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College Algebra Exam Review 219

College Algebra Exam Review 219 - 4.4 LINEAR ISOMETRIES 229...

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4.4. LINEAR ISOMETRIES 229 4.3.3. Consider a plane P that does not pass through the origin. Let ˛ be a unit normal vector to P and let x 0 be a point on P . Find a formula (in terms of ˛ and x 0 ) for the reflection of a point x through P . Such a reflection through a plane not passing through the origin is called an affine reflection . 4.3.4. Here is a method to determine all the products of the symmetries of the square tile. Write J for J r , 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 that J commutes with the eight rotation matrices; that is, D ˛J for all rotation matrices ˛ . (c) Check that the information from parts (a) and (b), together with the multiplication table for the rotational symmetries, suffices to compute all products of symmetries.
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