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39 views

Looking for a “Neat” Transform to Yield a Convex Set

Optimizing on a unit sphere $\mathbb{S}^n$ is almost a convex problem (if the function is convex in the new set) if we make our "new" set $\mathbb{R}^n$, via the stereographic projection. Clearly ...
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2answers
130 views

Fourier, Laplace, … and other Integral-transformations

I know Laplace, Fourier and Mellin-Transformation. Is there a general theory of transformations? My main interest is about classification of transformations satisfying specified properties like ...
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1answer
203 views

Mirror anamorphosis for Escher's Circle Limit engravings?

You are probably familiar with "mirror anamorphosis," the rendering in a painting of a distorted figure that can be undistorted by viewing in an appropriately tilted or curved mirror. The skull in the ...
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1answer
71 views

Determine all the linear transformation of $S^3$

I encounter the following problem: Let $S^3$ denote all the $3\times 3$-real symmetric matrices. If $F:S^3\to S^3$ is a linear mapping, with $F(OAO^T)=F(A)$, for any ...
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1answer
103 views

Transformation of Cubic Polynomial

I'm stuck on transforming this equation and am not sure where to begin. I know I need to define $x$ as some multiple of $u$ and somehow cancel the coefficient of the $x^2$ term but am not sure how to ...
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1answer
223 views

Transform between cartesian coordinate system and abstract coordinate system

I am trying to find a transformation that takes me between Cartesian coordinates and a pseudo-coordinate-system I have developed which is described as follows: Please first see the diagram below. ...
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1answer
117 views

condition for upper triangular matrix

Consider the following condition from this other post Define $S_k = {\rm span} (e_1, \ldots, e_k)$, where $e_i$ the standard basis vectors. Clearly, the linear map $T$ is upper triangular if and ...
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0answers
93 views

Update rotation matrix

Imagine you have a two noded beam in space, defined by extreme nodes 1 and 2. Image is owned by Jean-Marc Battini. To ...
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3answers
147 views

Prove an equality between dimensions of kernels

Let $V$ be a inner product space over field $\mathbb{R}$ with $\dim(V)<\infty$, and $T\in \text{Hom}(V,V)$. I'm trying to prove:$$\dim(\ker T)=\dim(\ker T^*)=\dim(\ker TT^*)$$ Also, as a conclusion ...
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1answer
177 views

Homothetic transformation

suppose that,we have Homothetic,in rectangular coordinate system,with center origin $(0,0)$ and $k$ ,and this homothetic send point $A(2,3)$ to point $B(2*x-1,x)$.aim is to find $k$,first i have ...
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1answer
79 views

Range of transformation $T: C[0, 1] \to \mathbb R$ defined by $T(f) = \int_0^1 f(x)\,dx$

What is the range of the transformation $T: C[0, 1] \to \mathbb R$ defined by $T(f) = \int_0^1 f(x)\,dx$ $C[0, 1]$ is the vector space of all continuous functions $f: [0,1] \to \mathbb R$. So the ...
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1answer
151 views

Formula for Adding Time expressed as decimal

If I record the duration of an event that was 97 minutes as 1.37 and then record the duration of another event that was 162 minutes as 2.42 and then add the two together to get 3.79, is there a ...
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1answer
81 views

Distribution of the Inverse of a Random Variable

I am trying to figure out how to find the distribution of the inverse of a random variable. Say, $Y=X^{-1}$ where X can take negative values. The two ways I know to find the distribution of a random ...
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1answer
94 views

log transformation for dummies

I have a question which is probaly very simple to answer for most people here: We have a formula: y = -log(x) Then this happens to x: ...
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0answers
131 views

Rewriting a formula (hopefully just basic algebra..)

I have a question which actually involves statistics, but I think it comes down to some basic algebra, so hopefully someone here can help me out (as I am obviously not a mathematician). Let’s say I ...
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1answer
35 views

Calculating transformation from origin to point

I have an icosahedron of radius $x$ with 12 vertices at known coordinates. If I have a point at $(0,0,x)$ where $x > 0$ and a vertex of this icosahedron at $(a,b,c)$ how can I find the rotation ...
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1answer
24 views

Algebraic Transformation query…

I'm boning up on Algebra, and I'm looking into Algebraic Transformation. I understand the basic concept - but I'm confused by two self assessment questions. The two questions, from what I can see, ...
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2answers
130 views

Transformation between the same rotation expressed in different coordinate systems

