
this space are represented by row matrices [α
0
α
1
… α
n
]. The dual of the
vector |vi = [v
1
v
2
. . . v
n
]
T
is represented by hv| = [v
∗
1
v
∗
2
… v
∗
n
] where the
∗
denotes complex conjugation. Thus the matrix representation of the dual
vector hv| is the complex conjugate transpose of |vi, denoted by |vi
†
.
The inner product of vectors |φi and |ψi is defined as the complex number
hφ|ψi. (This bracket h·|·i for inner product is the origin of the Dirac bra-ket
notation: the ket vector |·i has a dual bra vector h·| and their product gives
the “bra(c)ket”.)
If |ψi = [α
1
α
2
. . . α
n
]
T
and |φi = [β
1
β
2
. . . β
n
]
T
then their inner
product is
hφ|ψi = β
∗
1
α
1
+ β
∗
2
α
2
+ … + β
∗
n
α
n
. (3.1)
Some of the consequences of this definition are:
• Norm of a vector is defined as kvk =
p
hv|vi. A vector is said to be
normalized if it has unit norm. An arbitrary vector can be normalized
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