Reflection is a transformation that fixes a line or plane or a more general subset. Reflections appear in geometry, linear algebra, complex analysis, differential equations, etc -- therefore, this tag must be used with a tag describing the area of mathematics.

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Reflection principle for the modulus of the Brownian Motion

I have the following question. Suppose we define $M(t)=\sup_{0\le s\le t}|B(s)|$, where $B$ is an ordinary Brownian motion in $\mathbb{R}$. How can we compute $P(M(t)\ge a)$? Is it $2P(|B(t)|\ge a)$? ...
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Clarification on a reflexive function

$R_1 = \{(a, b) | a ≤ b\}$ is a reflexive function, but I'm confused on why it is. $a≤b$ but doesn't that not necessarily mean that there is an $a$ that will equal $b$? Couldn't all of the $a$ very ...
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The expression for reflection of a ray line $ax+by+c=0$ reflected by a mirror whose normal is given by $a'x+b'y+c'=0$.

Using vectors I tried obtain the expression for reflection of a ray line $ax+by+c=0$ reflected by a mirror whose normal is given by $a'x+b'y+c'=0$. The point of intersection is ...
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Reflection Principle in multi-variable complex functions

I was wondering if the reflection principle: $\bar{f(x)}=f(\bar{x})$, covers also the multivariable case, e.g: $\bar{F(z,w)}=F(\bar{z},\bar{w})$ and if yes, to which extent. More presicely, in the ...
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Find a vector V that is parallel to the reflection of a matrix about a line.

Specific Question: Let $M=(1/53)⋅[[45, 28], [28,-45]]$. M is the matrix of the reflection about a line. Find a vector v of magnitude (53)^0.5 parallel to that line. Currently in class we are ...
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38 views

Reflect point across line with matrix

What is the transformation matrix that I multiply a point by if I want to reflect that point across a line that goes through the origin in terms of the angle between the line and the x-axis? In other ...
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Inverse of a reflection matrix?

What does it mean when I say that the inverse of a reflection matrix is the reflection matrix itself? What does it mean, intuitively or geometrically, to invert a reflection matrix? Explanation with ...
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Relation between reflection group and coxeter group

Reflection group is defined see https://en.wikipedia.org/wiki/Reflection_group. An abstract Coxter group is defined to have generators $s_1$, $s_2$, ..., $s_n$ and relations $s^2_i=e$, ...
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Composition of two reflections (non-parallel lines) is a rotation

I am trying to prove that the composition of two reflections in non-parallel lines (i.e. lines that intersect) is a rotation. From observation I can see that using $L_1$ as the $x$-axis and $L_2$ as ...
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Determining whether an orthogonal matrix represents a rotation or reflection

The exercise asks us to determine whether the given orthogonal matrix represents a rotation or a reflection. If it is a rotation, give the angle of rotation; if it is a reflection, give the line of ...
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Give the line of reflection or angle of rotation of an orthogonal 2x2 matrix

Determine whether the given orthogonal matrix represents a rotation or a reflection. If it is a rotation, give the angle of rotation; if it is a reflection, give the line of reflection. ...
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Group generated by two reflections order 2

I have two reflections $r$, $s$ with $r^{2}=s^{2}=e$. Also $rs \neq sr$ and $(rs)^4 = (sr)^4= e$. I know $r$ and $s$ generate a group of order $16$ but that was by writing each matrix multiplication ...
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Isometric transformation

If $S$ belongs to the line $p$, prove that each of the isometrics $\rho_{S,\omega}$ ◦ $ \sigma_p$ and $ \sigma_p$ ◦ $\rho_{S,\omega}$ should be axial reflection. Can anyone please help me with this ...
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Realizing a reflection group as a Weyl group

Question 1. Suppose given a compact, connected Lie group $G$ and a subtorus $S$ (not maximal) such that the effective image $N$ of the $\mathrm{Ad}$-action of the normalizer $N_G(S)$ on the Lie ...
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Method of determining symmetries in an irregular polygon (2D or 3D)?

