Choosing a representation means choosing an orthonormal basis in the state space E.  Assume the set {|u_i>} is a discretely labeled basis and the set {|ω_a>} is a continuously labeled basis for E.  
<u_i|u_i> = δ_ij , <ω_a|ω_a'> = δ(a - a’).
Any ket |ψ> may be written as |ψ> =  ∑_ic_i|u_i>,  where  c_i = <u_i|ψ>.
|ψ> = ∑_i<u_i|ψ>|u_i> = ∑_i|u_i><u_i|ψ> = I|ψ> = |ψ>.
∑_i|u_i><u_i is the projector into the subspace spanned by {|u_i>}.  If {|u_i>} is a basis then ∑_i|u_i><u_i| = I is the identity operator.  This is called the closure relation.
Similarly, any ket |ψ> may be written as
|ψ> = ∫da c(a)|ω_a>, where  c(a) = <ω_a|ψ>.
|ψ> = ∫da <ω_a|ψ>|ω_a> = ∫da |ω_a><ω_a|ψ> = I|ψ> = |ψ>.
∫da |ω_a><ω_a| is the projector into the subspace spanned by {|ω_a>}.  If {|ω_a>} is a basis then ∫da |ω_a><ω_a| = I is the identity operator.  This is called the closure relation for a continuously labeled basis.

In the basis {|u_i>} the ket |ψ> is completely specified by its components c_i = <u_i|ψ>.  The corresponding bra is completely specified by its components c_i* = <ψ|u_i> in the basis {<u_i|}.
By convention, we represent a ket as a one column matrix of its components in a given basis.
Representation of |ψ>: 
We represent a bra as one row matrix of its components in a given basis.
Representation of <ψ|: (<ψ|u₁>  <ψ|u₂>  <ψ|u₃>  ... ) = (c₁*   c₂*   c₃*   ... )

The column matrices are representations of the kets |Φ> and |ψ> and the square matrix is a representation of the operator Ω in the {|u_i>} basis.  The Ω_ij = <u_i|Ω| u_j> are the matrix elements of the operator Ω.  The matrix elements are scalars.
By knowing the components of |ψ> in the basis {|u_i>} and the actions of the operator Ω on the basis vectors, we can calculate the components of |Φ> = Ω|ψ> in the same basis.
The matrix elements of Ω are Ω_ij = <u_i|Ω|u_j>.
The matrix elements of Ω^† are Ω^†_ij = <u_i|Ω^†|u_j> = <u_j|Ω|u_i>^* = Ω_ji^*.
If Ω is Hermitian, Ω = Ω^†, then Ω_ij = Ω_ji^*.  In particular, Ω_ii = Ω_ii^*.  The diagonal matrix elements of a matrix representing a Hermitian operator are real.

• The set {i, j} forms a basis for a two-dimensional vector space.  Let i = |x₁>, j = |x₂>.
  The vector A may be written as  A = A_xi + A_yj = A_x|x₁> + A_y|x₂>.
  with A_x = A₁ = |A|cosφ  and A_y = A₂ = |A|sinφ.
  
  In the {|x_i>} basis |A> is represented by the matrix  
  Let Ω be the projector onto the x axis.  We know the action of Ω on the basis vectors. 
  Ω|x₁> = |x₁>,  Ω|x₂> = 0.  We therefore know what Ω does to |A>.
  Ω|A> = |B>.  Ω|A> = A₁Ω|x₁> + A₂Ω|x₂> = A₁|x₁> = B₁|x₁> + B₂|x₂>.  B₁ = A₁, B₂ = 0.
  .
  
  is the matrix of Ω in the {|x_i>} basis.
  
  Ω₁₁ = <x₁|Ω|x₁>,  Ω₁₂ = <x₁|Ω|x₂>,  Ω₂₁ = <x₂|Ω|x₁>,  Ω₂₂ = <x₂|Ω|x₂>.
  Ω is a Hermitian operator, Ω_ij = Ω_ji^*.