**Dirac notation. Bra-ket notation** N-dimensional complex vectorial spaces can be described using the Dirac notation as invented by Paul Dirac. The notation is also called the bra-ket notation.

Let V be the N-dimensional complex vector space. Let |a> be any vector that is element of V and let us call it ket. As any vector can be described by a lineair combination of a basis set of kets, {|i>}

_{N} as:

|a> = a

_{1}|1> + a

_{2}|2> + a

_{3}|3> + ... + a

_{N}|N>

a

_{i} will be the component of the vector |a> in the direction |i>.

Note the sum is for i=1 to N; there are N dimensions and the basis set has N basis vectors, directions.

Since V is a complex vector space, the adjoint of any vector in the space can be defined named bra: bra = <a| = (|a>)

^{adjoint}.

Correspondingly there exist a bra basis set {<i|}

_{N} in which <i| = (|i>)

^{adjoint}.

The bra vector <a| can be developed as a lineair combination of the bra basis set:

<a| = a

^{*}_{1}<1| + a

^{*}_{2}<2| + a

^{*}_{3}<3| + ... + a

^{*}_{N}<N|

The representations chosen as vectors in the chosen bra basis set or ket basis set is arbitrary. An infinite number of sets of N vectors can be chosen as basis sets.

A compact representation of a ket vector in a given basis set is a column vector:

**a** = (a

_{1}, a

_{2}, a

_{3}, ..., a

_{N})

^{T} Note that we need to transpose the row vector to make it a column vector. This is purely due to the notation. A column vector is written vertically and takes much space.

A compact representation of a bra vector in a given basis set is a row vector:

**a**^{adjoint} = (a

^{*}_{1}, a

^{*}_{2}, a

^{*}_{3}, ..., a

^{*}_{N})

__See also:__ Bra-ket notation

http://en.wikipedia.org/wiki/Bra-ket_notation The adjoint of a vector

http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1200239019 Dirac Notation

http://people.seas.harvard.edu/~jones/ap216/lectures/ls_2/ls2_u1/dirac_notation/
dirac_notation.html