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), ...

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3answers
27 views

How can you enlarge a shape about a point other than (0,0), using matrices?

If I want to enlarge a shape, $A$, by scale factor $k$ about $\left(0,0\right) $ I multiply each point (in the form $\begin{bmatrix}x\\y\end{bmatrix}$) by $kI$. However, I can't work out a general ...
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0answers
7 views

Transforming UV Region to XY Bounded By Hyperbolas and Lines

Suppose I have a region in the x-y plane bounded by: $y=\frac{1}{x}, y=\frac{4}{x}, y=x, y=4x$ We see that: $1\leq yx \leq 4$, and $1\leq \frac{y}{x} \leq 4$ If I let $u=yx$ and $v = ...
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1answer
14 views

Inverse Z-transform of $\frac{10}{z-5}-\frac{2}{z-1}$

I must calculate the inverse z-transform of $\frac{10}{z-5}-\frac{2}{z-1}$. I decided to use the known formula $H(n-1)a^n\rightarrow \frac{a}{z-a}$, where $H(n)$ is the heaviside signal. I finally get ...
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2answers
33 views

How to determine that the 3 points given in homogeneous coordinates are collinear? [on hold]

How do I prove that the 3 points given in homogeneous coordinates are collinear? $$A=(1,3,2)^T, B=(0,6,8)^T, C=(3,3,-2)^T$$
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1answer
18 views

Translate a Rectangle Position from 1 Image to another [on hold]

I have a Large Size Image.Since its too large for processing within a small time, i need to resize it.I have the coordinates of a rectangle in the resized image.Is there a way i can translate this ...
0
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0answers
22 views

Finding transform matrix from resulting multiplypoint function

Two matrix transformation functions exist within the Unity3D API: 1) MultiplyPoint 2)MultiplyPoint3X4 3X4 matrix (2) preforms a standard transform against a vector (And ofc is easily replicated ...
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1answer
559 views

Inverse rotation euler angles

I have three angles representing a rotation (Pitch, roll and yaw). I need the inverse rotation (working on coordinate system transforms). What I do now is transforming these angle to a rotation matrix ...
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0answers
23 views

Fourier transform of $H(-t)e^{5t}$

i have to calculate the Fourier transform in the title. My professor says the result is $\frac{1}{5-2\pi i f}$. I start from $H(t)e^{\alpha t}$, and i calculate the transform $H(t)e^{5t}\rightarrow ...
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1answer
16 views

Can you kindly explain me in detail this Fourier transform?

I've this function to transform not using the general formula, but just substituting the known transform (i.e. $\text{rect}(t)\rightarrow \text{sinc}(f)$): $\frac{\sin(6\pi t)}{t}$ I know the ...
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0answers
17 views

Transforming parts of functions

I have a function in the form: $$ \mathrm{e}^{-t\lambda} \cdot \left[t\lambda - {(t\lambda)^2 \over 2}\right] $$ If one were to plot this for say $\lambda = \frac{2}{3}$ and $t$ from $0$ to $20$, ...
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0answers
21 views

Logarithmic function transformations

The standard log function form is $a \log[k(x-d)] + c$ Where $a$ vertically stretches or compresses $k$ horizontally stretches or compresses $d$ translates left or right $c$ translates up or ...
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0answers
12 views

Perspective correction from 3 points and foreshortening factor

I'm working on creating a homography 3x3 matrix to do a perspective correction of a photograph 2D piece of paper. The paper contains 3 markers (like the 3 corner markers of a QR code) in its corners, ...
2
votes
1answer
41 views

Fourier transform of $\frac{\sin(6\pi t)}{t}$

I have to calculate the fourier transform of this function in time domain: $\frac{\sin(6\pi t)}{t}$. First I tough to use the definition of $\operatorname{sinc}$ function as ...
2
votes
3answers
46 views

Image of a family of circles under $w = 1/z$

Given the family of circles $x^{2}+y^{2} = ax$, where $a \in \mathbb{R}$, I need to find the image under the transformation $w = 1/z$. I was given the hint to rewrite the equation first in terms of ...
1
vote
1answer
40 views

$Z$ coordinates disappear in the general rotation transformation matrix.

I wanted to generate the general rotation transformation matrix ($3D$). But when I did the multiplication the result didn't include the original $Z$ coordinates,I don't know why the $Z$ disappeared. ...
0
votes
1answer
13 views

Prove a transformation is injective if its restrictions are injective.

