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-...

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3answers
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Distribution of $R^2 = X^2 +Y^2$ where (X,Y) is a point on the unit circle

So I have a point $(X,Y)$ chosen from the unit disk with uniform distribution. And I'm attempting to find the distribution of $R^2$, where $R$ is the distance from the point to the origin. Now ...
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2answers
77 views

Can a linear transformation $\mathbb{R}^n \rightarrow \mathbb{R}^n$ have a complex eigenvector? [closed]

Can a transformation $\mathbb{R}^n \rightarrow \mathbb{R}^n$ have a complex eigenvector? Similarly, can a transformation $\mathbb{C}^n \rightarrow \mathbb{C}^n$ have a real eigenvector?
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1answer
22 views

Defining a region in $\mathbb{R}^2$

I was trying to do this exercise but my answer doesn't match with the solution and I'm wondering why: Consider the coordinates transformation defined by $x=2u+v$ and $y=u^2-v$. Being $T$ the ...
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2answers
30 views

Transformations of points in the plane

Hopefully somebody understands what I mean here, If take a polynomial with complex numbers as input, then I will get a complex number as an output. If the input and output are plotted on an Argand ...
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1answer
16 views

Transformation of a real number 4 by 4 matrix to a 3 by 3 matrix

Are there any transformation methods that could be used to represent a $4\times 4$ matrix of real constants as a finite product/sum of $3\times 3$ (numeric: real or complex) matrices?
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0answers
59 views

True or false: Every transformation T: Cn --> Cn (n ≥ 2) has n distinct eigenvectors

True or false: 1) Every transformation T: Cn --> Cn (n ≥ 2) has n distinct eigenvectors 2) Every transformation T: Cn --> Cn (n ≥ 2) has at least 1 eigenvector 3) Every transformation T: Rn --> Rn (...
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2answers
22 views

Polar Coordinate Conversion (Integration)

I want to convert some integrals to use polar coordinates as my differentials, my problem is getting the limits. So this is the first concept I am not understanding: If I have a circle in the xy-...
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1answer
18 views

Matrix transformations on objects

I am trying to solve the following question: I have created the scaling, translation and rotation matrices that I feel will transform the left figure to the figure on the right: Scaling $$ \...
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1answer
23 views

Aligning 2 Coordinate Systems

I have a camera and a table and I want to align the camera to co-exist in the same coordinate system as the table. Here is an image of the setting. What type of mathematical transformations I need to ...
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0answers
13 views

Prove that $l$ and $l_{AB}$ are parallel if and only if $\sigma_B \sigma_l \sigma_B \sigma_A \sigma_l \sigma_A = id$

Prove that $l$ and $l_{AB}$ are parallel if and only if $\sigma_B \sigma_l \sigma_B \sigma_A \sigma_l \sigma_A = id$ I imagine that this proof has to be along the lines of a proof by contradiction, ...
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0answers
19 views

Linear transformations and possible dimension mismatch

The problem: Let $L: R_4 \to R_3$ be defined by $$L([u_1, u_2 ,u_3 ,u_4]) = [u_1 ,(u_2+u_3), (u_3 + u_4)]$$ Let S and T be the natural bases for $R_4$ and $R_3$, respectively. Find the ...
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1answer
47 views

How to you find out what a matrix does to an equation.

Lets say I have an equation of a plane, $$x-3y+2z=0 $$ and I get matrix to transform it with say a 3x3 matrix with just a-i as place holders for the values in the matrix. How would I find what the ...
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1answer
32 views

Get vertex points of transformed rectangle knowing bounding box and transform matrices

(I'm not a mathematician so talk down to me). I have a rectangle that has been transformed by a series of matrix transforms. I can recover the transform matrices and get the x,y coordinates of each ...
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1answer
30 views

Finding the area between two curves using a set of transforms and their Jacobian

I have the following transforms: $\begin{align} x &= u^2 - v^2 \\ y &= 2uv \end{align}$ and am tasked with finding the area between the following curves: $\begin{align} x &= 4 - \frac{y^...
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4answers
58 views

Negation with De Morgan’s law

I'm having a hard time getting my head around transformation proofs. There is one particular example demonstration in the material I'm studying which I can't make sense of From this ¬ (¬ (¬ p) ∨ ¬ ...
3
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1answer
39 views

What is known about the space of measure-preserving transformations?

I started reading about measure-preserving transformations, the ergodic theorems and mixing, but I was also wondering what is known about the space of measure-preserving transformations. The books ...
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1answer
24 views

Transforming an integral to a different domain

For a given $v(x)$ with $x\in[0,1]$, use the variable transformation $x=g(\eta)=\frac{1}{2}\eta+\frac{1}{2}$ to transform the integral $I=\int_0^1v(x)dx$ to an integral over $[-1,1]$. My doubts: ...
0
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1answer
25 views

Image under $T_j$ of the basis vectors $e_1$ and $e_2$.

