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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Can the z-transform/laplace transform be generalized?

The fourier transform is a special case of z/laplace where the countour being projected on is the unit circle. The ztransform/laplace transform then generalizes this to a circle of any radius. Is ...
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2 views

Does the amount of translation for depends on whether it goes before or after dilation?

One question in my practice book, asks me to describe $y=f(-ax+b)$ based on y=f(x). According to the book, the order of transformation must be listed in this order: Reflection Dilation (scaling) ...
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2answers
29 views

transformation of differential equation

1.$$ x^2(1+u)dx + x^3(1-u)du = 0 $$ 2.$$ \frac {1-u}{1+u}du + \frac {dx}{x} = 0 $$ I know from 1 to 2 the equation is divided by $x^3(1+u)$, but i don't understand how that happens,was reading ...
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1answer
29 views

Corresponding matrix field basis

Hi people, I'm reviewing my notes for an exams and this is a question which I was unable to wrap my head around for many months. It should be fairly simple but I might be lacking a crucial piece of ...
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1answer
27 views

Is there a concise, specific name for a transform that consists of rotate, scale and translate?

I'm working on software that involves transforming between different mapping coordinate systems. In one part of the maths/logic, I have to derive, then apply a transform between two cartesian ...
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1answer
19 views

What is this method of scaling called? Can it be generalised?

Consider the problem of finding the values of $\alpha_1, \alpha_2, ..., \alpha_k$, subject to constraints, such that the following equation is satisfied \begin{equation} \alpha_1 x_1 + \alpha_2 x_2 + ...
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30 views

Tranformation of random variables

Let $X$ have the p.d.f $f(x)= e^{-x}$, $ x > 0$, $0 $ otherwise,find the pdf of $Y = X^2$ and space range $Y$. I use the change of variable formula The inverse is $${x} = \sqrt{{y}} $$ The ...
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25 views

fourier transform of a function that is only partially known

I'm interested in periodicity of a function that is not known completely, one way to think about it: our function $x(t)$ is defined at discrete points $t_1, t_2, ...$ but the values at these points ...
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20 views

Find me a sigmoid function with fixed point at the point of inflection in the unit interval

I am interested in finding sigmoid (S-shaped) functions $f$ on the unit interval $[0,1]$ such that For some $a \in (0, 1)$, $f(a) = a$ and $f'(a^-) > 0$ and $f'(a^+) < 0$. $f(0) = 0$ and $f(1) ...
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22 views

Before and after a transformation apply another transformation and its inverse?

So I have something like $y(x) = x\cdot5$, so whatever $x$ is, it's being scaled up by $5$. But I could center that scaling to a different point reference other than $0$. For example $10$. So for ...
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1answer
13 views

Method for transforming one curve around another?

I'm working with a complex problem involving waveforms. Essentially I want to bend a given waveform around a circle. At it's most basic, I want to take one curve on a linear graph and map it onto a ...
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73 views

Best sources on complete transforms (classic orthonormal transforms) and overcomplete transforms in signal processing

In the introduction section of a thesis I read a little about classic orthonormal transforms such as Fourier, discrete cosine and wavelet transforms and their application in signal processing. Then ...
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1answer
13 views

Z transformation with $k$ from non-zero

we known that $Z$ transformation of $f_k$ is defined as $$F(z)=\sum_{k=0}^{\infty}f_k z^k$$ My problem is if $k$ starts from $m$, where $m >0$, then $\sum_{k=m}^{\infty}f_{k+m} z^{k+m}$ is still ...
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29 views

How to get the transformation matrix from 3D normal Vector

I have one plane object with a 3D normal vector $(X,Y,Z)$. How to get the transformation matrix from this object to my 3D origin frame $(x_0,y_0,z_0)$. The normal vector should be aligned with $z_0$. ...
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2answers
4k views

RYB and RGB color space conversion

I am working on a project where I need to convert colors defined in RGB (Red, Green, Blue) color space to RYB (Red Yellow Blue). I managed to solve converting a color from RYB to RGB space based on ...
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1answer
93 views

Linear Transformations finding matrix in respect to a basis and coordinate change matrix.

