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Mathematic Methods HW Solutions 17

# Mathematic Methods HW Solutions 17 - Chapter 3 17 13.5 The...

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Chapter 3 17 13.5 The 4’s group 13.6 The cyclic group 13.7 The 4’s group 13.10 If R = 90 rotation, P = reflection through the y axis, and Q = PR, then the 8 matrices of the symmetry group of the square are: I = parenleftbigg 1 0 0 1 parenrightbigg , R = parenleftbigg 0 - 1 1 0 parenrightbigg , R 2 = parenleftbigg - 1 0 0 - 1 parenrightbigg = - I, R 3 = parenleftbigg 0 1 - 1 0 parenrightbigg = - R, P = parenleftbigg - 1 0 0 1 parenrightbigg , PR = parenleftbigg 0 1 1 0 parenrightbigg = Q, PR 2 = parenleftbigg 1 0 0 - 1 parenrightbigg = - P, PR 3 = parenleftbigg 0 - 1 - 1 0 parenrightbigg = - Q, with multiplication table: I R - I - R P Q - P - Q I I R - I - R P Q - P - Q R R - I - R I - Q P Q - P - I - I - R I R - P - Q P Q - R - R I R - I Q - P - Q P P P Q - P - Q I R - I - R Q Q - P - Q P - R I R - I - P - P - Q P Q - I - R I R - Q - Q P Q - P R - I - R I 13.11 The 4 matrices of the symmetry group of the rectangle are I = parenleftbigg 1 0 0 1 parenrightbigg , P = parenleftbigg - 1 0 0 1 parenrightbigg , parenleftbigg 1 0 0 - 1 parenrightbigg = - P, parenleftbigg - 1 0 0 - 1 parenrightbigg = - I This group is isomorphic to the 4’s group. 13.14 Class I ± R - I ± P ± Q Character 2 0 - 2 0 0 13.20 Not a group (no unit element) 13.21 SO(2) is Abelian; SO(3) is not Abelian. For Problems 2 to 10, we list a possible basis. 14.2 e x , xe x , e - x , or the three given functions
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