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Unformatted text preview: MA 35100 LECTURE NOTES: WEDNESDAY, FEBRUARY 3 Reflection Say that we have a line L in the plane R 2 . We have seen that any vector ~x R 2 can be written in the form ~x = ~x || + ~x where ~x || is parallel to the line, and ~x is perpendicular to it. We consider the reflection of ~x through this line. Definition. Consider a line L : ax + by = c in the coordinate plane running through the origin. For a vector ~x R 2 , denote ~x || = proj L ( ~x ) and ~x = ~x- ~x || . The reflection of ~x through L s defined as the vector ref L ( ~x ) = ~x ||- ~x where ~x = ~x || + ~x . We state a few facts about this transformation. Theorem. Consider the transformation T ( ~x ) = ref L ( ~x ) (1) ref L is a linear transformation. (2) ref L ( ~x ) = 2( ~x ~u ) ~u- ~x . (3) The matrix of this transformation is in the form A = a b b- a where a 2 + b 2 = 1 . We explain why these statements are true. It is clear that the first follows once we show the third....
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