Define $T: Poly_2 \ to\ Poly_2$ by $$T(at^2+bt+c)=3ct^2 +2at-b$$ 1) Show that T is a linear transformation and give a matrix A for T with respect to the basis $B=\{t^2,t,1\}$. 2) Give ...
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23 views

Transformations of discrete random variable

I understand how to do transformations. Well, I thought I did. But I can't seem to comprehend how to do a discrete to discrete. Here is the question I am working with. (I am currently studying for an ...
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33 views

f(2x + 1) transformation

I was working on a problem that asked: If there is a vertical asymptote at $x = 5$ for $f(x)$, where is the vertical asymptote for $f(2x + 1)$? The correct answer is at $x = 2$, but this ...
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1answer
32 views

Find Fourier transform of triangular function based on a Fourier results of rectangular

I have a triangular pulse given by $$x\left(\frac{t}T\right) = \begin{cases} 1-\frac {|t|}T, & \text{if $T\ge t$} \\ 0, & \text{otherwise} \end{cases}$$ Given that ...
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1answer
46 views

Order of Operations for Horizontal Transformations

We know that when we want to combine two horizontal transformations, specifically that of translating and stretching a function, we have to translate $f(x)$ first, and then afterwards stretch it. ...
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107 views

Show random variables are mutually independent

Let $X_1$,$X_2$, $X_3$ denote a random sample from the distribution having p.d.f $$f(x) = e^{-x},\,\, 0<x <\infty,$$ zero elsewhere. Show that $$y_1 = \frac{x_1}{x_1 + x_2} $$ $$y_2 = ...
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1answer
37 views

PDF for $\frac1a Uniform(−a,a)$

Problem: Let $X$ be $Uniform(−a,a)$ distrubuted. Calculate the PDF for $Z = \frac1a abs(X)$. Attempt: I think graphically here. $X$ is $U(-a,a)$ with $PDF = \frac1{2a}$ so $abs(X)$ is $U(0,a)$ with ...
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Variance Stablizing Transformations on Kernel Density Estimates

Suppose if I had a data set and performed a non-parametric kernel density estimate and generated a probability distribution function - using the delta method/taylor expansion (as seen in: ...
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143 views

Calculate centers of circles from their ellipse perspective.

Originally there are 4 circles in a plane and after perspective transform we get four conics. Now I know the equation of those ellipses. How could I get the origin of those four circles ? I know ...
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2answers
35 views

Using the dimension formula to prove isomorphism

Let V be a finite-dimensional vector space and $T: V\rightarrow V $. T is a linear transformation. Use the dimension formula to prove that if T is injective, it must also be surjective; if T is ...
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59 views

Normalize a diagonal matrix such that each element belongs to $[0,1]$.

Let $\mathbf{x}\in\Bbb{R}^n$ be an $n$-dimensional real vector and $C\in\Bbb{R}^{n\times n}$ be a diagonal real matrix. Suppose that two vectors $\mathbf{r}_\min,\mathbf{r}_\max\in\Bbb{R}^n$ ...
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17 views

Deriving the formula for Fixed-rate mortgage using z-transform

So, using z-transform, one could easily derive the formula for fixed-rate mortgages. However, when I tries this, I noticed one thing I'm very uncertain of. The relationship for fixed-rate morgages ...
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2answers
17 views

how to map series of coordinates onto a series of coordinates with different resolution

I have a set of target coordinates and a set of actually clicked coordinates which should be approximately the same, but not identical. The y coordinates are equal, however, the x-coordinates differ, ...
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2answers
880 views

Approximate a polynomial function using a sum of sine waves

I have a polynomial function which I need to approximate by a sum of sine waves with constant amplitude along a given domain. From what I hear, this might be a good time to make use of Fourier ...
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1answer
26 views

Find inverse $z$-transform of $\frac{5}{z^{2}-z-6}$

How can I find inverse z transform of $$X(z)=\frac{5}{z^{2}-z-6}$$ What I did: first i factored denominator and i got (z+2)(z-3), now we get A(-2^{n}) + b(3^{n}). To get A and B i used Partial ...
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336 views

Converting fractional coordinates to cartesian coordinates

I have a set of fractional coordinates with the following vectors: 2.950 -1.475 0.0000 0.000 2.5547 0.0000 0.000 0.0000 77.5379 with the ...
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1answer
41 views

Find inverse $z$-transform of $\dfrac{(z-1)^2}{z^3}$

How can I find inverse z transform of $$X(z)=\frac{(z-1)^{2}}{z^{3}}$$ What I did: I am thinking to do Partial Fraction Decomposition or long division. Is there another method ?
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1answer
35 views

Help tranformation of random variables?

Let $X$ have the p.d.f $f(x)= \frac{x^2}9$ , $0 < x < 3$, $0$ otherwise, find the pdf of $Y = X^3$ I have this exercise, but I do not know how to start, how do I know if it is a one to one ...
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2answers
353 views

How to define an affine transformation using 2 triangles?

