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Questions tagged [transformation]

Transformation has many meanings in mathematics. If using this tag, add another tag related to the object being transformed. If there is a tag for your specific kind of transformations, use that one instead: e.g., (laplace-transform), (fourier-analysis), (z-transform), (integral-transforms), (rigid-transformations).

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Transform an equation to linear form [on hold]

How do I convert the equation: $$T = 2\pi\sqrt{\frac{h^2 + k^2}{gh}}$$ to linear form?
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Transformation of a system of differential equations

Consider the system: $$ \begin{split} \dot W &= i u(t)W(t)\\ \dot S &= e^{-rt}\sqrt{(1-u(t))W(t)} \end{split} $$ I would like to examine a system: $$ \begin{split} \dot x &= u(t)x(t)\\ \...
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Transforming a nonlinear system to a new system that has an equilibrium point at the origin

Take a look at the following system $$ \begin{align} \dot{x}_1 &= x_2\\ \dot{x}_2 &= -x_1 + x^3_1 - x_2 \end{align} $$ which has three equilibrium points (0,0),(1,0), and (-1,0). In the book ...
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How can I calculate projection of a point through pinholes?

I have a point light source, pinhole plate and plane(subdivided into small pixels) as shown in below figure. Imaging Setup Light from the source passes through pinholes and fall on the plane. Then we ...
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Linear transformation for matrix using Einstein convention [duplicate]

I feel like I am stuck. How can I interpret the equation with Einstein summation? Here is the exercise: you should see that r′ can be written as a linear transformation of r. This means we should ...
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1answer
53 views

What is the characteristic equation of this linear ODE?

I am trying to solve the following linear ODE for $y(x)$: $$y^{\prime\prime\prime}+y^{\prime\prime}+\mathcal{H}[y^{\prime\prime}]+y^\prime-cy=0$$ subject to the boundary conditions $y\rightarrow0 $ ...
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1answer
45 views

Why do the components of the metric contain basis vectors?

I'm learning about the transformation rules for vectors. In the image above, the author addresses that the components of the metric transform covariantly. However, I'm confused as to why the metric ...
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$|X| \sim $ Exponential distribution with mean $\theta$.

Suppose it is given that $|X| \sim $ Exponential distribution with mean $\theta$. Then show that $X$ follows Double exponential $\theta$. My approach: Let $Y=|X|$. So,$f(y)=\frac{1}{\theta}e^{-\frac{...
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Tranforming a fixed random variable into an arbitrary one

Say I have two random variables $Z$ and $X$ taking values in $\mathbb{R}^d$ and $\mathbb{R}^n$ both with continuous cdf. Is there a theorem that guarantees the existence of a continuous function $G:\...
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1answer
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Easy Heaviside differential equation with sin(u(t))=y'

The problem is the following: $ y''* \theta(t) = \sin(\theta(t))$ This is how I have tried to solve it: $ y''* \theta(t) = y'* \theta(t) ' = y' * \delta(t) = y'$ (since everything convoluted with ...
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1answer
29 views

Linear Algebra, Finding matrix for transformation

I am revising for a Linear Algebra exam and am quite stuck on this question-any help, guidance or tips appreciated! So I have a mapping $T:V\rightarrow V$ where $V$ is a finite dimensional vector ...
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1answer
58 views

Insight on the polar decomposition of a shear?

I recently learned it, and really love the polar decomposition of a matrix, because it was the first time I actually could picture what it meant to "apply a transformation to space" (a phrase I kept ...
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44 views

How to calculate a double integral

I’ve got the following integral $$\int\int _D \frac{dxdy}{x+y}$$ D is the region bounded by $x+y = 1, x+y = 4, y=0, x=0$ and I have to use the transformation $x = u-uv, y=uv$ Anyone know what domain ...
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How to create a transformation matrix for a M22 → M22 transformation

I have a linear transformation, T, such that; T:${M_{22}}$→${M_{22}}$: T$\left(\begin{bmatrix}{x_{11}} & {x_{12}}\\{x_{21}} & {x_{22}}\end{bmatrix} \right)= \begin{bmatrix}{{x_{12}}-5{x_{21}...
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Image of a straight line under $w = Log(iz)$.

