Transcription of Orthogonal Transformations - U-M LSA
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Orthogonal TransformationsMath 217 Professor Karen Smith(c)2015 UM Math Deptlicensed under a Creative CommonsBy-NC-SA International :A linear transformationRnT Rnisorthogonalif|T(~x)|=|~x|for all~x :IfRnT Rnis Orthogonal , then~x ~y=T~x T~yfor all vectors~xand~ Last time you proved:1. An Orthogonal transformation is an The inverse of an Orthogonal transformation is also The composition of Orthogonal Transformations is with your table thegeometric intuitionof each of these statements. Why do they makesense, for example, SupposeT:R2 R2is given by left multiplication byA=[a cb d].AssumingTisorthogonal,illustrate an example of such aTby showing what it does to a unit square. Label your picturewitha,b,c,d. What do you think detAcan be?
so x= 53=5. G. TRUE OR FALSE. Justify. In all problems, Tdenotes a linear transformation from Rn to itself, and Ais its matrix in the standard basis. 1. If Tis orthogonal, then xy= TxTyfor all vectors xand yin Rn. 2. If T sends every pair of orthogonal vectors to another pair of orthogonal vectors, then T is
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