3320hw4sol

3320hw4sol - Math 3320 Problem Set 4 Solutions 1 1. Find a...

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Unformatted text preview: Math 3320 Problem Set 4 Solutions 1 1. Find a formula for the linear fractional transformation that rotates the triangle ( / 1 , 1 / 2 , 1 / 1 ) to ( 1 / 1 , / 1 , 1 / 2 ) . Solution : A rotation preserves orientation hence has determinant + 1, so we want to use only matrices of determinant + 1 if possible. A transformation of determinant + 1 that takes the edge ( / 1 , 1 / 2 ) to ( 1 / 1 , / 1 ) is parenleftBigg 1 1 1 parenrightBigg parenleftBigg 1 − 1 2 parenrightBigg − 1 = parenleftBigg 1 1 1 parenrightBigg parenleftBigg 2 − 1 1 parenrightBigg = parenleftBigg 2 − 1 3 − 1 parenrightBigg Here we inserted a minus sign in the first column of the second matrix in order to make its determinant + 1. One can check that the final matrix takes the third vertex of ( / 1 , 1 / 2 , 1 / 1 ) to the third vertex of ( 1 / 1 , / 1 , 1 / 2 ) , but this check isn’t really necessary since a transformation of determinant + 1 is uniquely determined by where it sends an edge. (An alternative approach would be to change the sign of one column of the first matrix rather than the second matrix, so that the final matrix would again have determinant + 1.) 2. Find the linear fractional transformation that reflects the Farey diagram across the edge ( 1 / 2 , 1 / 3 ) (so in particular, the transformation takes 1 / 2 to 1 / 2 and 1 / 3 to 1 / 3)....
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This note was uploaded on 11/09/2009 for the course MATH 332 at Cornell.

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3320hw4sol - Math 3320 Problem Set 4 Solutions 1 1. Find a...

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