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Do involutions suffice to find reflected vectors in a reflection group representation?

Consider a reflection group $W$ acting by isometries on a Euclidean space $V$. I want to understand the union of $(-1)$-eigenspaces for this action, the set $$\{v \in V : \exists w \in W\ (w\cdot v = ...
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1answer
37 views

Reflections - Understanding

I have been asked Does the set of all reflections in the plane form a group? Explain. I thought that the answer was that it wouldn't as if I reflected the parabola $y=x^2$ through the y axis, I ...
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0answers
13 views

Complete Triangle Given 3 Parallel Planes and 2 Points

I have a problem where a point B connects to a point C at a known angle and distance. Both point B and C are on two separate parallel axis, GH and JK respectively. I need to find a third point, A, on ...
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2answers
21 views

Reflecting and Translating Points

Point R (-4,5) is reflected about the x-axis onto point R'. It is then reflected about the line y=-x to find R''. It is then translated using the vector <2,-2> to find R'''. Find R'''
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5answers
283 views

Tangent and angle bisectors [closed]

The tangent to the incircle of a triangle ABC is reflected about the external angle bisectors. Show that the triangle formed by the resulting 3 lines is congruent to ABC .
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0answers
31 views

Fixed point set of orthogonal Transformation

I need some help with this problem. Let $g$ be an element of the orthogonal group and $s$ a reflection. Then the dimension of the fixed point set of $g$ and $gs$ differ by $\pm 1$. Since that ...
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2answers
28 views

Prove that this transformation is a reflection

Let $ v \in V$ be a unit vector in a Euclidean vector space. Prove that the endomorphism $$\phi: V \to V, \qquad \phi(x)=x - 2<x,v> v$$ is a reflection. I know that a reflection is an ...
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1answer
28 views

The composition of two different glide reflections is a rotation

Denote by $G_{XY}$ a glide reflection which reflects around the $XY$-axis and then takes the point $X$ to $Y$. I would like to prove that the composition of two different glide reflections $G_{XY} ...
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1answer
20 views

Order of rotation

For $\alpha \in \mathbb{R}^n$, let $s_\alpha$ be the reflection in $\alpha$, i.e. $s_\alpha$ sends $\alpha$ to $-\alpha$ and fixes pointwise the hyperplane orthogonal to $\alpha$. Then if ...
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1answer
86 views

angles between simple roots are obtuse, problem with proof

Let $\Phi$ be a root system in the following sense: (1) $\Phi \subset \mathbb{R}^n$ consists of a finite number of nonzero vectors, (2) for each $\alpha \in \Phi$, $\Phi \cap \mathbb{R} \alpha = ...
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1answer
44 views

Angle between roots in a root system

Let $\Phi$ be a root system in the following sense: (1) $\Phi \subset \mathbb{R}^n$ consists of a finite number of nonzero vectors, (2) for each $\alpha \in \Phi$, $\Phi \cap \mathbb{R} \alpha = ...
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2answers
151 views

Reflection in a plane.

What is the exact definition of a reflection through the plane $a.r=0$ for a given vector a and $r=(x,y,z)$. Of course I know what it is but I don't know what's part of its definition and what's part ...
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1answer
24 views

Rotation-Reflection-Symmetries

I have the following exercise: Do a rotation of $2 \cdot 72 ^\circ$. Then do a reflection of the axis $d4$. Then do a reflection of the axis $d3$. Then do a rotation of $2 \cdot 72 ^\circ$. ...
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3answers
59 views

What is this and why is it so important $(x,y) \to (y,x)$?

When doing a $y=x$ reflection the notation is $(x,y) \to (y,x)$. My teacher told us to find out what it is, what it is called, and why it is important? Please help ?
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3answers
21 views

Reflect the plane in the $x$-axis, and then in the line $y = \frac12$. Show that the resulting isometry sends $(x,y)$ to $(x,y+1)$

I have a hard time proving this without using any numbers. How do I show that the point $(x,y)$ reflected across $y=\frac12$ is $(x, 1-y)$ ?
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2answers
60 views

Image of a point reflected over $y=mx+b$ using dot product

So, I know that the image for a generic point is $$\small((1-m^2/1+m^2)x + (2m/1+m^2)(y-b), (2m/1+m^2)x - (1-m^2/1+m^2)(y-b)+b))$$ when you reflect it over the line $y=mx+b$. It's straightforward ...
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0answers
23 views

