For a choice of orthonormal basis{ej}, i.e., satisfying
〈ej,ek〉=δjk
a useful choice of label is the indexj, so
|j〉=ej
Because of orthonormality, coefficients of vectors|α〉with respect to the basis
{ej}are
〈j|α〉
and the expansion of a vector in terms of the basis is written
|α〉=
∑n
j=1
|j〉〈j|α〉 (4.4)
Similarly, for elements〈α|∈V∗,
〈α|=
∑n
j=1
〈α|j〉〈j|
The column vector expression for|α〉is thus
〈 1 |α〉
〈 2 |α〉
..
.
〈n|α〉
and the row vector form of〈α|is
(
〈α| 1 〉 〈α| 2 〉 ... 〈α|n〉
)
=
(
〈 1 |α〉 〈 2 |α〉 ... 〈n|α〉
)
The inner product is the usual matrix product
〈α|β〉=
(
〈α| 1 〉 〈α| 2 〉 ... 〈α|n〉
)
〈 1 |β〉
〈 2 |β〉
..
.
〈n|β〉
IfLis a linear operatorL:V →V, then with respect to the basis{ej}it
becomes a matrix with matrix elements
Lkj=〈k|L|j〉
The expansion 4.4 of a vector|α〉in terms of the basis can be interpreted as
multiplication by the identity operator
1 =
∑n
j=1
|j〉〈j|