# Thread: Show the properties of σ

1. 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?

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 $\sigma_a$ ?

2.

3. Originally Posted by 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.
.

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?

Originally Posted by mathmari
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?

Originally Posted by Klaas van Aarsen
The symmetry group is the group of bijections from a set to the same set.
Is it a bijection?
Do we check that using the definitions? As we do that when we consider functions?

Originally Posted by Klaas van Aarsen
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?
The one will be the vector x and the other the vector v?

5. Originally Posted by mathmari
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?
And additionally it proves that $\sigma_x$ is invertible, which implies it is bijective, doesn't it?

Originally Posted by mathmari
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?

Which one is the vector $x$?

Originally Posted by Klaas van Aarsen
Suppose we verify that $\sigma_x(\sigma_x)=\operatorname{id}$.
Then that proves that $\sigma_x^{-1}=\sigma_x$ doesn't it?
And additionally it proves that $\sigma_x$ is invertible, which implies it is bijective, doesn't it?
Knowing that the map is invertible do we know that the map is bijective or injective?

Originally Posted by Klaas van Aarsen
Isn't $v$ the parameter to the function?
It's not a constant vector, is it?

Which one is the vector $x$?
I haven't understood that part. Could you explain it further to me?

7. Originally Posted by mathmari
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?

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

Originally Posted by mathmari
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.

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.

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

Originally Posted by Klaas van Aarsen
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.

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.

Is $x$ a normal vector to the line of reflection, or is it a directional vector along the line of reflection?
$x$ is a directional vector along the line of reflection, or not?

9. Originally Posted by mathmari
$x$ is a directional vector along the line of reflection, or not?
Do we have $\sigma_x(x)=x$?
What is $\sigma_x(x)$?

Originally Posted by Klaas van Aarsen
Do we have $\sigma_x(x)=x$?
What is $\sigma_x(x)$?
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?

11. Originally Posted by mathmari
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.