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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88 views

image of a circle under conformal trasformation

Consider a circle: $C_R=\{w=(x,y): |w|^2=x^2+y^2=R^2\}$ Prove that $A(C_R)$ remains a circle if $A$ is either a conformal or an anticonformal matrix. My attempt: I defined the complex number $z:=x ...
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36 views

Identity for reducing the power in the parameters of Hypergeometric functions

Is there any identity/formula for reducing/increasing the power in the parameters of the Gauss Hypergeometric function $ _2F_1(a,b;c;z^d)$ (d is a real) let's say to z? Is there also any identity for ...
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194 views

Hodograph transformation and implicit solution of a non-linear PDE

I am trying to understand how can one apply the Hodograph transformation to a non-linear PDE. I read that this transformation implies the representation of the solution in the implicit form . So, if I ...
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1answer
69 views

How to find a new point on rectangle based on an known point on the same?

I have rotated a rectangle a certain amount of degree and got the point(x,y)=(130,40) which was previously (152,60). Now i want to find the x,y(marked as red) value at another location based on the ...
2
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1answer
25 views

(Kleiner) transform preserves smoothness class

Consider the transform of nonnegative continuous concave positive homogenuous of first order function $f(x)$, $x \in \mathbb R^n_+$, $f \not\equiv 0$ given by $$ f^\times(y)= \inf \left\{ \left. ...
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1answer
87 views

Finding a hyperbolic isometry that fixes the point $x = 2$ and $x = 17$

I know that a Möbius transformation is hyperbolic if the trace is $> 2$ which is $a + d$. But I'm not sure of the next steps involved to arrive at the answer.
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2answers
159 views

Find an orientation preserving isometry $f (z) = \frac{az+b}{cz+d}$ such that $f (i) = 17 + 3i$

This is probably a very simple questions but I am not clear on Möbius transformations and how to solve this problem. I'd appreciate if somebody can point me towards a method to do these sort of ...
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0answers
47 views

transforming a straight band into a logarithmic spiral

I want to plot the labels and the graduations of an historical timeline onto a logarithmic spiral. If this timeline is on the $x$-axis, $-\infty$ would project to the center of the spiral, $+\infty$ ...
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2answers
205 views

Find the standard matrix of the transformation $T:\mathbb{R}^2\to \mathbb{R}^2$ that corresponds to the reflection through the line

Find the standard matrix of the transformation $T:\mathbb{R}^2\to \mathbb{R}^2$ that corresponds to the reflection through the line $x_2=2x_1$ followed by reflection through the line $x_1=3x_2$ I am ...
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2answers
7k views

Value range of normalization methods? min-max, z-score, decimal scaling

I am working my way through Normalization (data transformation) of data and was curious about four methods: min-max normalization, 2. z-score, 3. z-score mean absolute deviation, and 4. decimal ...
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1answer
78 views

Non-linear transformation preserving stereographic Riemannian metric on the sphere of radius R

I have been given a the Riemannian metric of a sphere of radius R in stereographic coordinates: $$G=4R^4\frac{du^2+dv^2}{(R^2+u^2+v^2)^2}.$$ I have shown that this metric is preserved under rotation, ...
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3answers
2k views

transformation of integral from 0 to infinity to 0 to 1

How do I transform the integral $$\int_0^\infty e^{-x^2} dx$$ from 0 to $\infty$ to o to 1 and. I have to devise a monte carlo algorithm to solve this further, so any advise would be of great help
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1answer
68 views

Prove a transformation is a variational symmetry for J

The following problem is from The Calculus of Variations by B.von Brunt (page 215, Exercise 9.2.1) Let $$ J(y)=\int_a^b xy'^2\mathrm{d}x. $$ Show that the transformation $$ X=x+\epsilon2x\ ...
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1answer
370 views

extract [0 … 2PI] rotation from 3x3 homogeneous 2D transformation matrix

I'm trying to extract the angle [0 ... 2PI] from a 3x3 homogeneous 2D transformation matrix. In fact I've found 2 very helpful posts at stackoverflow and stackexchange. ...
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2answers
46 views

Please, I need a more detailed explanation of the particular solution of the problem with vectors

Here is the problem and its solution (link to the source if you are interested): Two different points $A$ and $B$ are given. Find a set of such points $M$, that ...
3
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1answer
161 views

Finding the dimensions of subspaces of a Vector space and S-cyclic subspaces using minimal poynomials

I've been staring at a chapter in Bill Cooperstein's Advanced Linear ALgebra for some time now and one section is giving me trouble. It is about elementary divisors and invariant factors. My ...
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1answer
72 views

Separating $x_1^{x_2}$ into sum of two terms by variable transformation

Given the function $f(x_1, x_2)$ in the form of product of two variables. $$f(x_1, x_2) = x_1^{x_2}$$ I want to apply a variable transformation on this function, so that I can write it sum or ...
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1answer
112 views

Why is it called the “Unscented” transform?

