sol1 - * g and g * f ? More generally, for f = ( m 1 ,b 1 )...

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Algebra , Assignment 1 Course website (bookmark!): www.cs.nyu.edu/cs/faculty/spencer/algebra/index.html 1. List with description the symmetries of the square. (There are eight of them.) Give the table for the products of the symmetries. Give the inverse of each symmetry. Solution: (Please save this as this will be a standard example. We will stick to the letters below.) The symmetries are I,R,S,T,V,H,D,A . (In words: do nothing, rotate π/ 2, rotate π , rotate 3 π/ 2, Fip vertical, Fip horozontal, Fip diagonal, Fip other diagonal.) The table is * I R S T V H D A I I R S T V H D A R R S T I D A H V S S T I R H V A D T T I R S A D V H V V A H D I S T R H H D V A S I R T D D V A H R T I S A A H D V T R S I All elements are their own inverse except R - 1 = T , T - 1 = S . 2. Let G denote the set of linear functions f ( x ) = mx + b on the real line with m n = 0. Denote such a function by ( m,b ). De±ne a product f * g as the function h ( x ) = g ( f ( x )). ²or f ( x ) = 3 x +7 and g ( x ) = 5 x 2 what are f
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Unformatted text preview: * g and g * f ? More generally, for f = ( m 1 ,b 1 ) and g = ( m 2 ,b 2 ) nd f * g and g * f . or f = ( m,b ) nd a formula for that g so that f * g = (1 , 0). Solution. g ( f ( x )) = g (3 x + 7) = 5(3 x + 7) 2 = 15 x + 33 f ( g ( x )) = f (5 x 2) = 3(5 x 2) + 7 = 15 x + 1 so f * g = (15 , 33) and g * f = (15 , 1). More generally g ( f ( x )) = g ( m 1 x + b 1 ) = m 2 ( m 1 x + b 1 ) + b 2 = m 2 m 1 x + m 2 b 1 + b 2 so f * g = ( m 1 m 2 ,m 2 b 1 + b 2 ) and, similarly, g * f = ( m 1 m 2 ,m 1 b 2 + b 1 ). Setting f * g = (1 , 0) and solving for m 2 ,b 2 we get rst m 2 = 1 m 1 and then b 2 = b 1 m 1 . We can check that setting g * f = (1 , 0) we get the same values for m 2 ,b 2 ....
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This note was uploaded on 04/01/2011 for the course MATH 101 taught by Professor Josephs during the Fall '08 term at NYU.

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