Finding a Vector Perpendicular to the Line of Reflection

[mathmari]

Hey!

Let $1\leq n\in \mathbb{N}$. For $0_{\mathbb{R}^n}\neq x\in \mathbb{R}^n$ we define the map $$\sigma_x:\mathbb{R}^n\rightarrow \mathbb{R}^n, \ v\mapsto v-2\frac{x\cdot v}{x\cdot x}x$$

Show that:

1. The map is linear.
2. It holds that $\sigma_x\in \text{Sym}(\mathbb{R}^n)$ and $\sigma_x=\sigma_x^{-1}$.
3. For all $v,w\in \mathbb{R}$ it holds that $v\cdot w=\sigma_x(v)\cdot \sigma_x(w)$.
4. Let $n=2$. Determine a vector $v\in \mathbb{R}^2$ such that $\sigma_v=\sigma_a$, where $\sigma_a$ is
   the reflection on the straight line through the origin, where $a$ describes the angle between the straight line and the positive $x$-axis.

I have done the following:

1. $\sigma_x$ is additive:
   \begin{align*}\sigma_x(v+w)&=(v+w)-2\frac{x\cdot (v+w)}{x\cdot x}x=(v+w)-2\frac{\sum_{i=1}^nx_i\cdot (v+w)_i}{x\cdot x}x\\ & =(v+w)-2\frac{\sum_{i=1}^nx_i\cdot v_i+\sum_{i=1}^nx_i\cdot w_i}{x\cdot x}x=(v+w)-2\left(\frac{\sum_{i=1}^nx_i\cdot v_i}{x\cdot x}+\frac{\sum_{i=1}^nx_i\cdot w_i}{x\cdot x}\right )x \\ & =(v+w)-2\frac{\sum_{i=1}^nx_i\cdot v_i}{x\cdot x}x-2\frac{\sum_{i=1}^nx_i\cdot w_i}{x\cdot x}x=v-2\frac{\sum_{i=1}^nx_i\cdot v_i}{x\cdot x}x+w-2\frac{\sum_{i=1}^nx_i\cdot w_i}{x\cdot x}x \\ & = \sigma_x(v)+\sigma_x(w)\end{align*}
   
   $\sigma_x$ is homogeneous:
   \begin{align*}\sigma_x(rv)&=(rv)-2\frac{x\cdot (rv)}{x\cdot x}x=rv-2\frac{\sum_{i=1}^nx_i\cdot (rv)_i}{x\cdot x}x\\ & =rv-r2\frac{\sum_{i=1}^nx_i\cdot v_i}{x\cdot x}x=r\left (v-2\frac{\sum_{i=1}^nx_i\cdot v_i}{x\cdot x}\right ) \\ & = r\sigma_x(v)\end{align*}
   
   Therefore the map $\sigma_x$ is linear.
2. Could you give me a hint how we can show that? (Wondering)
3. \begin{align*}\sigma_x(v)\cdot \sigma_x(w)&=\left (v-2\frac{x\cdot v}{x\cdot x}x\right )\cdot \left (w-2\frac{x\cdot w}{x\cdot x}x\right )=v\cdot w -2\frac{x\cdot w}{x\cdot x}v\cdot x-2\frac{x\cdot v}{x\cdot x}w\cdot x+4\frac{x\cdot v}{x\cdot x}\frac{x\cdot w}{x\cdot x}x\cdot x \\ & =v\cdot w -2\frac{(x\cdot w)(v\cdot x)}{x\cdot x}-2\frac{(x\cdot v)(w\cdot x)}{x\cdot x}+4\frac{(x\cdot v)(x\cdot w)}{x\cdot x}=v\cdot w -4\frac{(x\cdot w)(v\cdot x)}{x\cdot x}+4\frac{(x\cdot v)(x\cdot w)}{x\cdot x}\\ & =v\cdot w\end{align*}
4. How can we find such a vector? Do we use the matrix of the map of $\sigma_a$ ? (Wondering)

 

[I like Serena]

Hey!

Let $1\leq n\in \mathbb{N}$. For $0_{\mathbb{R}^n}\neq x\in \mathbb{R}^n$ we define the map $$\sigma_x:\mathbb{R}^n\rightarrow \mathbb{R}^n, \ v\mapsto v-2\frac{x\cdot v}{x\cdot x}x$$

Show that:

1. The map is linear.
2. It holds that $\sigma_x\in \text{Sym}(\mathbb{R}^n)$ and $\sigma_x=\sigma_x^{-1}$.
3. For all $v,w\in \mathbb{R}$ it holds that $v\cdot w=\sigma_x(v)\cdot \sigma_x(w)$.
4. Let $n=2$. Determine a vector $v\in \mathbb{R}^2$ such that $\sigma_v=\sigma_a$, where $\sigma_a$ is
   the reflection on the straight line through the origin, where $a$ describes the angle between the straight line and the positive $x$-axis.

.

2. Could you give me a hint how we can show that?

Hey mathmari!

