# 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 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?

https://brainmass.com/math/linear-transformation/linear-transformations-composition-two-reflections-101524

#### Solution Preview

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First Stage

1. Let . We know that . Hence,

........................(1)

Also we know that . So,

.........................(2)

By (2), we have

..............(3)

Second stage.

From the first stage, we have

i.e.,

...

#### 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.