Is there a Focal Point/Area/Line of a Parabola for not perpendicular Lines

I'm not sure if this is mathematical enough for this forum, since it's my first post, but please don't be too harsh! So my question is: If the incoming lines of a Parabola come in perpendicular to ...
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1answer
28 views

Conformal Mapping with homeomorphic extension

Suppose that $$D=\{z:0<x<a,0<y<b\}$$ and that $$D'=\{w:0<u<c,0,v<d\}$$ Then there is a conformal mapping $f$ of $D$ onto $D'$ whose homeomorphic extension $\tilde{f}$ to ...
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2answers
67 views

Why does reflecting a point (x,y) about y=x result in point (y,x)?

I noticed that whenever reflecting a point (x,y) about the line y=x the x and y coordinates become swapped in order to give (y,x). However, I do not know why this is the case. Is there any way to ...
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1answer
45 views

Hermitian matrix the only diagonizable

During the last lecture one of my professors claimed that the hermitian matrix is the ONLY complex matrix which was diagonizable. This seems strange to mee (not to say a very very strong claim to ...
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1answer
18 views

Common elements Job - Groups and Algebras Lie

I need some help to prepare my final master thesis. It is difficult for me, because I want to use things/informations/notes/concepts which I usually use in my job. The maths concepts which I need ...
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2answers
113 views

How to find plane of reflection from transformation matrix

If you have an orthogonal matrix with a determinant of -1, how do you determine the plane of reflection? Thanks
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1answer
73 views

Composition of orthogonal projection

Given $\gamma: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ (rotation around $o$) and $\sigma: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ (reflection in one of the lines through the origin), I have to show that ...
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1answer
74 views

Harmonic function reflection

I'm learning some harmonic function theory by reviewing some problems. I came across two: 1) Prove that a real harmonic function $u$ from $\mathbb{R}^n$ to $\mathbb{R}$ such that $u(x, 0) = 0$ for ...
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3answers
143 views

Why are orthogonal matrices generalizations of rotations and reflections?

I recently took linear algebra course, all the I learned about orthogonal matrix is that matrices is that Q transposed is Q inverse, and therefore it has a nice computational property. Recently, to my ...
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2answers
100 views

Reflection question

This is a practice question to a test I will be taking soon. My conjecture is that it's none of the choices given. I tried reflections about y=x, y-axis, x-axis and it doesn't work. Does anyone ...
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2answers
93 views

Rotations/Reflections of a point

It's been $10$ years since I see some kind of geometry and I'm preparing for a test of these sort of questions. I need help figuring out the following problems: 1) A point $P(x,y)$ is rotated $180$ ...
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0answers
82 views

A light beam enters a closed room. What is the maximal number of reflections?

I have the following problem: a light beam enters a mirror room with integer coordinates in the plane (consider it as a polygon). One of the walls of the room is removed and the light beam enters the ...
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1answer
49 views

point deflecting off of a circle

I know that this is a very simple question, but I am stuck at the very last part of this process and can't find the solution elsewhere (I figured I'd find it on this site, but I didn't see it). I ...
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1answer
41 views

Show that every element on $O(\mathbb{R} ^2)$ is either a rotation or reflection

Where $O(\mathbb{R} ^2)$ is the orthogonal group of $\mathbb{R} ^2$ or; The set of all linear maps $g: \mathbb{R} ^2 \rightarrow \mathbb{R} ^2$ represented by an $n \times n$ matrix $M$ w.r.t. the ...
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3answers
101 views

Show that $(A',B',C')$ form the vertices of an equilateral triangle.

Let $ABC$ be a triangle with $AB = AC $ and $angle BAC = 30.$ Let $(A')$ be the reflection of A in the line BC $(B')$ be the reflection of $B$ in the line CA $(C')$ be the reflection of C in the line ...
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0answers
49 views

virtual cell mirror reflections

Please note that this is an extension of this question: Mirror reflection Questions Mirrors A and B with the space between them, including the bird, are part of the real world. Those are all a part ...
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1answer
375 views

Reflection Matrix linear algebra

I am practicing some linear algebra question to prepare for my test. I have come across one question that has given me much trouble. It states: If $\lVert u\rVert = 1$, then $Q = I - 2uu^T$ is a ...
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1answer
23 views

Why does knowing where two adjacent vertices of regular $n$-gon move under rigid motion determines the motion?