Suppose that $V$ is a vector space, and let $V → W$ be a linear map. $$V_0 ⊆ V_1 ⊆ · · · ⊆ V_i ⊆ V_{i+1} ⊆ · · · ⊆ V$$ are subspaces of $V$ (one for each $i = 0, 1, 2, \ldots$) and inclusions ...
1
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1answer
12 views

Can you stretch a function with a zero or undefined gradient?

If $y=f(x)$ is either $y=3$ (zero gradient) or $x=2$ (undefined gradient), is it possible to stretch $y=f(x)$ by graphing $y=af(x)$ or $y=f(ax)$? If it is possible to stretch them, can you only ...
1
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0answers
14 views

How to determine changing scale factors when performing coordinate transfomations?

To explain: I have two coordinate systems. One (x,y) and the other (x',y') as seen in this photo. Coordinate systems I am trying to convert the coordinate in the (x,y) system to the rotated red ...
1
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1answer
2k views

Negative & Positive Shear Factor

My question relates to constructional geometry & matrices aren't to be involved in the solution because stated Math level is up to O Levels... The figure below shows shear with y=3 as invariant ...
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0answers
20 views

Homography with line correspondences

When calculating a homography with line instead of point correspondences, what is the derivation of the formula: $$ l_i = H^T\cdot l^{'}_i $$ I know that: $$ l^T\cdot x = 0 \quad\text{and}\quad ...
2
votes
1answer
387 views

Variance stabilization for Poisson data

Intro Let $Z > 0$ be a random variable with the mean and variance defined as $\mathbb{E}\{ Z \}$ and $\operatorname{Var}\{ Z \}$, respectively. The variance stabilization transform (VST) $f(z)$ ...
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1answer
44 views

Find the density of $Z=\frac{X}{Y}$ for an exponential distribution?

We have the iid random variables $(X,Y)$ where $f_x(x)=\lambda e^{-\lambda x}$, $x>0$. We are given $Z=\frac{X}{Y}$ and asked to find the cdf and the density function. Here's my attempt. ...
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2answers
708 views

Should I trust Mathematica or numerous other sources on this Fourier transform

Assume $a>0$ So Mathematica claims $$F\{e^{-a|t|}\}(\omega) = \frac{a\sqrt{\frac{2}{\pi}}}{a^2+\omega^2}$$ However, I've read about another transform pair (page 3): $$F\{e^{-a|t|}\}(\omega) = ...
3
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2answers
4k views

building transformation matrix from spherical to cartesian coordinate system

How to arrive at the following from given $ x = r\sin \theta \cos \phi, y = r\sin \theta \sin \phi, z=r\cos\theta $ $$ \begin{bmatrix} A_x\\ A_y\\ A_z \end{bmatrix} = \begin{bmatrix} \sin ...
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0answers
18 views

affecting the final result of function depending on external factor

Suppose I have a function $f(x) = \frac{x}{x+y}$ whose range is in the interval $[0,1]$ and there is an external factor say $a$, such that $a$ is in the interval $[0,1]$, moreover, a predefined ...
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1answer
929 views

Finding reflection of a matrix

To 2 decimal places, what is the value of the lower-right entry in the reflection matrix $Q_a $if a = 1.05? Not even sure where to begin, is there a formula? This is what I could find in my textbook ...
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0answers
9 views

Solving a transformation problem in complex (Argand) planes

Below is an example in a text-book section on using complex-number arithmetic to represent transformations in Argand planes Example A transformation $T$ of the $z$-plane to the $w$-plane is ...
0
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1answer
22 views

What's the formulation of N-point radix-N for NTT

We can write the formulation for the buttlerfly function applied in FFT as \begin{align*}y_0 &= x_0 + x_1,\\ y_1 &= x_0 - x_1. \end{align*} As seen here. For FFT (Fast Fourier Transform) we ...
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1answer
498 views
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0answers
7 views

Characteristik values of linear transformation?

Consider this inner product space on C^2. T be a linear operatör from C^2 to C^2 s.t every (x,y) element of C^2 T(x,y)=(x+iy,y+ix). Find all characteristic values of T. And T is self adjoint?
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1answer
15 views

How to determine if an affine transformation would cause reflection?

I have a list of affine transformation matrices and I want to write a code to delete the transformation matrices that applying them on an image would cause reflection. after seeing this image in ...
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0answers
20 views

Distribution of discrete function of continuous random variable?