Define the linear transformation. Decide which of the mappings of $\mathbb R^2$ to itself given below are linear. $$\begin{align}T_1(x,y)&=(x+2y,y-2x)&T_2(x,y)&=(x,2x+y)\\T_3(x,y)&...
0
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2answers
77 views

Calculate Rotation Matrix to align k n dimensional vectors

I have a $k$ number of $n$-dimensional vectors written with respect to two rotated frames: $X= \{\vec{x}_1,\vec{x}_2,...,\vec{x}_k\}$ and the same rotated vectors: $X'= \{\vec{x'}_1,\vec{x'}_2,...,...
2
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1answer
38 views

Finding a Transformation for a Sum of Exponentials

I am looking to see if it is possible to find a transformation $T_i(f(x))$ such that $$T_1\left(e^x+e^{ix}+e^{-x}+e^{-ix}\right)=e^x-ie^{ix}-e^{-x}+ie^{-ix}$$ $$T_2\left(e^x+e^{ix}+e^{-x}+e^{-ix}\...
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1answer
40 views

What is wavelet tranform in simple words?

I have read wiki and other sources and have still problem understanding the wavelet transform. What is the basic idea in simple words? Does the Fourier uncertainty hold for wavelet transform?
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0answers
3 views

Derivate formula for Radon-transformation

For the Radon-transformation $\mathcal{R}f(r,\omega)=\int_{\{x:x\cdot\omega=r\}}f(x)\mathrm{d}\sigma(x)$ with $r\in\mathbb{R},\omega\in\mathbb{S}^{n-1}$ I want to prove the following derivative ...
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2answers
64 views

Laplace transform of $\sin(\sqrt t)$

How can I use this differential equation $$4tf''(t) +2 f'(t) + a^2 f(t)=0$$ to show that $$L(\sin(\sqrt{t}))=\frac{1}{2}\sqrt{\pi}\,\frac{1}{s^{\frac{3}{2}}}\,e^{\frac{-1}{4s}}$$
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0answers
12 views

Abel transformation - sources

I'd like to study the Abel transformation, that is, $$Af(x) = \int\limits_x^\infty\frac{f(t)t}{\sqrt{t^2 - x^2}}\ \mathrm{dt},\quad x\in(0,\infty).$$ I'm especially interested in estimates/...
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1answer
26 views

Area of Region using Transformation

Let R be the region bounded by the curves x = 0, y = sin(x)+1, y = sin(x), and y = 2 − x. Find the area of R. I need to use a transformation to find this, but I could not solve it using a ...
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1answer
33 views

A Deeper Understanding / Interpretation of Homographies

I currently understand that a homography matrix, which allows for a mapping between planes in 3-dimensions, is a $3\times3$ matrix of the following general form: $$\begin{bmatrix} \vert & \vert &...
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0answers
20 views

Zero Order Laguerre Transform of Sin(at)

Zero Order Laguerre Transform is given by $$L\{f(t)\}=\int_0^\infty e^{-t} L_n(t)f(t)dt $$ I've to prove $$L\{\sin(at)\}=\frac{a^n}{(1+a^2)^{\frac{n+1}{2}}}\cdot\sin\left[n\tan^{-1}(a^{-1})+\tan^{-...
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0answers
22 views

Transform matrix into constant diagonal matrix (or hollow matrix)

Does there exist a (possibly unique) orthogonal transformation, $U$, which will create a hollow matrix (or matrix with constant diagonal entries) from an arbitrary symmetric matrix, $A$? $\sum_{kl}...
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0answers
15 views

Conformal transformation of an annulus

I am given two circles in the complex plane, one of radius a, the other of radius b such that $a<b$. Their centres are separated by a distance h such that $a+h<b$. I need a conformal ...
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0answers
28 views

what is a ordinally quadratic function?

A function is ordinal equivalent to another means there exist a (unique) monotonic transformation between wiki definition of ordinal utility. I am a little confused, a function is ordinally quadratic ...
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0answers
17 views

What kind of transformation matrix should i use?

I am trying perform inverse kinematics on a 6 jointed robot, but is having a hard time determining how my transformation matrix should look like. I am using a piece of software to which you feed an ...
0
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1answer
36 views

Map an ellipsoid to a sphere

If I have a ellipsoid described by: $(\boldsymbol{x} - c)^T \boldsymbol{A} (\boldsymbol{x} - c) = 1$ How do I get the transformation to an unit sphere centered at the origin? From the principal ...
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0answers
35 views

How to obtain the given “analytical” solution to the 4th order ODE?

