Axiomatic Geometry - Mathematics 360 - Spring 2009

For Homework 13

1. Prove symmetry is invariant under Mobius transformations; first give a rephrasing of this statement that is precise. In particular, are we talking about an invariant set or an invariant function? If an invariant set, what are the figures in it? If an invariant function, what is the function, and what is its domain?
   The book proves that symmetry is invariant under Mobius transformations. If you've already read the proof before attempting this problem, make sure you understand and explain every step, and can reproduce the proof without looking at the book.
2. Given a Mobius transformation Tz = (az + b) / (cz + d), let A be the following 2x2 matrix (pretend I'm using "large parentheses" below!)
   (a b)
   (c d)
   We say "A is a matrix for T". (We don't say A is the matrix for T because there are infinitely many matrices for T; can you see why?)
   Prove that if A is a matrix for T, then its inverse A^(-1) is a matrix for T^(-1).
   Hint: use the fact that if A and B are matrices for transformations T and U respectively, then the product of the matrices is a matrix for the composition of the transformations (if done in the right order).