I have a profound question. We call spinor an element of a space which transforms with the irreps with semi-integer $j$ of $SU(2)$. But i can't find the meaning of this. How could it transform in other ways? I mean, i am working with $SU(2)$ matrices so it MUST transfom with them. If i use a random matrix it would transform in other ways, so what's the meaning? A tensor is defined by how it transforms under some action of a group, but how could it transform in other ways if we apply only that group?
In 3D space we impose the distance to be not-modified, so we found the group of transformation is $SO(3)$. Why we say "vectors are quantities that transforms with $SO(3)$ matrices"? I can use a general matrix and it would be still a coordinate in the space.
For example Lorentz group, we impose the $s^2$ to rest the same, so we get transformations left-right spinors like $(1/2,0)$ or $(1,1)$ but what does this all mean?