I have not been able to track down the reason the Unscented Transform has the name it has. Can anyone shed some light on the meaning of the term "unscented" in this context?
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217 views

Way to Tietze's Transformation Theorem

during our knot-theory lecture we have talking about the following theorem: Given two finite presentations of the same group, one can be obtained from the other by a finite sequence of Tietze ...
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1answer
206 views

What is the sense of bottom row of affine transform matrix?

Usually affine transform matrix (in 2D) is represented like where block A is responsible for linear transformation (no translation) and block ...
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0answers
54 views

Linear 2D transform in the sense of geometric figures?

Consider tranformation which turns one aligned rectangle to another: This tranformation can be written in matrix form in the following way where ...
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239 views

Geometric intuition for Jordan normal forms (invariant subspaces, shearing, scaling, etc.)

I'm trying to visualize what a linear operator does to a vector space if that operator can be put into Jordan normal form. For concrete motivation, let's take $V = \mathbb{R}^3$, with some linear ...
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1answer
86 views

Finding $u$ and $v$ in Jacobian substitutions

I've used Jacobians before in multivariable calculus to simplify integrals, but I'm lost when I need to find the substitutions myself. Today on the quiz, there was the problem $\int\int_{R} xy dxdy$ ...
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1answer
183 views

Interpret the graph of $\frac{ax+b}{cx+d}$ as a transformation of $y=\frac{1}{x}$

As part of a problem-set I'm self-studying, I'm trying to interpret the graph of $f(x)=\frac{ax+b}{cx+d}$ as a transformation of the graph of $y=\frac{1}{x}$, including determining what restrictions ...
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1answer
145 views

Local axis follows origin node rotation

I'd like to define a local axis (unit vectors l, m and n) which once defined follow the rotation of the origin node, i.e. regardless of the deformation the local axis should be basically the same as ...
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1answer
144 views

How to calculate a double integral over a triangle by transforming to polair coordinates & by using a transformation

Let T be the triangel with vetrices $( 0,0 ) , ( 1,0 )\mbox{ and } ( 0,1 ) $. Evaluate the integral : $$ \iint_D e^{\frac{y-x}{y+x}} $$ a) by transforming to polar coordinates b) by using the ...
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2answers
78 views

Invertability of a linear transformation

Given $T : \mathbb{R}^3 \to \mathbb{R}^3$ such that $T(x_1,x_2,x_3) = (3x_1,x_1-x_2,2x_1+x_2+x_3)$ Show that $(T^2-I)(T-3I) = 0$. Solution 1: I can very easily write down the matrix representing $T$, ...
2
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1answer
144 views

Transforming matrix-equation to overdetermined minimum problem

i have broken down my problem to plainmath and could really use some help. Basis: I have an image. In this image I have several UV-XYZ pairs. So i know the 3d position of serveral Pixels. Given the ...
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1answer
132 views

Change of coordinate matrices

Find the change of coordinate matrices: Wherein B is the standard basis for P2 $$B' = (t^2+2,t+3,t^2+t+1) \\B" = (2t^2+t+1, t^2, 2t+1) \\ B= (t^2,t,1) $$ $$P_{B'B}$$ means the transformation for the ...
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1answer
148 views

Transformation of a matrix, change of basis

Find the change of coordinate matrices: Wherein B is the standard basis for P2 $$B' = (t^2+2,t+3,t^2+t+1) \\B" = (2t^2+t+1, t^2, 2t+1) \\ B= (t^2,t,1) $$ $$P_{B'B}$$ means the transformation for the ...
2
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2answers
112 views

question on transformation

If a $2$d coordinate transformation function is given by $f(x,y)= 3x+1$, then what does it mean? How do I calculate the transformed coordinates for the points say $(3,4)$ in the initial space?
2
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1answer
84 views

