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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$K=\frac{1}{2}mV^{2}$, random variable transformation.

An object has random velocity $V$ and kinetic energy $K = \frac12mV^2$, where $m$ is the mass of the object. Suppose that the velocity has the Laplacian distribution with probability density ...
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24 views

How to go about finding a transformation $T$ in order to solve an integral.

I have the integral $$\int\int_R\left(2x+y\right)dA$$ Where $R$ is the region bounded by $$x+y=-1, x+y = 3, 2x=y,2x-4=y$$ So my first though was drawing the region, which gave me this odd region, so ...
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33 views

Projective transformation a parabola to a circle

Take the parabola $x^2 - y = 0$ in the cartesian plane. I'm not entirely sure about this, but we can express this using homogenous coordinates as $X^2 - Y = 0$ (the $W$ coefficient is $0$?) With the ...
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129 views

What do $a_0$ ,$a_m$ and $b_m$ terms mean in the Fourier series formula?

We know that a Fourier series for signal $x(t)$ is given as $$\frac {a_0} 2 + \sum \limits _{m=1} ^\infty (a_m \cos \frac {2 \pi m t} T + b_m \sin \frac {2 \pi m t} T)$$ So my question is what ...
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35 views

transform orthonormal coordinate system to another

I have one orthonormal coordinate system ABC that it's origin is the point p0. I would like to transform it to another orthonormal coordinate system A'B'C', that it's origin is p1. I know how to ...
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43 views

Show whether this linear transformation is one-to-one and onto.

$T:P_2$ $\rightarrow$ $R^3$ is a linear transformation defined by $$T(a+bx+cx^2) = \left[ \begin{array}{ccc} 2a-b \\ a+b-3c \\ c-a \end{array} \right]$$ This linear transformation is neither ...
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31 views

Eliminating 3rd term in $\frac{d^2u}{dx^2} + \frac{d^2u}{dy^2} + 3u = 0$ through change of variables?

I want to eliminate the third term in this pde $$\frac{d^2u}{dx^2} + \frac{d^2u}{dy^2} + 3u = 0$$ Here's what I thought of, is this ok? Let $a = \lim \limits_{k \to \infty} kx$ Let $b = \lim ...
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19 views

Transforming pde to nicer form?

I have a second order differential equation for $u$ $$\frac{d^2u}{dx^2} + \frac{d^2u}{dy^2} + 5u = 0$$ I am looking for a transformation $u(x,y) \rightarrow v(x,y)$ that gives $$\frac{d^2v}{dx^2} + ...
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20 views

Transformed pde but my answer doesn't match solution?

$$\frac{d^2u}{dx^2} + \frac{d^2u}{dy^2} + \frac{du}{dx} + 2\frac{du}{dy} + 3u = 0$$ Let $u = ve^{ax + by}$ and find $a, b$ such that we can transform to the following equation $$\frac{d^2v}{dx^2} + ...
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92 views

Have some queries about Fourier Transform

I have some queries about the Fourier transform In most of the cases, the Fourier transform of a signal is symmetric with respect to positive and negative frequency. I think the computational ...
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152 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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65 views

Proof that Determinant is Scale Factor

I've seen a lot of supposed properties of linear transformations that're never proven -- just often repeated. These include: The determinant is the scale factor between the volume of region in your ...
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113 views

Whether the job of Fourier Transform is just to convert signals from time domain to frequency domain only or more than it?

I am a beginner . We convert a signal in time domain to frequency domain by applying Fourier transform on the signal to obtain frequency and phase spectrum. So,whether the job of Fourier transform ...
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21 views

Education tool for learning 3D angles

I hope it is not an off-topic. I have started working on 3D frame transformation and transforming a vector such as acceleration or angular velocity from one coordination to earth coordination. My ...
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40 views

transforming $(A,B,C)$ to $(0, 0, 1)$ by rotations

I'm trying to reflect the "world" through a specified plane $p:Ax+By+Cz=0$. I know how to reflect the "world" through the $xy$-plane, so I want to rotate $p$ in the $3$ axes ($x,y,z$-axes) so it will ...
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41 views

Lens Conformal Map

Please help me find a conformal map of the set $ A = \left \{\; z: \; |z-1| < \sqrt{2} \; and \; |z+1| < \sqrt{2} \; \right \}$ one-to-one onto the open first quadrant. First, I noticed that ...
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34 views

Transformation of a Partial Differential Equation

How can we convert $$\frac{\partial c}{\partial t} = M\left[\frac{\partial}{\partial x}\left(c\frac{\partial c}{\partial x}\right)+\frac{\partial }{\partial y}\left(c\frac{\partial c}{\partial ...
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1answer
25 views

Distribution of Logistic of Normal

If $X \sim N(\mu_X, \sigma^2_X)$ and $Y= \frac{\exp(X)}{1+\exp(X)} $, what is the distribution of $Y$? I thought logit-normal would fit the bill, however the logit of $Y$ is ...
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What are the limitations /shortcomings of Fourier Transform and Fourier Series?

