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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.