# 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).

2,270 questions
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### Linear Algebra Question ( rank of matrix )

Let $\bf A$ be an $m \times n$ matrix. If $\bf P$ and $\bf Q$ are invertible $m \times m$ and $n \times n$ matrices, respectively prove $\operatorname{rank}(\mathbf{PA}) = \operatorname{rank}(\bf{A})$...
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### Scaling - Rigid or Non-Rigid Transformation

I am trying to look for a precise definition of what rigid and non-rigid transformation is, and to which categories does 'scaling' belong. This is connected to a Point-Set registration problem that I ...
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### Rotation of an ellipse fixed at two points

I have a situation for which I have made a very crude drawing. Let's say we have an ellipse in $\mathbb{R}^2$ that is fixed at $x_0 = -a$ and $x_1 = a$ (as if it were resting on two poles). I am ...
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### Transforming a square into a parallelogram

as an exercise I wanted to calculate the transformation matrix in order to make the square $ABCD$ into the parallelogram $A'B'C'D'$. I am able to get the matrix so that the square is first at the ...
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### Extract param of $\sin$ from expression $y=2(\sin b-\sin a)/(\sin c-\sin a)$

$y=2\frac{\sin b-\sin a}{\sin c-\sin a}$, where $a=q(n+0)$ $b=q(n+1)$ $c=q(n+2)$ $q=\frac{2 \pi f}{s}$ Is it possible to extract $n$ from this formula? I already try this on WolframAlpha but I do ...
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### Prove that for $x\neq 1, 0<y<\pi/2$ the system $u=\sin y/(x-1) ,v=x\tan y$ define a system of curvilinear coordinates.

Prove that for $x\neq 1, 0<y<\pi/2$ the system $u=\sin y/(x-1) ,v=x\tan y$ define a system of curvilinear coordinates. So this amounts to showing that the transformation is injective. ...
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### What does it mean for a Wavelet transform to “commute” with a translation?

I'm referencing this paper here: https://arxiv.org/pdf/1203.1513.pdf Within this paper, it states that "A wavelet transform commutes with translations, and is therefore not translation invariant". ...
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### From right-skewed to normal distribution

I have a variable that has this right-skewed (positive skew) distribution below: I aim to transform it in order to get a normal distribution. I have tried standard transformations (log10, natural ...
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### 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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### If $T$ is an invertible linear transformation and $\vec{v}$ is an eigenvector of $T$, then $\vec{v}$ is an eigenvector of $T^{-1}$

I saw there is a proof for invertible matrices, but I don't know how to put this mathematically for a transformation. How do I prove an invertible linear transformation has the same eigenvectors as ...
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### If $X$ is an exponentially distributed variable with mean $\lambda$, $Y=−3\ln(X)$ has Gumbel distribution?

Let X be a random variable which follows an exponential distribution with parameter $\lambda$ ($\lambda>0$), find the distribution of the random variable $Y = −3\ln(X)$. So this is my answer for ...
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### Getting a transformation matrix from a normal vector

I'm trying to randomly generate coordinate transformations for a fitting routine I'm writing in python. I want to rotate my data (a bunch of $(x,y,z)$ coordinates) about the origin, ideally using a ...
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### Image under billinear transformation

What is a image of $x+y>4$ under billinear transformation $B(z)=\frac{z-4-8i}{z-4}$? I got that $B(z)=1-\frac{8\sqrt{2}e^{i\frac{\pi}{4}}}{z+4}$, but I cannot conclude image correctly (it should ...
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### Find joints positions in 3D robotic manipulator?

I have been trying to solve this 3D mechanics problem, but can't seem to be able to figure out what the best way to do it is. I have this $3D$ robot manipulator with $3$ rotary joints $(B, C, D)$. I ...
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### Acos 90 degree matrix transformation.

I'm writing a program that transforms a matrix of points by 90°. In it, I have two vectors from which I am performing the rotation. Both vectors are normalized: ...
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### Using transformation to evaluate double integral

Given the transformation $T(x, y) = (x - y, x + y)$, evaluate the double integral $\iint_R (x^2+y^2) dA$, where $R$ is the rectangle in the $xy$-plane with vertices $A(1, 1)$, $B(2, 2)$, $C(-1, 5)$ ...
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### Monotonic transformation to smooth the probabilities

I am studying some event for a set of objects that can be plotted on a square $[0, 100] ^ 2$. I have used logistic regression to calculate probabilities that event occur for different objects and the ...
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### Calculate transform horizontal/vertical skew and scale from 2d coordinate

I'm currently working on a javascript which allows to create a 3D rotating cube. I successed to create the 3D cube thanks to 8 points coordinates. However, I need to add an image on one the cube face. ...
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### Show that if $T$ is surjective and spans $V$, then $T(S)$ spans $W$.

Given that $T: V \to W$ is a linear transformation from $V$ to $W$. Show that if $T$ is surjective and $S\subset V$ spans $V$, then $T(S)$ spans $W$. I think the main thing stumping me right now is ...
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### Prove that $\sin^2(\pi x)$ is chaotic

My approach is based on the following from the book Chaos and Fractals: New Frontiers of Science, by Peitgen, Heinz-Otto, Jürgens, Hartmut, Saupe, Dietmar. To be more specific: "If $f$ is chaotic and ...
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### Transforming $\left(\begin{smallmatrix} A^{T} \\ -A^{T} \end{smallmatrix}\right)^{T} x = -b$ after using Farkas Lemma

Let $A \in \mathbb{R}^{m\times n}$ and $u, b \in \mathbb R^{m}$. I am close to proving: $A x =b$ has a solution $\iff$ $b^{T}u \leq 0$ and $A^{T}u=0$. Using Farkas Lemma I get to the point where ...
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### What is image of $f(z)=\tan(z)$ where $\Im(z)=cst$?
Can you help me figure out what is the image of line segments ${z =x+iy: -π/2<x<π/2, y=const}$ under $f(z)=\tan(z)$. I've got \$tan(x+iy) = sin(2x)/(ch(2y)+cos(2x)) + i sh(2y)/(ch(2y)+cos(2x)) ...