In Sean Carroll's book, "Spacetime and Geometry", p. 17,
... the vector $V$ itself (as opposed to its components in some coordinate system) is invariant under Lorentz transformations. We can use this fact to derive the transformation properties of the basis vectors.
$V = V^\mu \hat{e}_{(\mu)} = V^{\nu'} \hat{e}_{(\nu')} = \Lambda^{\nu'}_{\,\,\,\, \mu} V^{\mu} \hat{e}_{(\nu')} (1.40)$
then the author introduces
$\hat{e}_{\mu} = \Lambda^{\nu'}_{\,\,\,\, \mu } \hat{e}_{(\nu')}. (1.41) $
From this answer, a spinor transforms differently (with a projective representation).
Does a spinor also have bases/components? If yes, is the projective representation for the transformation of components or bases or? Will the spinor itself (with basis and component together, if we chose a basis) be invariant under Lorentz transformations? Any reference would be helpful.
