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    Linear Transformations : The composition of any two reflections, whose lines of reflection are orthogonal, is a half-turn.

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    Matrix Theory - Homework 7
    Prove the following in several stages:
    The composition of any two reflections,whose lines of reflection are orthogonal,is a half-turn.
    We will work in the vector space ℝ
    First Stage
    Refer to the diagram provided.The line l makes some angle with the x-axis.
    Let us suppose that the point c ,d ∈l .(It is clear that,if we were given ,we could find such a
    point c ,d ,and if we were given c ,d ,we could find .)
    Informally,we define :ℝ2 ℝ2
    to be the action of reflecting any point x through l .
    Now consider any arbitrary x =x 1,x 2 .(Ignore e 1,e 2 in the diagram for now.)
    1.Using the notation of projections,derive an expression for the vector x .
    You may suppose that c is the vector starting at the origin and going to c ,d .
    Do not attempt to evaluate it further at this stage;just leave things in terms of c and x .
    Second Stage
    1.Evaluate your expression further,using the formula for the projection of a point.Your final
    expression must be given in the form A x ,for some matrix A whose entries are in terms of c and d .
    Notice that you have constructed a formula for the action of reflecting ℝ2
    through the line going
    through the origin and c ,d ,for any c ,d ∈ℝ.
    Third Stage (sanity check Using your formula,let k ∈ℝand evaluate kc ,kd .
    1.What do you get?
    Fourth Stage
    Consider a specific situation and draw a diagram of it:x =2,0 ,c ,d =3,3 .
    Take e 1,e 2 to be the point of intersection of l and the line orthogonal to l intersecting x .
    1.What is ?
    2.What is e 1,e 2 ?
    You should notice something interesting about the triangle 0,0 ,2,0 ,2,0 .
    3 Figure out 2,0 using standard geometry and this observation.
    4.Now apply your formula to determine 2,0 .Do you get the same answer?
    Fifth Stage
    Now rotate your point c ,d ninety degrees counterclockwise.
    1.What coordinates does it have?
    2.Define the line m to be the line through this point and the origin.Define g :ℝ
    to be
    the action of reflecting points through m .So g x =B x ,for some matrix B .
    3.According to your formula from the Second Stage,what is B ?
    Sixth Stage
    Now we note that g x =g A x =B A x =BA x .
    1.What is BA ?Is this the matrix for the half-turn linear operator?

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    Solution Preview

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    First Stage

    1. Let . We know that . Hence,
    Also we know that . So,
    By (2), we have

    Second stage.

    From the first stage, we have


    Solution Summary

    That the composition of any two reflections, whose lines of reflection are orthogonal, is a half turn is proven. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.