It has been quite some time that I did statistics, and I am not sure how to figure out the distribution of a function of a random variable if the function itself discretizes (if that is a word) the ...
0
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1answer
54 views

Bring system in normal form up to the second order

Bring the system $$ x'=y+xz,\quad y'=x^2+y^2+z^2,\quad z'=-2z+xy $$ to a normal form up to the second order (kill all non-resonant quadratic terms). The equilibrium is $(0,0,0)$ ...
0
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1answer
19 views

Invariance properties of transformations

In Gentle's Matrix Algebra (2007, p. 175), he presents a table of what features of vectors various transformations preserve. What does it mean to say a transformation T preserves some property of a ...
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1answer
24 views

How to approximate linear relationship between two timeseries? [closed]

I have two time series A and B I would like to solve for the equation in the form $$y_t = m x_t + b$$ that transforms a point $y$ at time $t$ from series $B$ to the corresponding point $x$ at time ...
0
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0answers
100 views

PDFs of Piecewise Transformations: why doesn't it apply in this case?

This is from Casella and Berger's Statistical Inference, although it is more of a probability question than a stats question. Theorem 2.1.8 Let $X$ have pdf $f_{X}$, let $Y = g(X)$. Define the ...
0
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1answer
22 views

The correct order for applying transformations?

I had a line, y=-1/2 x I wanted to reflect everything onto the line so as you can see by my steps in the picture below(sorry, I dont know the coding) I first rotated the line theta degrees then ...
1
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1answer
28 views

rational integral with a quartic function in the denominator

Let's say I've got the integral: $\int [ 2 Q^4 - 5 Q^2 + 3 ]^{-1} dQ$ This integral evaluates to: $\int [ 2 Q^4 - 5 Q^2 + 3 ]^{-1} dQ = \tanh^{-1}\left( Q \right) + \sqrt{ \frac{2}{3} } ...
51
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3answers
2k 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 number of them. Is there a unified ...
1
vote
1answer
31 views

Transform PDE to ODE (3 variable case) with given boundaries

How can I transform the following PDE into an ODE? I tried using three different functions $H(x),G(y)$ and $F(t)$ but that didn't help hence I did not post it here. I really hope someone can help me ...
0
votes
1answer
10 views

Three Variable Transformation and Independence

If $X_1$, $X_2$, and $X_3$ are independent, identically distributed random variables and $Y_1$, $Y_2$, and $Y_3$ are functions of them, how do I show that the $Y$ variables are mutually independent? ...
1
vote
1answer
20 views

Integral with more-dimensional substitution variables

Good day, In the lecture of partial differential equations we had the following transformation: $$\int_{||\nu||=1} h(x+\nu c t, \tau) d\nu = \frac{1}{c^2 t^2} \int_{||y-x||=ct} h(y,\tau) dy$$ for ...
2
votes
2answers
1k views

Matrix representation of the dual space

Let $V$ be an $n$-dimensional vector space over $F$, with basis $\mathcal{B} = \{\mathbf{v_1, \cdots, v_n}\}$. Let $\mathcal{B}^{*} = \{\phi_1, \cdots, \phi_n\}$ be the dual basis for $V^{*}$. Let ...
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0answers
33 views

Is there a product integral that preserves zeroes?

The integral essentially takes the arithmetic mean of the range of a function multiplied by the domain, adding together each possible output weighted by the amount of the domain accounted for by that ...
3
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2answers
92 views

Cayley's Theorem for Semigroups

I've read and fully understand Cayley's theorem for groups, however when I get to the theorem for semigroups I come to a complete stop. I've figured that the identity and cancellative properties are ...
0
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1answer
534 views

How can I transform a 3D triangle to xy plane

Suppose I am given a triangle ABC and its corresponding vertex coordinates in 3D. I want to transform ABC in such a way so that vertex A lies on global (0,0,0) coordinate, B lies on (dist, 0, 0) ...
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0answers
29 views

(Answered) Transformation between AutoCAD OCS and WCS coordinate systems

This question originates in a problem I am having with transforming between two coordinate systems, that come from AutoCAD. This question is purely about the math of the problem, I just include the ...
2
votes
0answers
22 views

Fourier Transform of triangle function

I have a question regarding the FT of the triangular function: How does $e^{-j\omega t}$ becomes the cosine function in the first line? What happened to the sine when you go from $e^{j \omega t}$ ...
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3answers
32 views

Transformation of exponentials

Find the transformation that takes $y=3^x$ to $y=\textit{e}^x$. I have tried: Let $y=3^x$ to $y=e^{x'}$ $$\log_{3}(y)=x\quad\text{hence}\quad\log_{3}(y)=\frac{\log_{e}(y)}{\log_{e}(3)}$$ ...
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0answers
26 views

Graph Transformation

This question seems trivial to me but I still cannot figure out how to approach it. For any $x \gt 0$ and $f(x) \le 2^x$, how do I prove the following? $$ f(2x) \le f(x)^2 $$