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

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 &#8477;
....
First Stage
Refer to the diagram provided.The line l makes some angle &#57534;with the x-axis.
Let us suppose that the point &#57502;c ,d &#57503;&#8712;l .(It is clear that,if we were given &#57534;,we could find such a
point &#57502;c ,d &#57503;,and if we were given &#57502;c ,d &#57503;,we could find &#57534;.)
Informally,we define :&#8477;2 &#57484;&#8477;2
to be the action of reflecting any point &#57496;x through l .
Now consider any arbitrary &#57496;x =&#57502;x 1,x 2 &#57503;.(Ignore &#57502;e 1,e 2 &#57503;in the diagram for now.)
1.Using the notation of projections,derive an expression for the vector &#57502;&#57496;x &#57503;.
You may suppose that &#57496;c is the vector starting at the origin and going to &#57502;c ,d &#57503;.
Do not attempt to evaluate it further at this stage;just leave things in terms of &#57496;c and &#57496;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 &#57496;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 &#8477;2
through the line going
through the origin and &#57502;c ,d &#57503;,for any c ,d &#8712;&#8477;.
Third Stage (sanity check Using your formula,let k &#8712;&#8477;and evaluate &#57502;&#57502;kc ,kd &#57503;&#57503;.
1.What do you get?
Fourth Stage
Consider a specific situation and draw a diagram of it:&#57496;x =&#57502;2,0 &#57503;,&#57502;c ,d &#57503;=&#57502;3,&#57485;&#57502;3 &#57503;&#57503;.
Take &#57502;e 1,e 2 &#57503;to be the point of intersection of l and the line orthogonal to l intersecting &#57496;x .
1.What is &#57534;?
2.What is &#57502;e 1,e 2 &#57503;?
You should notice something interesting about the triangle &#57517;&#57502;0,0 &#57503;,&#57502;2,0 &#57503;,&#57502;&#57502;2,0 &#57503;&#57503;.
3 Figure out &#57502;&#57502;2,0 &#57503;&#57503;using standard geometry and this observation.
4.Now apply your formula to determine &#57502;&#57502;2,0 &#57503;&#57503;.Do you get the same answer?
Fifth Stage
Now rotate your point &#57502;c ,d &#57503;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 :&#8477;
...
to be
the action of reflecting points through m .So g &#57502;&#57496;x &#57503;=B &#57496;x ,for some matrix B .
3.According to your formula from the Second Stage,what is B ?
Sixth Stage
Now we note that g &#57502;&#57502;&#57496;x &#57503;&#57503;=g &#57502;A &#57496;x &#57503;=B &#57502;A &#57496;x &#57503;=&#57502;BA &#57503;&#57496;x .
1.What is BA ?Is this the matrix for the half-turn linear operator?

#### 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.

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