Why do the Pauli matrices 'transform like a vector'? Maybe I still have trouble understanding what physicists mean when something is a vector but here is how I see it. I use the Einstein summation convention throughout.
Given a set of basis vectors $B = \{\vec{x}_i\}$ and a set of coordinates $C = \{c_i\}$, we represent a physical quantity $\vec{v} = c_i \vec{x}_i$. Now I am free to change my basis to some $\vec{x}'_i = R_{ij}\vec{x}_j$. In this case, I must have $c'_i = R^{-1}_{ji}c_j$ so that I still refer to the same physical quantity after the transformation. 
With this in mind, I'm trying to understand what it means when people say that one can make a vector out of the Pauli matrices as explained here https://en.wikipedia.org/wiki/Pauli_matrices#Pauli_vector. 
What is special about the Pauli matrices such that when I apply the transformation $R$ on basis of Pauli matrices, it "works"? What exactly wouldn't work if I took some other arbitrary set of 2x2 matrices?
 A: Take 
$$
R=\left(
\begin{array}{cc}
 e^{-\frac{1}{2} i (\alpha +\gamma )} \cos \left(\frac{\beta }{2}\right) & -e^{-\frac{1}{2} i (\alpha -\gamma )} \sin \left(\frac{\beta }{2}\right) \\
 e^{\frac{1}{2} i (\alpha -\gamma )} \sin \left(\frac{\beta }{2}\right) & e^{\frac{1}{2} i (\alpha +\gamma )} \cos \left(\frac{\beta }{2}\right) \\
\end{array}
\right)=R_z(\alpha)R_y(\beta)R_z(\gamma)\, .
$$
Then, for instance,
\begin{align}
R\cdot\sigma_z\cdot R^{-1} = \sigma_z\cos\beta +
\sigma_x\cos\alpha\sin\beta +\sigma_y\sin\alpha\sin\beta
\end{align}
which is the same as the rotation $R$ applied to $\hat z$.  The other Pauli matrices also transform into linear combinations of themselves as the corresponding basis vectors do.
A: Suppose you start out looking for a vector $\vec{ \sigma}$, that you want to satisfy:
$$ {\vec{\sigma} \times \vec{\sigma}} = 2i \vec{ \sigma} $$
So it's a priori a vector with a non-zero cross product with itself.
Expanding the cross product:
$$ \sigma_a\sigma_b - \sigma_b\sigma_a = 2i\epsilon_{abc}\sigma_c  $$
You can solve that with 2 indices ($\alpha, \beta \in (+\frac 1 2, -\frac 1 2)$):
$$ [\sigma_a]_{\alpha\beta} = (\alpha + \beta)\delta_{a3} 
+|\alpha-\beta|\delta_{a1} + i(\beta-\alpha)\delta_{a2}$$
which are the Pauli matrices, forming a vector. The challenge now is convincing yourself that they describe an internal degree of freedom that is spin 1/2.
A: The formulae you see in textbooks for the spin matrices are just one representation of an algebra satisfying $\sigma_a\sigma_b =\delta_{ab}I_2+i \sum_c \epsilon_{abc}\sigma_c$. This condition is preserved under the usual rotational transformation, which obtains a different representation of this algebra. The same one-among-many disclaimer applies to other matrix-valued constants in physics (although the preserved rule will be different), such as Gell-Mann matrices & gamma matrices.