EDIT Lets assume my transformation does the following mapping: x = -y y = -x z = -z Which produces this transformation matrix $R_t$: $ \begin{array}{lll} 0 & -1 & 0 \\ -1 & 0 & 0 \\ ...
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1answer
237 views

my plane is not vertical, How to update 3D coordinate of point cloud to lie on a 3D vertical plane

I have a bunch of points lying on a vertical plane. In reality this plane should be exactly vertical. But, when I visualize the point cloud, there is a slight inclination (nearly 2 degrees) ...
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1answer
109 views

Dependence of vectors : before and after linear transformation

I have a pretty simple question that confused me: V is a vector space of a finite dimension. $T: V \to V$ is a linear transformation. The information that's been given in question: ...
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3answers
74 views

geometry: linear transformation

I know I do it wrong but where is the mistake??? In E3* are given the points $A(1,0,0,0)$, $B(0,1,0,0)$, $D(0,0,1,0)$, $O(0,0,0,1)$ and $E(1,1,1,2)$. The linear transformation $\Phi$ operates ...
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1answer
63 views

Interpretation of Linear Algebra and superposition

I have been told and have seen that knowing Linear Algebra is foundational in pursuing advanced mathematics. After dealing with it enough I have gotten "used to it" (I still need a lot more practice!) ...
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2answers
61 views

Explanation of an example of linear transformation

This is an example from a text (Linear Algebra, Freidberg). I am trying to follow along, and I feel like I should know this from vector calc but I am missing something silly. The example is: ...
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1answer
48 views

Image of $A \subset \mathbb{R}$ under transformation $(x,y) \rightarrow (u,v)$

What is the image of the set $$A=\{ (x,y) : 0\le x \le a\ , \ 1\le y\}$$ under the transformation $(x,y) \rightarrow (u,v)$ where $$u=x/y$$ $$v=x$$ The parameter $a$ is positive. I got a 'triangle' ...
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4answers
92 views

Showing that a transformation $T:\mathbb R^3 \to \mathbb R^2$ is linear

OK, I am trying to prove the following transformation is linear, and find the basis for $\ker(T)$ and Im$(T)$ (also denoted in our textbook by $N(T)$ and $R(T)$ ). Then we're suposed to find the ...
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4answers
131 views

Why do we define a linear transformation to have the property that $f(cW)=c f(W)$?

Why we define a lin tranfs to have the property that $f(cW)=c f(W)$ ? let $V,T$ be any two vector spaces and let $f:V\rightarrow T$ be a linear transformation between $V $and $T $ why do we ...
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1answer
127 views

Fourier transform for a 2D curve

I am stuck on the following problem about a Fourier transform of a 2D curve: I have to calculate the Fourier transform (using 1D complex FT) (and the opposite of it) for a 2D curve z(t). The curve is ...
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1answer
56 views

Function to map a range of $[-1,1]$ to a range of $[0,1]$ [duplicate]

I'm not capable of 'rigorously' defining the problem I have but this is the best I can do. If I have a set of points that range from $-1$ to $1$ inclusive and I have to transform the data so that it ...
2
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0answers
55 views

Stabilize Variance for Statistics (Transformation)

Problem: When $Y (> 0)$ has mean and variance equal to $\mu$ and $\mu/n$ respectively, it is shown in the textbook that the appropriate transformation of Y to stabilize variance is the square root ...
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1answer
75 views

Möbius transformation question

Möbius transformation copies the annulus $\{z:r<|z|<1\}$ to the domain between $\{z:|z-1/4|=1/4\}$ and $\{z:|z|=1\}$ Please help me to find what is $r$.
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2answers
293 views

Understanding perspective transform matrix elements interpretation

I am representing 3D points (vectors) in the following way: ...
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0answers
37 views

Inverting a discrete linear transformation

Consider the transformation from the set $\{a_n\}_{n=0}^N$ to the set $\{p_j\}_{j=0}^N$: $$ p_j = \sum_{n = 0}^Na_n\phi_n(x_j)$$ where $\{\phi_n(x)\}_{n=0}^N$ is a set of basis functions (linearly ...
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1answer
89 views

Finding an Orthogonal Transformation with 2 given vectors

There are two possible orthogonal transformations of $\mathbb{R}^2$ that leave the origin fixed and send the point $(0,13)$ to $(5,12)$. Find their matrices and describe them geographically. Can ...
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1answer
21 views

From parameterform to parameterfreeform

I have got the answer $\frac{1}{8}(30x-2y+21z+20)=0$ to an equation of a plane but I want the answer to be in parameter free form
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1answer
68 views