I am reading the book Abstract Algebra by Dummit and Foote. In the section about the group $D_{2n}$ (of order $2n$) the authors claim that knowing where two adjacent vertices move to, completely ...
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3answers
43 views

Why $Hx=x-(\rho u^Tx)u$?

A Householder reflection is a matrix of the form $$H=I-\rho uu^T$$ with $\rho=2/\|u\|^2$. Obviously, $Hx=x-\rho uu^Tx$. Textbook http://www.mathworks.se/moler/leastsquares.pdf says that ...
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2answers
157 views

Expressing an isometry [closed]

Let $s$ denote reflection of the plane about the vertical axis $x=1$. Also let $r$ dentoe the reflection with respect to the horizontal component of the basis in $\mathbb R^2$. Find an isometry ...
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1answer
26 views

Linear Algebra, reflected linear image

If I have a linear image of the room where v1 and v2 is an image of theirselves and v3 is an image of the null vector. If that gives me the matrix A=(a, b, c; d, e, f; g, h, i;) then A^n = A because ...
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2answers
96 views

Prove linear operator is a reflection

Prove that a linear operator on $\mathbb{R}^2$ is a reflection if and only if its eigenvalues are $1$ and $-1$ and the eigenvectors with these eigenvalues are orthogonal. $\Rightarrow$: Let $r: ...
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1answer
47 views

Reflection along subspace

A symmetric space is a Riemannian manifold M with the following property: For every point $p \in M$ there is an isometry $\phi: M \rightarrow M$ such that $\phi(p) = p$ and $\phi_*(v_p) = -v_p \in ...
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1answer
314 views

About the use of Schwarz reflection principle in the proof of the mapping formular between the upper half plane to a given polygon

In Chap 8 of Stein's complex analysis, he proved that all the conformal maps $f$ from the upper half plane $\mathbb{H}$ to a given polygon $P$ is of the form $c_1S(z)+c_2$, where $c_1, c_2$ are ...
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3answers
160 views

Two Parabolic Mirrors Opposite of Each other

Suppose we have two parabolic mirrors opposite of each other (e.g. $x=y^2$ and $x= -y^2+10$). Also suppose the first mirror is smaller than the second mirror. If a light ray enters into the opening ...
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2answers
735 views

Finding reflection transformation matrix

I have two 3 dimensional points. $A [x_1, y_1, z_1]$ and $B [x_2, y_2, z_2]$. I need to find a transformation matrix which when multiplied to $A$ will give me $B$ and when multiplied by $B$ give me ...
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2answers
721 views

The Schwarz Reflection Principle for a circle

I'm working on the following exercise (not homework) from Ahlfors' text: " If $f(z)$ is analytic in $|z| \leq 1$ and satisfies $|f| = 1$ on $|z| = 1$, show that $f(z)$ is rational." I already know ...
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2answers
66 views

How to find what point a wave is reflected off

If a wave is reflected off a surface, the angle of reflection is equal to the angle of incidence. But, how can we use this to find the actual path of the incident and reflected waves if we only know ...
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3answers
132 views

Reflecting a golfball off a wall to a hole and compensating for the balls radius

Problem: I'm struggling to compensate for the radius of a ball when reflecting it off a wall towards a target. (sorry I cannot yet post images) What I want is to do this: golf reflections but this ...
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1answer
424 views

Using the three reflection theorem

There is this sample question from my book that I dont know how to go about. please help out Use the three reflections theorem to show that the only transformations of the Euclidean plane are ...
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1answer
113 views

Householder reflections

Let $x=\begin{bmatrix} 1\\ 2\\ 3 \end{bmatrix}$ I want to use a Householder reflector U to keep only first element in vector x, and make everything else zero but I'm doing something wrong... ...
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2answers
270 views

Reflecting a point by a line in $\mathbb R^3$

I would like to know if it's possible, given the vector equation of a line and the coordinates of a point, whether it's possible to reflect the point by the line.
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0answers
39 views

Invariants of parabolic subgroups

Let $G$ be a finite reflection group acting on $R^n$. Then for each point $x\in R^n$ we can look at its stabilizer $G_x$. Since $G$ is a finite reflection group, its ring of invariants is a polynomial ...