I am fond of Fourier series & Fourier transform. But every approach has some outcomes and some shortcomings. It's limitations lead to innovation of new approach. So, can anybody explain about ...
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21 views

Proving a theorem on a rotation about a line followed by the inversion to show that it is a reflection

A theorem in my textbook is : A rotation about a line followed by the inversion about a point on that line is a reflection or a rotary reflection. I can picture this theorem in my head on a 3D space ...
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1answer
41 views

Rotation in high dimension in the direction of given vectors

Given two vectors $A$ and $B$ (with high dimension), and an angle $\alpha$. How can one find the vector $C$ which is $A$ rotated over $\alpha$ in the direction of $B$? If it changes anything: the ...
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13 views

Transforming coordinates according to Point of View.

Imagine I have two cameras (A and B) in the same axis and looking at the same white wall with a black dot. Both cameras see the black dot. I'd like to know how to proceed in order to derive the black ...
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47 views

Dynamical System transformation

How can the system $$\frac{dx}{dt}=-y+\epsilon x(x^2+y^2)$$$$\frac{dy}{ dt}=x+\epsilon y(x^2+y^2)$$ be transformed into $$\frac{dr}{dt}=\epsilon r^3$$ $$\frac{d\theta}{dt}=1$$ via polar coordinates? ...
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106 views

How to determine when this two variable transformation is invertible?

I am given: $$ X= U \cos(V) \tag{1}\\ $$ $$ Y = U \sin V \tag{2}$$ Now, I need to: a) Give the respective ranges for $U$ and $V$ in order that the transformation defined is one to one. and ...
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78 views

How do i scale 2D vector using matrix

I know that scale matrix is 2x2 { x, 0, 0, y } basis. My vector { 100, 2 } and i want to scale it using custom 2x2 matrix. I've read that if left operand is 2D row vector, then multiplying it on a ...
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11 views

Elementary Transformation of Mean density

Sir Im confused about dealing with some transformed mean density. I cant follow the logic how the author derived a specific result out from the following given: $$p(t) = \frac{1}{nπ} ...
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26 views

transformation of mean density

Sir Im confused about dealing with some transformed mean density. I cant follow the logic how the author derived a specific result out from the following given: $$p(t) = \frac{1}{nπ} ...
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21 views

What are The applications of Fast Walsh–Hadamard Transform.

There is a problem requiring the expect value of the intersection of two random subsets selected from a universal set, with the values and the probabilities of subsets given. My friend said it could ...
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20 views

Effects of Isomorphic Transformations on Vector Spaces.

Let $V$, $W$ be finite-dimensional vector spaces and let $T: V\rightarrow W$ be an isomorphism. Let $X$ be a subspace of $V$. Show that $T(X)$ is a subspace of $V$. My attempt: I know two vector ...
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Transforming differential equation to polar coordinates, example.

... For example, if we write $\dfrac{\operatorname dy}{\operatorname dx} = \dfrac{-x}{y}$ in polar coordinates, we obtain the equation $\dfrac{\operatorname dr}{\operatorname d\theta} = 0$ whose ...
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Fourier transform from Laplace transform

So what I did was Laplace transform $f(t)$ to $F(s)$ and then plug in $s=jw$. However, when I tried this for $cos(3t)$, $$F(jw)={jw\over9-w^2}$$ was the answer. I don't know if this is correct, and ...
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65 views

Strange results on training ANN by BP to “do” a kind of simple Discrete-Fourjer-Synthese

Here's the result of an experiment I'm trying to do atm. I've checked it, It really does a FourjerSythese! Can someone please explain me how these strange matrixs' can do that ? For those of you ...
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28 views

Compute the solutions of the following equation in Fourier space:

$$\frac{d^3u}{dx^3} − αxu = 0, x ∈ R, $$ where $ α > 0$ is some constant and $u(x)$ is assumed to satisfy $\int_R u(x) dx = π.$ I know this is a ODE so this is what I came up with so far: ...
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59 views