The symmetry group is the group of bijections from a set to the same set.
Is it a bijection? (Wondering)

3. How can we find such a vector? Do we use the matrix of the map of $\sigma_a$ ?

Suppose $\sigma_x$ is a projection. What would be a vector along its line of reflection?
And what would be a vector perpendicular to its line of reflection? (Wondering)

 

[mathmari]

The symmetry group is the group of bijections from a set to the same set.
Is it a bijection? (Wondering)

Do we check that using the definitions? As we do that when we consider functions? (Wondering)

Suppose $\sigma_x$ is a projection. What would be a vector along its line of reflection?
And what would be a vector perpendicular to its line of reflection? (Wondering)

The one will be the vector x and the other the vector v? (Wondering)

 

[I like Serena]

Do we check that using the definitions? As we do that when we consider functions?

Suppose we verify that $\sigma_x(\sigma_x)=\operatorname{id}$.
Then that proves that $\sigma_x^{-1}=\sigma_x$ doesn't it? (Wondering)
And additionally it proves that $\sigma_x$ is invertible, which implies it is bijective, doesn't it? (Wondering)

The one will be the vector x and the other the vector v?

Isn't $v$ the parameter to the function?
It's not a constant vector, is it? (Worried)

Which one is the vector $x$? (Wondering)

 

[mathmari]

Suppose we verify that $\sigma_x(\sigma_x)=\operatorname{id}$.
Then that proves that $\sigma_x^{-1}=\sigma_x$ doesn't it? (Wondering)
And additionally it proves that $\sigma_x$ is invertible, which implies it is bijective, doesn't it? (Wondering)

Knowing that the map is invertible do we know that the map is bijective or injective? (Wondering)

Isn't $v$ the parameter to the function?
It's not a constant vector, is it? (Worried)

Which one is the vector $x$? (Wondering)

I haven't understood that part. Could you explain it further to me? (Wondering)

 

[I like Serena]

Knowing that the map is invertible do we know that the map is bijective or injective?

Do we have any propositions or some such saying so? (Wondering)

Or can we prove it?
That is, can we prove that $\sigma_x$ is both injective and surjective given that it is invertible? (Wondering)

I haven't understood that part. Could you explain it further to me?

We have the map $\sigma_x$ and it is given by a formula of the form $\sigma_x:v\mapsto \sigma_x(v)$.
In this formula $x$ is a fixed vector.
And $v$ is an arbitrary vector that is mapped to a vector in the image. In particular $v$ is not a fixed vector. (Nerd)

Assuming the $\sigma_x$ is a reflection in $\mathbb R^2$, there must be a vector $n$ normal to the line of reflection such that $\sigma_x(n)=-n$, mustn't it?
And there must be a directional vector $d$ along the line of reflection, such that $\sigma_x(d)=d$, which is a fixpoint. (Thinking)

Is $x$ a normal vector to the line of reflection, or is it a directional vector along the line of reflection? (Wondering)

 

[mathmari]

We have the map $\sigma_x$ and it is given by a formula of the form $\sigma_x:v\mapsto \sigma_x(v)$.
In this formula $x$ is a fixed vector.
And $v$ is an arbitrary vector that is mapped to a vector in the image. In particular $v$ is not a fixed vector. (Nerd)

Assuming the $\sigma_x$ is a reflection in $\mathbb R^2$, there must be a vector $n$ normal to the line of reflection such that $\sigma_x(n)=-n$, mustn't it?
And there must be a directional vector $d$ along the line of reflection, such that $\sigma_x(d)=d$, which is a fixpoint. (Thinking)

Is $x$ a normal vector to the line of reflection, or is it a directional vector along the line of reflection? (Wondering)

$x$ is a directional vector along the line of reflection, or not? (Wondering)

 

[I like Serena]

$x$ is a directional vector along the line of reflection, or not?

Do we have $\sigma_x(x)=x$?
What is $\sigma_x(x)$? (Wondering)

 

[mathmari]

Do we have $\sigma_x(x)=x$?
What is $\sigma_x(x)$? (Wondering)

We have that $\sigma_x(x)=x-2\frac{x\cdot x}{x\cdot x}x=x-2x=-x$.

That means that $x$ is a normal vector to the line of reflection, right? (Blush)

 

[I like Serena]

We have that $\sigma_x(x)=x-2\frac{x\cdot x}{x\cdot x}x=x-2x=-x$.

That means that $x$ is a normal vector to the line of reflection, right?

Yep. (Nod)

 

[mathmari]

Yep. (Nod)

Ok! Can we get a specific vector? Maybe using the matrix of map $\sigma_a$ ? (Wondering)

 

[I like Serena]

Ok! Can we get a specific vector? Maybe using the matrix of map $\sigma_a$ ? (Wondering)

The question only asks for a fixed vector.
We know now that we need a vector perpendicular to the line of reflection.
The line has an angle a with respect to the positive x-axis.
So $(\cos a,\,\sin a)$ is a directional vector along the line.
Can we find a vector perpendicular to it? (Wondering)