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# hw3sol - Homework 3 Solutions 1 Prove that the set of all...

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Homework 3 Solutions 1. Prove that the set of all collineations in Trans ( R n ) is a subgroup of Trans ( R n ). Solution: We need to show that the collineations satisfy the identity, inverse, and clo- sure properties (under function composition). It is immediate from the definition that ι is a collineation. In Problem 2 of Homework 2 it was proved that if α Trans ( R n ) is a collineation, then α - 1 is a collineation, so the inverse property is satisfied. Finally, if α, β are collinations, then let be a line and set 0 = β ( ) and 00 = α ( 0 ). Now 0 is a line since β is a collineation and then 00 is a line since α is a collineation, thus we find that α β ( ) = α ( β ( )) = α ( 0 ) = 00 . Thus, the image of under αβ is a line and since was arbitrary, we find that αβ is a collineation. 2. Find an involution in Trans ( R 2 ) which is not the identity, a reflection, or a rotation. Solution: The function f : R 2 R 2 given by f ( x, y ) = ( x, y ) if ( x, y ) 6 = (0 , 0) , (1 , 1) (1 , 1) if ( x, y ) = (0 , 0) (0 , 0) if ( x, y ) = (1 , 1) is one of many solutions.

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