Find a vector from a given vector transformation matrix [closed]

if anyone can give an explanation on how to solve 10.f, that would be great, thanks! Find an: $$\vec x$$ such that: $$T(\vec x) = \begin{bmatrix}2\\1\\3\end{bmatrix}$$ The whole question is here ...
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0answers
130 views

Canonical form of a curve (geometry)

I am bothering with this geometric problem more than half a day and couldn't understand it yet. Here it is: In orthonormal coordinate system K=Oxy we have a curve C: $9x^2 - 4xy + 6y^2 + 6x - 8y + ...
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3answers
867 views

Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z…: unified treatment of transforms?

I understand "transform methods" as recipes, but beyond this they are a big mystery to me. There are two aspects of them I find bewildering. One is the sheer multiplicity of such transforms. Is ...
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1answer
73 views

What is the image of $D=\{z:0<\operatorname{Re}z<\pi\}\setminus\{\pi/2\}$ under $f(z)=\tan z$?

What would be the image of the domain $D = \{z:0<\operatorname{Re}z<\pi\} \setminus \{\pi/2\}$ under $f(z) = \tan z$? I havn't met with tan(z) transformation so I don't really know how to ...
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1answer
682 views

Is perspective transform affine? If it is, why it's impossible to perspective a square by an affine transform, given by matrix and shift vector?

I'm a bit confused. I want to program a perspective transformation and thought that it is an affine one, but seemingly it is not. As an example, I want to perspective a square into a quadrilateral (as ...
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1answer
69 views

Inverting an infinite sequence transformation

Consider a sequence $\{b_k\}$ define via: $$ b_k = \sum_{n=0}^\infty \frac{(n+k)!}{n!}a_n. $$ I would like to invert this transform. That is, I would like to know the coefficients $c_{nk}$ such that ...
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1answer
48 views

Can't figure out this transformation matrix

So basically I want to write a transformation matrix to take me out of one coordinate system and into another. The transformation has to be as follows: 1) The positive z axis normalized as ...
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3answers
93 views

Find $Y=f(X)$ such that $Y \sim \text{Uniform}(-1,1)$.

If $X_1,X_2\sim \text{Normal} (0,1)$, then find $Y=f(X)$ such that $Y \sim \text{Uniform}(-1,1)$. I solve problems where transformation is given and I need to find the distribution. But here I ...
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1answer
68 views

Calculating Fourier Transform of $1/|t|^n$

I have found the Fourier Transform of $x(t)=|t|^{n}$ and i can't calculate the Fourier Transform of $x(t)=|t|^{-n}$. Any suggestions?
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1answer
240 views

Simple Graph Transformation Question $\rightarrow$ $1/f(x)$

for the graph: such that the function is : $ y = \frac{a+x}{b+cx} $ where a = -2, b = 1 and c = 1/2 how do you sketch the graph of $ y = |\frac{b+cx}{a+x}| $ ?? i got that the VA of the new ...
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1answer
45 views

Need help understanding a transformation

I know this might be an unusual question, but please bear with me. In the book ¨An Introduction to Maximum Principles and Symmetry in Elliptic Problems¨ There is the following example of ...
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1answer
123 views

What are the units of Singular Value Decomposition components?

I have a symmetric variance/covariance matrix $A$ which is of size (27 x 27). I know that it's rank deficient (rank = 21). I also know that the units of $A$ are $m^2$. I am trying to use Singular ...
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1answer
87 views

Basis of kernel and image of a linear transformation - verification

The transformation matrix I found is: $$\begin{pmatrix} 1 & -1 \\ 1 & 1 \\ 0 & 0\end{pmatrix}$$ Is this how a basis for $\ker$ and $\mathrm{im}$ is calculated? $$\begin{pmatrix} 1 & ...
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1answer
106 views

Harmonic Function Transformation Help

Consider the harmonic function $$u(x,y)=1-y+\frac{x}{x^2+y^2}$$ on the upper half plane $y>0$. What is the corresponding harmonic function on the first quadrant $x>0$, $y>0$, under the ...
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0answers
31 views

Transformation of binary data

I have a function that I try to optimize using Particle Swarm Optimization. Objective function gets a binary string. So these binary strings are candidate solutions of the subject function. I can ...
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1answer
57 views

Special linear transformations

Special linear transformations are matrices with determinant equal to 1. What additional properties do such transformations have compared to "regular" linear transformations?