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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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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30 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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9 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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2answers
43 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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1answer
101 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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31 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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9 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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23 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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11 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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1answer
18 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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44 views

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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26 views

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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56 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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26 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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40 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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1answer
20 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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104 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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21 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
17 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
42 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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1answer
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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1answer
57 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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53 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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21 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
44 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
39 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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1answer
34 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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1answer
28 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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94 views

If $\frac{x-1}{e^x-1} = y$ then $x=?$

I have following equation: $$\frac{x-1}{e^x-1} = y$$ I want to solve this equation such that I have the value of $x$ in the term of $y.$ i.e. inverse of the equation
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1answer
44 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
26 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 ...
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1answer
35 views

The curvatures of a transformed surface under a similarity transformation

Setup: Let $f:\mathbb R^3\to\mathbb R^3$ be a similarity transformation. Then $f=rA+b$ for some fixed orthogonal matrix $A$, vector $b$ and nonzero real $r$. Suppose $S$ is a surface, and $S'=f(S)$. ...
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1answer
40 views

What is this subclass of the class of monotonic transformations?

Let $u$ be a continuous function from $R$ to $R$. Then $v$ is called a positive monotonic transformation of $u$ if $u(x) < u(y)$ if and only if $v(x)<v(y)$ and similarly for greater than and ...
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42 views

The Adrian Transformation of a function in $\mathbb{R}^{2}$

Recently I came upon a problem (if you would call it that, more of a thought experiment), which was phrased something like this: Rotate the area formed by $\int_{-1}^12dx$ around the curve ...
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39 views

Is this chain of inequalities correct?

Is this chain of inequalities correct? If not how to make it works? $$\frac{\ln \left( 1+x^3+y^3 \right)}{\sqrt{x^2+y^2}} \le \frac{\left( x^3+y^3 \right)}{\sqrt{x^2+y^2}} \le \frac{ \left( ...
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1answer
28 views

How to prove $\sqrt{Y}$ can be variance stabilizing transformation of poisson distribution?

I am studying constant variance checking when conducting ANOVA. I know that $\sqrt{Y}$ is one of the common transformations for a Poisson distribution, but I can't prove it. I also read Anscombe ...
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1answer
21 views

Identifying translations and rotations as compositions.

I am having trouble understanding the below which are the ones underline in red and blue. For the red: Why is that $R_{A,90}(A)=A$ and that $\tau_{AB}(A)=B$ As for the blue: Why is that ...
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15 views

What is or how do you get the rotational matrix of 4-D vector onto the xyz-space?

which would make the 4-D component 0. To be honest I'm not really sure how 4-D rotations work. I know about the simple rotations but not the mechanism in how it rotates, and I'm not sure whether to ...
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1answer
25 views

Does the following series of transformations of inequalities holds?

I am to calculate limit of the function $f(x,y)$ i am trying to apply squeeze theorem. Is the following series of transformations of this inequality correct? If not how to do this correctly? i.e. are ...
3
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1answer
21 views

Let $R$ be any rotation and $P$ any reflection then $R \circ P$ and $P \circ R$ are both glide reflections

Let $R$ be any rotation and $P$ any reflection then $R \circ P$ and $P \circ R$ are both glide reflections I am having trouble showing $P \circ R$ is a glide reflection, I manage to get $R \circ P$, ...
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29 views

Solids of Revolution around other functions.

Recently I've been thinking about solids of revlution, and thought about an interesting experiment. Can you rotate functions around, for example, the line $f(x)=x$? And consequently, could you rotate ...
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9 views

Transformation of graphs, finding the values of unknowns

I am a second grade IB student using "Mathematics Standard Level for the IB Diploma, Cambridge" book.This is the question I have a problem with: "Let f(x)=(3x-5):(x-2) a) Find the value of constants ...
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46 views

Find a basis for Kernel and Image of a Linear Transformation

Given: $$A = \left\{\begin{bmatrix} 0 & 1 \\ 0 & 2 \\ 0 & 1 \end{bmatrix}\right\}$$ Find a basis for $ImT_A$ and $kerT_A$ So far, I've ...
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1answer
32 views

generating system of the kernel of a module-transformation

Let $G ≠ 1$ be a group and A a commutative ring. Now, the group ring $A[G]$ is naturally an A-module. Next, let's consider the transformation: $$\phi: A[G] \to A, \sum_{g \in G} a_gg \mapsto \sum_{g ...
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18 views

How can I transform coordinate systems based on quaternion data?

I have a single rigid body object, and its orientations in quaternion with respect to two coordinate systems, each is called original and prime, respectively; therefore, I have two quaternions ...
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2answers
22 views

Why cannot the homogeneous coodinates be zero?

Given a point (x, y) on the Euclidean plane, for any non-zero real number Z, the triple (xZ, yZ, Z) is called a set of homogeneous coordinates for the point. Why can't Z be zero?
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1answer
47 views

How to transform function values to specific interval

I'm doing a project at university about scientific computing and I'm stuck. As in: I seem to lack quite a bit of mathematical background for this project. The program has as input an array of $x$ ...
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11 views

transformation of variables in Melnikov method

Supposing there is a system of non-autonomous non-linear differential equations with small damping and small forcing. The unperturbed system (zero damping and forcing) is Hamiltonian but neither has a ...
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24 views

Is the transformation $T: (r, \theta) \to (r, \theta + \phi)$ linear? Here $\phi$ is a given angle

Let $T$ rotate every point through the same angle $\phi$ about the origin, $i.e.$ $T: (r, \theta) \to (r, \theta + \phi)$ where $\phi$ is given. If in addition that $T(O) = O,$ namely, if $T$ maps the ...