I have $2$ triangles ($6$ dots) on a $2D$ plane. The points of the triangles are: a, b, c and x, y, z I would like to find a ...
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2answers
27 views

Image of $A \subset \mathbb{R}^2$ under general transformation

If I have a transformation $\varphi: \mathbb{R}^2 \to \mathbb{R}^2$ which doesn't have any particular property, for example, which is not linear, how do I know what is the image of a subset $A \subset ...
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2answers
25 views

Show the solution for $\mathcal{F}(e^{-\left | t \right |})$ [closed]

I'm trying to show that $$\mathcal{F}(e^{-\left | t \right |}) = \frac{2}{\sqrt{2\pi}(1 + w^2)}$$ Knowing that the Fourier transform is in the form $$\mathcal{F}(f(x)) = ...
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1answer
80 views

How can I show that the AR process is nonstationay if x(n) has nonzero mean?

This is a first-order-real-valued autoregressive (AR) process $y(n)$ that satisfies the real-valued difference equation $y(n)+a_1y(n-1)=x(n)$ where $a_1$ is a constant and x(n) is a white noise ...
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175 views

Transform square region to triangular region

How do you express x and y in terms of u and v so that the region $\{(u,v): 0\le u, v\le 1\}$ is mapped to the triangular region in the $xy$-plane with vertices $(0,0)$, $(1,0)$, and $(0,1)$? Now, ...
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Where do $H(3,1)$ and $H(3,2)$ come from when building Homography?

So you're given 4 pairs of points, including the original and where it maps to, i.e. $(x_1, y_1)\Rightarrow (x_1', y_1')$, $(x_2, y_2)\Rightarrow (x_2', y_2')$, $(x_3, y_3)\Rightarrow (x_3', y_3')$, ...
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2answers
107 views

Real Linear vs. Complex Linear

I recently started a new math course and got hung up on a particular problem from the book "Linear Algebra Done Wrong". Specifically, problem 1.3.6 (c). I am an engineer, and I believe I simply lack ...
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1answer
58 views

Determine shift between scaled rotated object and additional scale step

I am trying to find the amount to move an object so that, when it rotates and resizes, the resize would be smooth. I have a Qt program, where I have to rotate objects around center, and resize based ...
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47 views

Find linear transformation of a given matrix of linear transformation

I have two basis. The first is a $R2$ basis ${(1,0) , (0,2)}$. Lets call it basis of $U$. The second is a $R3$ basis ${(1,0,-1), (0,1,2), (1,2,0)}$ Lets call it basis of $V$. Is given a matrix of a ...
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864 views

Finding the transformation when given transformation matrix

Lets say, there is a transformation: $T:\Re ^{n}\rightarrow \Re ^{m}$ transforming a vector in $V$ to $W$. Now the transformation matrix, $A=\begin{bmatrix} a_{11} & a_{12} &...&a_{1n} \\ ...
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46 views

Prove something is not a linear transformation

As I understand it, the two things that define a transformation as linear are: 1) $T(u+v) = T(u)+T(v)$ 2) $T(cu) = cT(u)$ I want to prove that $T(x,y) = x+y+1$ (where $T: \mathbb{R}^2\to ...
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17 views

Inverse z-transform of $z^4+1.827z^3+2.338z^2+1.827z+1$

I need to transform the following $H(z)$ back to time domain: $$ H(z)=(z-e^{j\frac{8}{15}\pi})(z-e^{-j\frac{8}{15}\pi})(z-e^{j\frac{12}{15}\pi})(z-e^{-j\frac{12}{15}\pi}) $$ I did the following steps ...
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1answer
59 views

Shear matrix simple explanation

I can understand translation, dilation and rotation matrices, but the shear one is still obscure to me (despite understanding what shearing means graphically). This is the matrix: ...
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36 views

Transformations of the form $f(ax+b)$

Suppose I wanted to sketch the graph $y=\sqrt{5x-10}$ for $5x-10 \ge0$ Is there a direct method? I know that I can define first $g(x)=\sqrt{x}$ and consider $y=g(5x)$ or $y=\sqrt{5x}$ This is a ...
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19 views

Is a signal summable when given a z-transform?

The output of a system is given as z-transform: $$ Y(z)=\frac{1+z^{-2}}{(1+\frac{1}{4}z^{-1})(1-\frac{1}{2}z^{-1})} $$ I want to know if the signal in the time domain is summable, meaning that the ...
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2answers
185 views

Del operator in Cylindrical coordinates (problem in partial differentiation)

I am currently reviewing basic vector analysis and trying to understand every single detail, however, I got stuck in some derivation. What I want to show is the following: Given the del operator ...
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1answer
32 views

If $f:\mathbb{R}^{2}\to \mathbb{R}$ continuous, strictly increasing with $f(x(y),y)=0$ for unique $x(y)$ for each $y$

A claim: If $f:\mathbb{R}^{2}\to \mathbb{R}$ is a continuous function, $f(x,y)$, strictly increasing in each of its arguments, with $f(x(y),y)=0$ for a unique $x(y)$ for each $y$, then $f$ must be a ...