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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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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27 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
29 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
53 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) ∨ ¬ ...
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37 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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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: ...
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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)&...
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2answers
74 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,...,...
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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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39 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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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
61 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
11 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
25 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
30 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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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
14 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
26 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 ...
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1answer
35 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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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 ...
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1answer
52 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 ...
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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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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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25 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): ...
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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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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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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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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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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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16 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 ...
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1answer
28 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
27 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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19 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 ...
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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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21 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 ...
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1answer
27 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 ...
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17 views

finding the equation of a graph with 2 given characteristics

I need to find the equation for the graph of $y = x^2$ with the following characteristics 1. congruent to $4x^2 + 8$ 2. shares the same translations as $\frac{1}{3x - 9}$ I have attempted this ...
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19 views

Choose a Daubechies wavelet to approximate a polynomial function

Assume I have a polynomial function $s(x) = \sum\limits_{i=0}^d a_i x^i$, and $\boldsymbol{s}_{d}$ is a discretized sample from $s(x)$, i.e. $\boldsymbol{s}_{d}=(s(0), s(\Delta x), \ldots, s((n-1)\...
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1answer
38 views

Let $\hat{g}=f.$ Is $g$ continuous?

Let $f: \mathbb R \to \mathbb C$ such that there exists $g\in L^{1}(\mathbb R)$ with $\hat{g}=f.$ Then by Riemann-Lebesgue Lemma, we have $f$ is continuous and vanishing at infinity. My ...
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2answers
25 views

Translating a Plane

I am trying to understand plane equations but am finding it a bit confusing. My understanding of the plane equation says that for points that lie in the plane they will give an output of $0$ i.e. $f(...
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3answers
62 views

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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1answer
18 views

Order of affine transformations on matrix

I am trying to solve the following question: Apparently the correct answer to the question is (a) but I can't seem to figure out why that is the case. The only way I can seem to replicate the ...
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20 views

Transformation matrix from three points in 3-D space

I have three points in $3$-D space, say ${\bf p}_i, i=1,2,3$. A new co-ordinate system, $\{2\}$, is defined such that: The origin is located at the circumcentre, ${\bf p}_0$, of the triangle given ...
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19 views

Transformation matrix under a basis

I'm trying to find transformation matrix T under the basis {1, x, x^2} if \begin{align} T(a_0+a_1x+a_2x^2)=a_0-a_2x^2 \end{align} So \begin{align} T(1)=1; 1(1)+0(x)+0(x^2) \end{align} \begin{align}...
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1answer
29 views

Prove polynomial transformation is linear

Suppose a polynomial transformation: How do I prove the "closed under addition" property of linearity? I am trying this: I try to expand the equation on the left hand side, but I don't get ...
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8 views

Same sample points but get the different result of Fourier transform

Fourier Transform formula:$G(f) = \int_{-\infty}^\infty g(t)e^{(-i2 \pi ft)}dt$ I want to transform the following two equations: $ cos(2\pi f_ct)rect(\frac{100}{101}(t-\frac{1}{2}))\sum_{n=-\infty}^\...
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1answer
17 views

How to construct the combined transformation matrix that adjusts this rectangle.

Like what is shown above, dashed line constructed rectangle is the original one, and the solid constructed rectangle is the target we want to transform. So we can see that points of the original one ...