The 4th order ODE is $$(D^2-k^2)^2f=0, \qquad (1)$$ where $f=f(y)$, $D\equiv\frac{d}{dy}$, and $k$ is a constant. It is subject to the boundary conditions $f(0)=f'(0)=f(1)=0$. A solution to (1) is ...
0
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1answer
53 views

How to skew a box to fit inside another under certain conditions

I guess my question is either fairly simple or impossible to solve. I have two boxes. One (I'll call child box) inside another (I'll call parent box). The parent box has width x and height y. The ...
0
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1answer
22 views

Find the matrix of ortogonal reflection

Let $e_1, e_2, e_3$ be an orthonormal basis for $R^3$ and consider the plane with equation $x_1 + 2x_2 - 2x_3 = 0$. Find the matrix of orthogonal reflection in that plane with respect to the given ...
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0answers
17 views

Partial Deviation of a Langrange-Function $L$?

$E$ is a transformation matrix which shall only do a rotation. Thus, $E^T * E = 1$. This requirement leads to an optimizing problem with a restriction which can be solved by Lagrange-optimization. To ...
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0answers
30 views

Applying rotation matrix on inclined plane

I want to rotate an inclined plane to a flat surface. I think I can use the Euler angles to perform this operation. Using following points (Matlab): ...
0
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1answer
9 views

How to calculate z transform of $x(n)=(n-1)(\frac{1}{2})^{n-2}u(n-2)$

Let $x(n)=(n-1)(\frac{1}{2})^{n-2}u(n-2)$, where $u(n-2)$ is shifted unit step function. How can I calculate z transform of this function? By definition, $X(z)=\sum_{n=-\infty}^{n=\infty}x(n)z^{-n}=\...
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0answers
39 views

I'm looking for a rotation matrix for following transformation

I'm working with a 3D camera and I found out the formula to transform the camera measurements to real world coordinate system when you have a rotation around x and y (no z rotation). http://i.imgur....
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1answer
29 views

Using Fourier Transform to solve an ODE

Consider the differential equation $$f^{iv}+3f^{''}-f=g$$ I have read that taking the Fourier Transform of both sides gives $$\left(i\lambda\right)^{4}F\left(\lambda\right)+3\left(i\lambda\right)^...
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0answers
38 views

Notation of the transformations in Linear Algebra

I am very confused with succinct notations of the transformations in Linear Algebra. When do we write each of the ways? What is the difference? In the lecture notes it says: T(x) = Ax = b in R^m, ...
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2answers
74 views

Get the four corners of a rectangle

I have a boundary given ($xMin$, $yMin$, $xMax$, $yMax$) and the two points of a reference line of a rectangle. The begin point is at $(x_b, y_b)$ and the end point is at $(x_e, y_e)$. This reference ...
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0answers
17 views

Apply homogeneous transform to line parameters

I have a 2D range scanner mounted on a robot. This scanner is tilted around its x and y axes (meaning its scanning plane is not horizontal) with some unknown small angles. I initialize those roll and ...
0
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1answer
30 views

Concatenating two Rotation-Matrices

I have two $2\mathrm{D}$-planes in $3\mathrm{D}$-space with orientation parameters expressed as rotation $R_1$ and translation $T_1$ and rotation $R_2$ and translation $T_2$ with respect to some ...
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1answer
28 views

Mobius transforms which map the region ¨ (C \ D(1, 1)) ∩ D(0, 2) into the strip {|Im z| < 1}.

So far I have thought about first having my transformation, $T$, map $i$ to $\infty$ so that I get two parallel lines. But then I am not sure where to proceed from there.
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2answers
31 views

Transforming $f$ into $|f(-x)|$ and $\frac{1}{f(x)}$

I apologize if this is extremely straightforward. So I've been given a drawn graph of $f$, and it is asking me to draw the transformed graph $|f(-x)|$ and $\frac{1}{f(x)}$. For $|f(-x)|$, I ...
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0answers
21 views

finding the standard matrix for the transformation.

So assuming u=(cos x, sin x), where x is theta of course, is the direction of a line through (0,0) in R^2. And T is the operator which reflects vectors about the line. Using T(e1) and T(e2), How would ...
0
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1answer
22 views

Laplace: exponential transformation!

What is the inverse transformation of exponential? L(e^(at)) <--> 1/(s-a) However, if I have to I have to do the inverse transform of e^(-s)?
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0answers
22 views

Transforming points between two polar coordinate systems

I have 2 dimensional points (r, theta) defined in a polar coordinate system A, and a second polar coordinate system B with a known homogeneous transform T transforming between A and B in a Cartesian ...
0
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1answer
28 views

Gaussian integration involving operators

I was just wondering if someone could explain the following series of equalities: The Gaussian integral may be evaluated using an orthogonal transformation $R$ to diagonalise the real symmetric ...