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1answer
11 views

Give an example of a relation on the set A with 4 elements which is relfexive, but not transitive.

let A = {1, 2, 3, 4, 5 ,6} be the set with 6 elements. I have worked out the relation R = {(1,1),(2,4),(4,2),(5,5)} i believe this is reflective because they are all elements of A but not not ...
3
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3answers
43 views

Should a reflection matrix of a vector have the same form as a rotation matrix?

According to the book: I know that it is not possible to write a reflection as a rotation, but from the text it seems that the matrix of the form $$ A=\left[ \begin{array}{ c c } a & ...
2
votes
1answer
55 views

Group Duality with respect to Generators and Relations

Although the following question is not phrased in the most accurate way, I would like to ask it in the same way it rushed to my mind: "Looking at some basic examples of group theory with geometrical ...
0
votes
1answer
25 views

Reflection of a vector across a line

For homework, I need to find the reflection of the vector <1,1,1> over the line defined by all the scalar multiples of <2,1,2>. I tried looking at the other questions about similar topics here, ...
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3answers
53 views

Every reflection is an isometry proof

The theorem is that every reflection $R_{S}$ in an affine subspace $S$ of $\mathbb{E}^{n}$ is an isometry: $R_S:\ \mathbb{E}^{n} \rightarrow \mathbb{E}^{n}:\ x \mapsto R_{S}(x) = x + 2 ...
0
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1answer
26 views

Reflecting a line from plane

I have a plane given by equation $0=10x+2y-3z$ and a line described by vector with variable $t\ge0$ $(1-t, 1+2t, 1+t)$ How to calculate reflected vector of this line from plane? We treat line as ...
0
votes
1answer
65 views

Constructing a 2x2 matrix R which represents reflection in the x-y plane,

Construct a 2x2 matrix R which represents reflection in the x-y plane through the line $$(cos(\theta)x+(sin(\theta)y=0$$, where $\theta$ is any real number. (Let's call this line "L".) Write an ...
5
votes
4answers
157 views

Reflections on a sphere

There is a sphere located in a point s with radius r. The Sphere is a perfect mirror. If i'm sitting in the point c, I want to cast a ray to the sphere such that I hit the point p after bouncing in ...
0
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3answers
36 views

Matrix with linear transformation with reflection

Find the matrix of the linear transformation A which is the reflection in the line $y = \sqrt{2}x$ with respect to the standard basis in $\mathbb{R^2}$. I Have no idea how to approach this problem... ...
0
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0answers
26 views

Householder reflector sign error

I am studying Householder reflectors from Trefethen and Bau but am having trouble creating a simple example for it. I am given the equation for the vector v that the Householder reflector H is based ...
2
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0answers
64 views

Proving two reflections across a line are commutative if and only if the two lines are perpendicular

Please note brackets () denote inner product - for some reason the traditional symbols for inner product do not show up when I type them in! Hi, I am trying to prove that the two reflections in ...
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0answers
37 views

ODE with Reflection

Some Background: I come from a computer science background and differential equations was not part of my course work. I'm still in contact with my professor that I ...
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0answers
20 views

Reflection to get within convex polygon

Let $P$ be a convex polygon, and let $A_1$ be a point on the same plane as $P$. Prove that we can find an integer $n$, and points $A_2,A_3,\ldots,A_n$, such that $A_{i+1}$ is a reflection of $A_i$ ...
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vote
2answers
64 views

Why isn't $f(x)=\sqrt{2-x}$ reflected across the y-axis?

If I try to graph this function, it does not appear to reflect across the y-axis when it comes time to do the reflection. Rather, it is reflected around the point where the function begins on the ...
0
votes
1answer
26 views

Simple systems are the “smallest”

Let $\Phi$ be a root system of a finite reflection group $W$. Let $\Delta$ be a simple system in $\Phi$. I want to prove that $\Delta$ is "the smallest" set which generates $W$, more precisely: There ...
0
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0answers
30 views

reflection(reflection) = rotation

Lel $\alpha$ and $\beta$ be two distinct simple roots in a root system $\Phi$. How to prove that i) $S_{\alpha} S_{\beta}$ is a rotation in $\mathbb{R}\Phi$ ii) Composition of two reflection is a ...
1
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0answers
39 views

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 = ...
1
vote
1answer
53 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
20 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 ...
0
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2answers
25 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'''
9
votes
5answers
345 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 .
2
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0answers
38 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 ...
1
vote
2answers
35 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 ...
0
votes
1answer
68 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} ...
0
votes
1answer
26 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 ...
1
vote
1answer
97 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 = ...
1
vote
1answer
58 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 = ...
0
votes
2answers
177 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 ...
1
vote
1answer
42 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 $2 \cdot 72 ^\circ$. ...
0
votes
3answers
61 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 ?
0
votes
3answers
26 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 $(x, 1-y)$ ?
0
votes
2answers
97 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 ...
2
votes
0answers
28 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 ...
1
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0answers
44 views
1
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1answer
39 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 ...
0
votes
2answers
92 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 ...
1
vote
1answer
51 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 ...
0
votes
1answer
19 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 ...
0
votes
2answers
230 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
1
vote
1answer
108 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 ...
2
votes
1answer
79 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 ...
4
votes
3answers
270 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 ...
0
votes
2answers
128 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 ...
1
vote
2answers
109 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 $180$ ...
4
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0answers
85 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 ...
1
vote
1answer
62 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 ...
0
votes
1answer
46 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 ...
-1
votes
3answers
119 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 ...
2
votes
0answers
52 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 ...
2
votes
1answer
446 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 ...