Where does a tensor/spinor live in? I learned that spinors are elements of $\mathbb C^2$ when it is regarded as a representation space for $SU(2)$.
The "when it is regarded as" part makes me wonder: where do the spinors live, really? Do they live in $\mathbb C^2$? But then suppose the $SU(2)$ disappears, would that magically make the spinors turn into vectors? How can the nature of a mathematical object depend on something external?
The problem comes with trying to understand tensors as well. A rank-2 tensor on $\mathbb R^3$ is an element of $\mathbb R^9$ regarded as a representation space for $SO(3)$. But why can't 2-tensors suddenly turn into 1-tensors (aka plain vectors), if we suddenly use the $\mathbb R^9$ to represent $SO(9)$?
 A: When we say that something is a vector/tensor/spinor, etc., the meaning of this sentence depends on the context, more precisely, on some external structure. In fact, generally speaking the only  way to make sense of the statement " is a vector" is to specify a vector space that $v$ belongs to. For instance, believe it or not, your shoes are vectors, more precisely, your pair of shoes forms a 1-dimensional vector space over the field ${\mathbb Z}_2$ of two elements, where, say, the left shoe is the zero vector and the right shoe is the nonzero vector. Given this information, there is only one way to prescribe the vector addition of your shoes and the multiplication of your shoes by scalars, elements of ${\mathbb Z}_2$. But, once this external structure is forgotten, your shoes turn into what they were before, just shoes.
The same is with, say spinors (more precisely, $Spin(3)$-spinors). The only way to say that something (say, a pair of complex numbers $(z,w)$, or something more exotic) is a $Spin(3)$-spinor is to give the vector space ${\mathbb C}^2$ that $(z,w)$ belongs to an extra structure, given by a faithful linear representation of the group $Spin(3)$, usually given by an isomorphism  $Spin(3)\to SU(2)$, where the group $SU(2)$ acts on ${\mathbb C}^2$ in the usual manner. Once this extra structure is forgotten then $(z,w)$ turns into what it was before: At best, a vector in the vector space ${\mathbb C}^2$. But, if we choose to forget the fact that the set of pairs of complex numbers has structure of a (complex) vector space, then $(z,w)$ is not even a vector anymore. It turns into, just a pair of complex numbers. (Just like your pair of shoes in the earlier example.)
A similar story is with tensors...
Hopefully, it makes sense. It took mathematicians a long time to come to understand these things. David Hilbert was one of the first (if not the first) at the turn of the last century.
