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Math · Advanced Math
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a Given a matrix A where a, b, c, d are compler numbers it c d acts on all of C via az b GA(z) cz d 1. Let D 0 1/2 Show that the map GD maps the unit circle EC: Izl 1) to the circle of radius 4, Hz EC: Iz 4) 1 w where w E C is arbitrary. Show that the image 2. Let U of the unit circle 12 EC: Izl 1) is the circle of radius 1 centered at w, that is, the collection of points (2 EC: Iz -wl 1Y 3. What is the image of the unit circle tz EC: Iz 1) under the map Guy o GD? 4. Show that the hyperbolic circle centered at i of radius t is a Euclidean circle centered at icosh t of radius sinht, that is, to verify the identity et i cos 0 sin 0 i cosh t sinh t et i sin 0 cos 0 5. Show that under the Mobius transform, the inversion matrix 10 takes C(2,r), the circle centered at z and of radius r, to the circle C( TETEFI). We assume 2 ER and 2 f tr Something to think about, what happens if z r or -r?

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