In detail, I have to find the image of the straight line parallel to the co-ordinate axes for the function $w = Log(iz)$. What I have tried is as follows (Since I am new to complex-analysis, I must be ...
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1answer
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Is there an easy way to remove scale from a squared linear transformation matrix

Given a linear transformation matrix $A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix}$, I know that one can use SVD or QR decomposition ...
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Trouble in mapping of möbius transformation

Question:- Show that the transformation $$ w = \frac{2z+3}{z-4}$$ maps the circle $x^2+y^2-4x=0$ onto the straight line $4u+3=0$ My attempt:- The circle $x^2+y^2-4x=0$ is $|z-2|=2$ . . .$(1)$ So ...
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1answer
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Translate point after rotation relative to different origin

I have 200x380 input image and coordinates (63,146) where (0,0) is top-left: I rotate about the centre some amount of degrees and expand the "canvas" to avoid cropping resulting in larger output ...
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Find best transform to map points

I have a system with a robot picking parts based on feedback from a distance sensor. I have found that errors in alignment etc can cause problems over the working range. I want to create a correction ...
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Volume preservation Liouville's Theorem , explanation of proof

I am trying to understand the following proof of Liouville's Theorem , that states that trajectories generated by Hamiltonian equations are volume preserving.The proof can be found in the following ...
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Calculating Ego-Motion from planar keypoints

I'm working on a module to estimate the ego motion from a robot with the help of a camera. In order to simplify that system I decided to track keypoints on the floor below the robot instead of solving ...
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1answer
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Joint distribution of $Y_1=X_1/X_2, \quad Y_2=X_2$ when $h(x_1,x_2) = 8x_1x_2$

I am having trouble finding the joint distribution of the following. Joint distribution of $Y_1=X_1/X_2, \quad Y_2=X_2$ when $h(x_1,x_2) = 8x_1x_2$ when $0 < X_1 < X_2 < 1$. I think I am ...
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1answer
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Find transformation matrix $[T]_{B,B}$ representing $T$ in the basis $B$

Info provided: $T:P_2→P_2$ given by $T(p(x)) = p(kx)$ where $k>0$ Find matrix $[T]_{B,B}$ representing $T$ in the basis $B$ Attempted Solution: Using the standard basis for $P_2$, {$1,x,x^2$}, ...
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If $\Sigma^{-1}=(A^{-1})^TA^{-1}$, then why does $|A^{-1}|=|\Sigma|^{-1/2}$?

In the derivation of the joint pdf of $f_\textbf{X}(\pmb{x})$, where $\textbf{X}=\pmb\mu+A\pmb Z$ and $\textbf{X}\sim~N_n(\pmb\mu,\Sigma)$, there is a step I do not understand. In particular, it is ...
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1answer
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Confusion over order of transformations of graphs

I did a search on the order of transformations applied to graphs, and mostly found the following, e.g. in this post. Given a function $f$ always perform transformations $$Af(Bx+C)+D$$ in the order $...
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Matrix transformation for finding eigenvalues

For each $m\in\{1, 2, ..., n\}$, is there a transformation $\phi_m$ that I can apply to a matrix $M\in \mathbb{R}^{n\times n}$ such that the $m^{th}$ largest eigenvalue $\lambda_m$ of $M$ is the ...
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1answer
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Logarithm transformation for a function in Machine Learning

I was looking for a clarification on why the function (the one raised to the power of M) was transformed to a logarithm function. I know it will be used to find p when L is maximised ( derivative of ...
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Transformation matrix in affine matrices

I'm studying this for my computer graphics course under the topic of affine transformations but I've searched everywhere and I don't get how to do this Given the transformation matrix: $$ M = \begin{...
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Pure pursuit algorithm - Transform Global to vehicle coordinates?

project: autonomous driving car I have a small RC car, a Raspberry Pi and a camera which are fixed on the car. Now I want to let the car drive autonomous with image processing. So far I can detect ...
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How is function transformation related to geometric transformation?

I know how $y = \mathrm{a} f(\mathrm{b} x + \mathrm{c}) + \mathrm{d}$ transforms $y = f(x)$. I wonder how function transformation is related to geometric transformation. The MathWorld page about ...
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How can I get local scale from a rotation matrix and a world scale?

I'm working on a tool for game modding, intended to help users create mesh morphs. This part of the tool concerns joint-based morphs. My problem is this: I want users to be able to enter delta ...
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Showing that a transformation is a variational symmetry

I'm trying to solve problem 9.2.1 in the book 'The calculus of variations' by Bruce Van Brunt. I was given the functional $$J(y)=\int_{x_0}^{x_1}xy'^2 dx$$ Now I'm supposed to show that the ...
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1answer
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How to make a transformation to the floor function to the right or left?

Assume a function called $f(x)$ , Then all of us know that of we draw $f(x+a)$ it will be a transformation to the left or right and $f(x)+b$ to up or down. But when I drew floor function on Desmos ...
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transformation of the Gaussian function

consider a gaussian function: $$ \frac{1}{\sqrt{\pi}} \exp \left\{-\frac{1}{2}\left(x^{2}+y^{2}\right)\right\} $$ And I have to prove that transformation of this function: $$ \exp \left\{-\eta\left(...
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General Transformation Matrices for Images

for a vector (x, y) the matrix (a, b; c, d) can represent arbitrary linear transformations, using the vector (x, y, 1) and a 3x3 matrix arbitrary transformations (projective?) Now I want to transform ...
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Problem with Graphing Transformation of Cartesian coordinate into Polar coordinate.

