Problem 1763 (difficulty: 4/10)

(a) Prove that all fractional linear transformations can be expressed as a composition of translations, rotations, dilations and conjugate inversion (inversion with respect to the unit circle followed by conjugation).

(b) Derive from this the basic properties of fractional linear transformations, they are bijective conformal maps of the Riemann sphere to itself that preserve the cross-ratio and circlines.

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