Thank you in advance for helping. Given a polygon with $n$ vertices, $$P = \begin{bmatrix} x_{1} & x_{2} & ... & x_{n} \\ y_{1} & y_{2} & ... & y_{n} \end{bmatrix}$$ how does ...
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Characteristics of $\vec u$ in equation of the reflection of $\vec x$ about the line $N$

The linear transformation of the reflection of $\vec x$ about line $N$ is $$\vec x = 2 \text{proj}_N(\vec x)\vec x - \vec x= 2(\vec x \cdot \vec u) \vec x- \vec x.$$ Is the unit vector, $\vec u$, ...
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Range of a standard brownian motion, using reflection principle

With a standard brownian motion $B_t$, I'm trying to find the distribution of the "range": $$R_{t} = \sup_{0 \leq s \leq t} B_s - \inf_{0 \leq s \leq t} B_s = \overline{M_t}-\underline{M_t}$$ The ...
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How to determine the reflection point on an ellipse

Here is my problem. There are two points P and Q outside an ellipse, where the coordinates of the P and Q are known. The shape of the ellipse is also known. A ray comming from point A is reflected by ...
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Reflecting coordinates over the line $x = -1$

I know how to reflect a coordinate over the $y$ and $x$ axis, but is there a rule I could use to help me find the reflected point over $x = -1$? This is what I know already: Over the $x$-axis: ...
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Finding the glide reflection using a compass or straightedge

Given two congruent triangles that are not a rotation, translation or reflection of each other; how can I find the glide reflection (the last remaining option) using only compass and straightedge. ...
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Possible eigenvalues of projection and reflection operators

What are the eigenvalues of an (orthogonal) projection operator that projects vectors onto some hyperplane passing through the origin? Well, for vectors $v$ orthogonal to this hyperplane, the ...
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Given $a,b \in S^n$, then there exists an isometry $f: S^n \rightarrow S^n$ such that $f(a) = b$

$S^n = \{x\in \mathbb{R}^{n+1} : \|x\| = 1\}.$ I am using this definition: isometry is a surjective function $f:M \rightarrow N$ between two metric spaces $(M,d)$ and $(N,\rho)$ such that $$\rho ...
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Isometries of reflections

Suppose $R_\rho \in M_2$ denotes a reflection across a line which is through the origin and at a anticlockwise angle of $\rho$ with the x-axis. The question asks that for any $n \in \mathbb{N}$, does ...
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Find Normal from Reflection

I have a vector A with direction $\vec D$. I also have another vector B with direction $\vec R$. (think of $\vec R$ to be the reflection of $\vec D$ at point B.) From the reflection formula:$$ \vec ...
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Linear algebra reflection or rotation

I have a matrix $$\begin{bmatrix}2&-\frac12\\1&1\end{bmatrix}.$$ How to define matrix is rotation or reflection?
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$s$ is a reflection of the plane about $x=1$. $r$ is a reflection across the x-axis.

$s$ is a reflection of the plane about the vertical line $x=1$. $r$ is a reflection across the x-axis. $g$ is an isometry and $grg^{-1}=s$. What is $g$?
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Express rotation of the plane about a point as the product of a translation and a rotation about the origin

I need help with the following problem: a) Let $s$ be a rotation of the plane with angle $\frac{\pi}{2}$ about the point $(1,1)^t$. Write the formula for $s$ as a product $t_a \rho _\theta $ Edit: ...
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Order of affine reflections (described with complex numbers operations)

Let be the affine reflection described as an operation with complex numbers : $$s_\beta,_\nu : z \mapsto \nu + \overline{\beta z},$$  where $z, \nu \in \Bbb C$ and $\beta \in \Bbb C^1 = \{x+iy \ ...
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Vector Reflection Matrix over line

I need to find a matrix that reflects vectors over any line y= mx + b. Furthermore, I need to be able to find the components of both of these vectors. Thanks!
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Regarding the Schwarz Reflection Principle, is getting the analyticity of f(z) on the real axis a consequence of the theorem itself,

or a consequence of Morera's Theorem? Basically, I want to be able to cite it correctly, e.g., can I say we have not only continuity of f(z) along $R$ (by assumption) but also it turns out that ...
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let X has a finite dimensional show that X is reflexive.

We know If a normed space X is reflexive, then $X'$ is reflexive.and also Reflexive normed spaces are Banach. but can you proof if X has a finite dimensional Then X is reflexive.
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Householder QR problem

Can somebody give me a hint or help me to solve this problem. Let V be a p×q matrix with orthonormal columns (p > q), and $M = I−2VV^T$ , with I being the p × p identity matrix. The matrix M can ...
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Find the Inverse Matrix of a Transformation

Let $ f: \mathbb R^3 \rightarrow \mathbb R^3$ be a linear mapping which reflects $\bar{x}$ over the plane $x_1+x_2+x_3 = 0$ . You are given the standard matrix for $f$ is: $$ ...
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Roots of height 1 are necessarily simple.