Transformations and coordinate Systems

I am working on some practice exercises (not homework) on transformations and need some intuition and help. One of the questions is: $(u,v)=f(x,y)$ where $ \quad u= { e }^{ x }\cos(y), \quad v = { e ...
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0answers
63 views

transformation function using genetic programming

If I have a set of points in two spaces, say set $A$ contains 50 points and set $B$ contains 50 points. I have to find a transformation function such that if I transform the points in set $A$ using ...
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1answer
79 views

Linear Transformation Concept Question

Consider that we have a given standard matrix of T and we are asked to find the image T(X) where X is a given vector. T is 4x3 and X is 3x1. is the solution of T(x) simply T*X?
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1answer
173 views

Regarding the kernel of a linear transformation and that of the associated representing matrix

Let $V, W$ be finite dimensional vector spaces over a field $F$. Let $\mathcal{B}_{V} = \{\mathbf{v_1, \cdots, v_n} \}$ and $\mathcal{B}_{W} = \{\mathbf{w_1, \cdots, w_m} \}$ be corresponding bases. ...
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0answers
57 views

Help in implementing of peak function in Fourier transform

I have a function Peak function I know how to implement it in time range just need to caclulate $r$. first I initial $x$, y with a range and meshgrid them, after it calculate $r$ ...
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1answer
108 views

Fourier Transform - Time Shift

Could someone help me understand how a simultaneous time shift on two separate functions is possible? I am having trouble linking a property to this solution. Given the function: $$x(t) = ...
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2answers
582 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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5answers
133 views

Linear map between duals induced by linear maps between vector spaces

Let $V, W$ be vector spaces over a field $F$ and let $\psi: V \to W$. Show that $\psi$ induces a linear map $\psi^{*}: W^{*} \to V^{*}$ naturally. Although the question asks for a naturally induced ...
2
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3answers
54 views

X has pdf $f(x) = \frac{x^{2}}{18}$ for -3<x<3, what is the pdf of $X^{2}$

So this was my solution: Say, $Z = X^{2}$, then $X=\pm \sqrt{Z}$ and, $$P(Z=z)=P(X = \sqrt{z}) + P(X = -\sqrt{z}) = \frac{z}{18} + \frac{z}{18} = \frac{z}{9}$$ for $$0<z<9$$ However: ...
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2answers
114 views

Basic questions regarding matrix algebra.

I had two true/false questions on my exam of which I missed. $1)$ The map $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by $T(x)=x+e_1$ is a linear transformation. I know this to be false, because ...
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1answer
116 views

Set with plane having two axes of symmetry with angle of intersection

Proof that a closed and compact subset of plane having two axes of symmetry with angle of intersection not a rational multiple of $\pi$ is either a disc or a whole plane. Proof The composite of two ...
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1answer
225 views

transformation between square and a polygon?

I have a square and a polygon. I want to transform all the points inside this square such that they are mapped inside the polygon. I was trying using scale and rotate matrices but I am not able to ...
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0answers
62 views

Is there an intuitive understanding of what a walsh coefficient is?

I am working with Walsh coefficients. I know the intuitive understanding is almost that that they are the degree of connectivity, but it is there a better way of thinking about it? What is the ...
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2answers
499 views

How to compute a matrix for rotating and centering rectangle in viewport?

I have a rectangle given by 4 points. I'm trying to compute a transformation matrix such that the rectangle will appear straight and centered within my viewport. I'm not even sure where to begin. ...
2
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1answer
3k views

How to find a transformation matrix having several original points and their respective transformed results?

I have three original points $pt_1, pt_2, pt_3$ which if transformed by an unknown matrix $M$ turn into points $gd_1, gd_2, gd_3$ respectively. How can I find the matrix $M$ (all points are in ...
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1answer
101 views

The Legendre Transform of Bernoulli r.v.s

This question is most likely related to calculus/algebra and tricks regarding supremums rather than actual understanding of large deviations and the Legendre transform, but anyway. For a random ...
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1answer
55 views

Transpose of 2 matrices together

So if I have an $m\times n$ matrix $A$ and I represent that matrix as $\displaystyle A = QR$, how do I write $A^{T}$ (transpose) in terms of the original $\displaystyle QR$? Does it become ...
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123 views

Proof for a summation-procedure using the matrix of Eulerian numbers?

I've discussed a procedure for divergent summation using the matrix of Eulerian numbers occasionally in the last years (initially here, and here in MSE and MO but not in that generality and thus(?) ...