Transform of the Cartesian plane that maps hyperbolic arcs $xy = C$ to line segments

I have the finite set of curves: $$y = \frac{C}{x}, \qquad C = 2, 3, \ldots, C_{\max},$$ with $C$ and $x$ positive integers, $2 \le x \le C$ ($x$ varies on a finite domain). Is it possible to apply ...
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21 views

Homomorphisms inbetween factor modules

Consider the Ring $\mathbb{Z}$ and the two ideals $(n), (m)$, where $n, m \in \mathbb{N}$, and consider $GCF(m, n)$ (the greatest common factor of $m, n$). Let p: $\mathbb{Z}/(m) \to \mathbb{Z}/GCF(m, ...
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116 views

Fourier Decompositon problem

have a look at this video of Fourier Decomposition of an image (otherwise you can also refer the image, which shows few plots of different extracted waves from an image). We also know that a Fourier ...
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1answer
16 views

Determine the Fourier transform of $f(x) &

f(x)=1 if |x| < a or f(x) = 0 if |x| > a We use the formula $$ {1\over 2\pi} \int_{\infty}^\infty f(\bar x)e^{i\omega \bar x} $$ So is $f(\bar x)$ the same as $f(x)$ ?? In an answer they ...
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23 views

Inverse z transform - help

How can I find inverse z transform of $$X(z)=\frac{5}{(z-2)^{2}}$$ ? It is known that discrete signal x[n] is causal. Here is what I have done. Since signal x(n) is causal, convergence of z ...
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1answer
21 views

Construct a triangulated adjunction from a right triangulated adjoint functor?

In the Lemma 40. of the note on triangulated categories by Daniel Murfet, one finds the construction of a triangulated adjunction from a left (triangulated) adjoint triangulated functor, whose proof ...
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1answer
46 views

An expression with gcd and abs is transformed magically!

There's a problem to calculate $\sum^{n}_{i=1}\sum^{m}_{j=1}\frac{|i-j|}{\gcd(i,j)}$, whose tutorial gives the following transformation I really don't understand. ...
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46 views

Explain why: If T a linear transformation that is also onto (surjective) from V to W, then range of T = W.

I learned a corollary to theorem today. In the proof of the corollary, it states exactly what I typed in the title. Just curious how to make sense of R(T) = W (the co-domain). Thanks!
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94 views

matrix for reflection across a given line

Say I have the equation y = mx + k and I want to reflect vertices with respect to that line. How would I go about finding that matrix? Is there an example of such a matrix anywhere? I've only found an ...
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56 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 ...
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40 views

Rotations/Transformations with Complex Numbers/Eulers Formula

Hello, I am not entirely sure how to do this question, as I understand a rotation in the complex plane can be described by using Euler's formula, $e^{i\theta}$. Since this is an equilateral ...
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2answers
52 views

Affine transformation that sends a conic to itself but does not preserve the focci or the axes [closed]

So I'm trying to find an affine transformation that sends a conic to itself but does not preserve the foci or the axes. I don't know if this can be done. I'm pretty sure that if it is possible then I ...
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1answer
46 views

Reflection matrix in $ \mathbb{R}^{3} $.

I need help in understanding how they got the transformation matrix $ Q_{L} $ from Theorem 2 and $ P_{M} $ at the bottom of the page. They skipped some steps and I find it confusing. Any help would ...
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43 views

Linear operators proof, projection and reflection matrices

I am trying to understand two parts from the picture below in my textbook, but I dont understand how they arrived at it. I am trying to understand the proof below and how they got $P_L(\vec{v}) = ...
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29 views

Transformation and matrices

Two sequences $y_t$ and $z_t $ satisfy $$y_t = ay_{t-1} + bz_{t-1}$$ $$z_t = cy_{t-1} + dz_{t-1}$$ Where $a = 6$, $b = -20$, $c = -17$ and $d = -12$. From the two given equations above, ...
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1answer
63 views

What will happen if we try to reconstruct signal using phase only or magnitude only?

I am studying Fourier Transform and it's inverse. We get phase and magnitude from Fourier transform and reconstruct it back from both together My question is that What will happen if we try ...
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1answer
30 views

Eigenvectors and geometrical transformation

$$A= \begin{pmatrix} 2/3 & 2/3 & -1/3 \\ 2/3 & -1/3 & 2/3 \\ -1/3 & 2/3 & 2/3 \\ \end{pmatrix}$$, I need to understand that kind of ...