Suppose $\Phi$ is a root system, $\Pi \subset \Phi$ is a fundamental system (let $\Pi = \{r_1,...,r_l\}$). Now any root $r \in \Phi$ is a linear combination of the elements of $\Pi$ with all the ...
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Reflection through a hyperplane, orthogonality propoerty

While I was studying for my exam, I found Stanford's EE263 course's old homework questions and this particular one attracted my attention, however couldn't solve it. If you can help me, I will really ...
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Find a 3x3 Matrix that reflects in an Arbitrary Line y=mx+c

I want to find a 3x3 Matrix that reflects in an Arbitrary Line y=mx+c I have a matrix with me but it doesnt take into account the '$c$' This is the matrix \begin{pmatrix} \frac{1-m^2}{m^2+1}& ...
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Finding the reflection that reflects in an arbitrary line y=mx+b

How can i find the reflection that reflects in an arbitrary line , $y=mx+b$ I've examples where its $y=mx$ without taking in the factor of b But i want to know how you can take in the factor of $b$ ...
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$PQRS$ is a convex quadrilateral.Find the area of $PQRS.$

Let $P$ be the point $(3,2).$Let $Q$ be the reflection of $P$ about the $x-axis.$Let $R$ be the reflection of $Q$ about the line $y=-x$ and let $S$ be the reflection of $R$ through the origin.$PQRS$ ...
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A subset of roots whose mutual angles agree with those of a simple system

I would appreciate help/hints solving the following exercise from Humphreys book "Reflection Groups and Coxeter Groups", page 11, exercise 1. Let $\Phi$ be a root system of rank $n$ of unit ...
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Can reflection across a line segment be done using the rational field?

Assume that I have a point and a line segment, all specified using rational coordinates. Can I compute the reflection of the point across the line segment using only rational numbers? This previous ...
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Every positive system of roots contains a unique simple system.

The following question is in effort to understand a proof to a theorem appearing in "Reflection Groups and Coxeter Groups" by Humphreys on page 8. Let $\Phi$ be a root system in the euclidean space ...
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Describing the symmetries of a $2n$-gon in $\Bbb R^2$ with matrices.

Problem: Consider a regular $2n$-gon in the Euclidean plane $\Bbb R^2$ centered at the origin $(0, 0)$ and with its $2n$ vertices equally distributed on the unit circle. Label the vertices from ...
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Proof of conformal mappings onto polygons (Stein)

Recently I am reading Stein and Shakarchi's Complex Analysis and I find a great difficulty in understanding the proof of theorem 4.6 in Chapter 8 (p.242-244), which talks about conformal mappings onto ...
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Linearity of Certain Reflections in $\Bbb R^n$

Let $f: \mathbb{R^n} \rightarrow \mathbb{R^n}$ be a reflection about a hyperplane passing through $\vec 0$. Is $f$ always a linear transformation? If so, how can the matrix of the reflection be ...
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$\mathbb{R}^3 \to \mathbb{R}^3$ transformation: reflection across a plane

Notation: $v$// is $v$ parallel symbol, $v\bot$ is $v$ perpendicular, and both are relative to plane $\sqcap$ Let $\sqcap \subseteq$ $\mathbb{R}^3$ be the plane whose equation is $x + y + z = 0$. ...
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How to reflect an object without changing its geometry

I have an object in which has its orientation defined by 3 vectors and a point and I need to reflect it across a plane without changing the geometry of the object (For example if there was text on the ...
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Schwarz reflection principle, understanding the conjugated function:

Given a symmetric region $\Omega$, say, symmetric w.r.t. the real line, and f(z) defined and analytic only on $\Omega^{+}$, we can analytically continue the function to $\Omega^{-}$ with the analytic ...
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How can i reflect position and direction vectors from a plane

I'm now working on a project that has mirrors. I'd like to reflect a virtual camera and the way which i can do this is to reflect two vectors - position and normalized direction vectors of the camera. ...
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Product of a Householder transformation and reflection through the origin in 3 dimensions

This came up doing some research in quantum information. Let us consider two orthogonal three-dimensional unit vectors $v$ and $w$ $v^T\cdot w=0$, and the Householder transformation ...
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Interesting property of reflection matrix

By inspection, is is evident that the reflection matrix is an orthogonal matrix, while its transpose is equivalent to its non-transpose. The reflection matrix can be represented by the square matrix: ...