Very soft question and I may be completely wrong about this, but does it make any sense to think about the Möbius transformation matrix as a change of basis for $\mathbb C$?
2026-03-28 02:59:41.1774666781
Soft Question about Möbius Transformations
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No, the Möbius transformation is not a linear transformation of $\Bbb{C}$ even if the associated matrix might have other action on the space. Hence, the Möbius transformation matrix should not be thought of as a basis change for $\Bbb{C}$.