Home_7 - Homework 7 Stephen Taylor June 4 2005 6 Evaluate...

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Homework 7 Stephen Taylor June 4, 2005 Pages 54 - 57 : 6. Evaluate the following cross ratios. We first note that the cross ratio is defined to be Definition 1. If z 1 C then ( z 1 , z 2 , z 3 , z 4 ) is the image of z 1 under the M¨ obius transformation which takes z 2 1, z 3 0, z 4 → ∞ where the transformation is given by: S ( z ) = z - z 3 z - z 4 · z 2 - z 4 z 2 - z 3 (c) (0 , 1 , i, - 1) S (0) = 0 - i 0 + 1 · 1 + 1 1 - i = 1 + i (d) ( i - 1 , , 1 + i, 0) S ( i ) = i - 1 - (1 + i ) i - 1 - 0 · ∞ - 0 ∞ - (1 + i ) = - 2 i - 1 = 1 + i 8. If T ( z ) = az + b cz + d show that T ( R ) = R iff we can choose a, b, c, d to be real numbers. ( T ( R ) = R ) Let a = d and b = c = 0. Then T is the identity function which is an automorphism of the extended real numbers. Hence we have chosen the desired a, b, c, d R ( a, b, c, d R ) Since a, b, c, d are real and T acts on the extended real numbers, we note that T is a real function. So if suffices to show that T is an onto function. Or for every x R there exists a y in R such that T ( y ) = x .
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