I was trying to map Rectangle from cartesian to polar coordinates. I started by making a rectangle in a Cartesian Plane. From the x and y coordinates of the rectangle, I calculate the radius and angle ...
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Obtaining the equations after scaling and shifting?

I am thinking about this coordinate shift. Suppose we have the two equations as below, where $x_0,y_1, x,y,\bar x,\bar y$ are variables while others are constants: $$\begin{aligned} \bar{x_0} &= ...
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1answer
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Find the CDF and PDF of $W$ where $W =.7 - b(.7 - y)^2, 0<b<1$, with $Y \sim U[0,1]$

A random variable $Y \sim U[0,1]$. Let $W = \frac7{10} - b\cdot(\frac7{10} - y)^2, 0<b<1$.Completely specify the CDF and PDF of $W$.Also show that the PDF of W integrates to $1$. So I have ...
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Normal distribution non linear transformation

I have the following problem : Given $X \sim N(\mu,\sigma^2)$ and $X' = h(X) = (\frac{x-\mu}{\sigma})^2$ Find $E[X']$ and $V[X']$. My reasoning is as follow : Since $X' \sim (\frac{x-\mu}{\...
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Orbit of a point and flow

Let $\varphi:X\times \mathbb{R}\to X$ be a continuous flow on compact metric space $X$ without singularity. For $\delta>0$ and $x\in X$, take \begin{equation} \Gamma_\delta(x)=\bigcup_{h\in\...
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Given that x~U[0,1] and y~U[0,1], derive the conditional CDF of W=x-b*(x-y)^2 where 0<b<1? Condition on x (i.e. treat x as a constant).

Given that $x\sim U[0,1]$ and $y\sim U[0,1]$, derive the conditional CDF of $W=x-b\cdot(x-y)^2$ where $0<b<1$? Condition on $x$ (i.e. treat $x$ as a constant). I am running into difficulties ...
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1answer
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How do I rotate a bitmap image?

I am trying to write an algorithm to rotate a bitmap image of $n$ by $n$ size by an angle $\alpha$. I know that I have to find a rotation matrix, then perform matrix multiplication of the rotation ...
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2answers
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A question on transformation

I am doing a transformation problem of getting the graph of $\sin (2x – \pi/6)$ by applying transformations to $F(x) = \sin x$ In the process, I let $f(x) = F(2x) = \sin 2x$. Next, I then let $g(x) ...
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1answer
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Connection between rotate / translate / scale as operations in 3D space

This feels like a silly question, what am I missing: I'm a 3D artist working on videogames. Sometimes I make 3D artworks like the ones here: http://samuelthomson.org/blog/2012/06/07/topologic-...
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Why heat kernel is a dirac sequence?

I want to show that the heat kernel $\displaystyle\gamma_t(x) := \frac{1}{(4\pi t)^{d/2}} \exp\left( - \frac{|x|^2}{4t} \right), \quad x \in \mathbb{R}^d, \ t > 0$ is a Dirac sequence. $\gamma_t \...
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Is there an appropriate transformation of weights?

Background Sometime back I managed to conjecture an interesting formula: $$ \lim_{k \to \infty} \lim_{n \to \infty}\ \sum_{r=1}^n \lambda_r \left( f(\frac{k}{n}r)\frac{k}{n} \right) = \lim_{s \to ...
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Back transforming the intersect of 2 log normal curves

Lets say i have a dataset that follows a double log normal distribution. these 2 functions intersect at $(x,y)$ when $y = p(\frac{1}{\sigma_1\sqrt{2\pi}}exp(-\frac{(x-\mu_1)^2}{2\sigma_1^2})) = (...
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Proving DTFT pair as special case of another.

Consider these 2 basic discrete-time Fourier transform (DTFT) pairs... $$ \require{extpfeil}\Newextarrow{\xleftrightarrow}{15,15}{0x2194} \begin{array}{rcl} u[n] & \xleftrightarrow{\mathscr{F}} &...
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How to integrate over arbitrary quadrilateral

I need to integrate the product of two polynomial functions defined on an arbitrary (convex) planar quadrilateral defined by 4 points in $\mathbb{R}^3$. I was trying to firstly rotate the system of ...
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2answers
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Kummer transform of the confluent hypergeometric function of second kind

I can see the kummer transformation of the confluent hypergeometric function of first kind throught the integral representation. However, I failed to see that for the second kind. More specificially, ...