How to find dimensions of a vector space for a quantum mechanics system? Space of state for a simple spin is supposed to be two dimensional. How do you generally find the dimension of vector space for a quantum system? 
For example for a simple spin, how do we know it is 2 dimensional?
 A: 
In more complex systems how can we know all the possible independent outcomes and hence determine the dimensionality of the system 

Tensor  Product
It could be  4  electrons, each with 2 states, so you take all the possible combinations of independent states.
Or something like this for spin states:
$${\displaystyle |\psi \rangle =c_{\psi }|\uparrow \downarrow\rangle +d_{\psi }|\downarrow \uparrow\rangle }$$
An example of the tensor product is:
$${\displaystyle S\otimes T}$$
For example, if V, X, W, and are all two-dimensional and bases have been fixed for all of them, and S and T are given by the matrices
$${\displaystyle {\begin{bmatrix}a_{1,1}&a_{1,2}\\a_{2,1}&a_{2,2}\\\end{bmatrix}},\qquad {\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\\\end{bmatrix}},}$$
respectively, then the tensor product of these two matrices is
$${\displaystyle {\begin{bmatrix}a_{1,1}&a_{1,2}\\a_{2,1}&a_{2,2}\\\end{bmatrix}}\otimes {\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\\\end{bmatrix}}={\begin{bmatrix}a_{1,1}{\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\\\end{bmatrix}}&a_{1,2}{\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\\\end{bmatrix}}\\&\\a_{2,1}{\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\\\end{bmatrix}}&a_{2,2}{\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\\\end{bmatrix}}\\\end{bmatrix}}={\begin{bmatrix}a_{1,1}b_{1,1}&a_{1,1}b_{1,2}&a_{1,2}b_{1,1}&a_{1,2}b_{1,2}\\a_{1,1}b_{2,1}&a_{1,1}b_{2,2}&a_{1,2}b_{2,1}&a_{1,2}b_{2,2}\\a_{2,1}b_{1,1}&a_{2,1}b_{1,2}&a_{2,2}b_{1,1}&a_{2,2}b_{1,2}\\a_{2,1}b_{2,1}&a_{2,1}b_{2,2}&a_{2,2}b_{2,1}&a_{2,2}b_{2,2}\\\end{bmatrix}}.}$$
The resultant rank is at most 4, and thus the resultant dimension is 4. Here rank denotes the tensor rank (number of requisite indices), while the matrix rank counts the number of degrees of freedom  in the resulting array.
