Equivalence of a ket to a vector and equivalence of states up to a global phase, question about notation The ket notation describes quantum states. We could also represent quantum states with vectors. In the wikipedia article (see link below) the equivalence of both representations is expressed as 
$$a|0\rangle + b|1\rangle \doteq \begin{bmatrix}a\\b\end{bmatrix}$$
Additional to this I usually see this symbol $\equiv$ to represent that two states are equal up to a global phase
$$|\phi\rangle \equiv e^{i\theta} |\phi\rangle$$
with $\theta\in[0,2\pi)$. Basically, expressing that both states can be treated equivalent. Now my question is: Are the symbols $\equiv$ and $\doteq$ interchangeable? Is this notation related to representation theory or some other part of math?
Link:
https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation#Bras_and_kets_as_row_and_column_vectors
 A: In the first case, the ≐ symbol is used to represent the equivalence between the notations. The LHS is equal and same as the RHS but written in different fashions (ket-bra notation and the vector notation).
  However in the second case, the LHS and RHS are not exactly equal. But because of the fact that a global phase cannot be observed in quantum mechanics, it does not make a difference whether we work with one or the other, for the sake of simplicity we just ignore the global phase altogether.
So in summary, the symbols are not interchangeable.  ≐ is used to represent an equality between two notations whereas ≡ is used to convey the message that the LHS can essentially be used instead of the RHS although they are not strictly equal.
A: I looked in Sakurai, J. J., & Commins, E. D. (1995). Modern quantum mechanics, revised edition. page 20.

where the symbol $\doteq$ stands for "is represented by."*

and further in the footnote

*We do not use the equality sign here because the particular form of a matrix representation
  depends on the particular choice of base kets used. The operator is different from a representa-
  tion of the operator just as the actress is different from a poster of the actress.

For example the state 
$$ |0\rangle \doteq \begin{bmatrix}1\\0\end{bmatrix}
\text{ for the standard basis, or }
|0\rangle \doteq \frac1{\sqrt{2}}\begin{bmatrix}1\\1\end{bmatrix}
\text{ for the basis }\{|+\rangle,|-\rangle\}$$
