4.4. LINEAR ISOMETRIES2294.3.3.Consider a planePthat does not pass through the origin. Let˛be a unit normal vector toPand letx0be a point onP. Find a formula(in terms of˛andx0) for the reflection of a pointxthroughP. Such areflection through a plane not passing through the origin is called anaffinereflection.4.3.4.Here is a method to determine all the products of the symmetries ofthe square tile. WriteJforJr, the reflection in the.x; y/–plane.(a)The eight products˛J, where˛runs through the set of eight ro-tation matrices of the square tile, are the eight nonrotation matri-ces. Which matrix corresponds to which nonrotation symmetry?(b)Show thatJcommutes with the eight rotation matrices; that is,J˛D˛Jfor all rotation matrices˛.(c)Check that the information from parts (a) and (b), together withthe multiplication table for the rotational symmetries, suffices tocompute all products of symmetries.
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