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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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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20 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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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, ...
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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). ...
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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 ...
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37 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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72 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 ...
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13 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
25 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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18 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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19 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
26 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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18 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, ...
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1answer
35 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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23 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. ...
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3answers
53 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
14 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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19 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} ...
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1answer
27 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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7 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 ...
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15 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 ...
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13 views

Transformation between camera velocity and projection of a tube's cut

I have a vision sensor / flying camera inside a circular tube as shown below: The camera is using the pinhole model and I am using edge detection to detect the "black" hole that is forming due to ...
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2answers
45 views

Integration similaring to Fourier transform of Gaussian function

I would like to calculate the integral: $$\int^{\infty}_{0}x\cdot \exp(-x^2)\cdot \exp(-ikx)dx$$ Are there some tricks to solve it? Many thanks.
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1answer
28 views

How to show $T$ satisfies $T^2=I$ if $T$ is such mapping that $T(A)=A^t$? [closed]

Here how do you find the matrix of tranformation. I guess it is $A^{t-1}$. But how is $A^{2(t-1)}=I?$ In addition, let's say $\lambda =1,-1$ in this case (as I think so). How do you find eigenspace ...
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9 views

Fourier transform of real part part of a complex signal

Let $$g(t) = g_r(t) + j g_i(t)$$ be a complex signal with $g_r(t)$ and $g_i(t)$ real signals. We can write the Fourier transform $$G(f) = G_r(f) + jG_i(f),$$ where $G_r(f)$, $G_i(f)$ and $G(f)$ are ...
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46 views

sum of singular vector dyadics derived from the matrix itself

I have an m*n (m>n) n-rank matrix (let's denote it by A), with nonnegative elements. SVD decomposition says, that A=UDV', where U and V are orthogonal matrixes, and their columns are the singular ...
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Transform and domain such that phase shift of the original function results in shift in transformed domain variable

It is known that the Fourier transform of a phase-shifted function results in a constant shift of the dependent variable of the phase spectrum: If $ F(x(t)) = X(w) = |A(w)| \cdot e^{-i \cdot ...
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19 views

Relative Motion between two rotating frames

I am looking for mathematical relations between equations of motion between two rotating frames. Relating the said motion by first going to a globally fixed frame is not an issue but how to approach ...
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17 views

Scale a rectangle about a point considering reflection

Given a rectangle of width $(w_0)$, height $(h_0)$, left $(x_0)$, top $(y_0)$. How do I scale it from an origin $(x_1,y_1)$ with a scale factor of $(w_1, h_1)$ taking into account reflection? This ...
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1answer
29 views

Find equation of log function given graph

How would you find the equation of this log graph with respect to transformation of the function $$ a\log[k(x-d)] + c$$ I had trouble solving this accurately.
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Certain symmetrized product of cosines - can it be transformed into more manageable form

I am interested in the following expression: $$ F_{k_1,\ldots,k_n}(t):=\sum_{\sigma\in S_n}\cos(\sigma(1)k_1t)\cos(\sigma(2)k_2t)\cdots\cos(\sigma(n)k_nt) $$ where $k_1, \ldots, k_n$ are natural ...
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29 views

Fast procedure to know which transformations are linear

I have the following transformations: How can I tell if the transformation is linear? Is there a fast way of knowing (by looking at equations)? or do I need to check through the theorems: ...
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1answer
42 views

Is there any way to find an explicit formula for the adjoint of a linear transformation?

I know that the definition of the adjoint of a linear transformation is defined to be $\langle T(x), y \rangle = \langle x, T^{*}(y) \rangle$ but is there any way to find an explicit formula for ...
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1answer
27 views

How to check if the following are isomorphism?

$a)$ $T: P_3(R)\rightarrow P_3(R)$ given by $\ T(p(x))=xdp(x)/dx$ $b)$ $T: P_2(R)\rightarrow R^3$ given by $T(p(x))=(p(0), p(1), p(2))$ The precondition I know is $\dim (V)=\dim (W) \iff $the ...
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1answer
18 views

Group of Motions

In Needham's Visual Complex Analysis on pdf page 57, book page 37, Needham says that the set of indirect/opposite motions (motions reflecting the angle of a vector input) does not form a subgroup of ...
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75 views

Understanding bounded linear operators

The definition of a bounded linear operator is a linear transformation $T$ between two normed vectors spaces $X$ and $Y$ such that the ratio of the norm of $T(v)$ to that of $v$ is bounded by the same ...
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1answer
39 views

Finding a transformation that yields a prescribed PDF

I am attempted to procure a function from a composition when given the PDF (I typed the full problem at the bottom in its entirety in case I left out details in my inquiry). I understand how to get ...
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1answer
26 views

Find conditions on $C$ and $C^{\prime}$ so that the spirals $r = Ce^{\varphi/a}$ and $r = C^{\prime}e^{\varphi/a}$ are the same

This question is related to one I asked here about the logarithmic spiral. In the linked problem, I had to find and sketch the image of the straight line $z=(1+ia)t+ib$, for $-\infty < t < ...
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Find and sketch the image of the straight line $z = (1+ia)t+ib$ under the map $w=e^{z}$

I need to find and sketch the image of the straight line $z = (1+ia)t +aib$, where $-\infty < t < + \infty$, $a,b\in \mathbb{R}$, and $a \neq 0$, under the map $w = e^{z}$. In order to ...
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44 views

Going from a point in a sphere from a point in an ellipsoid

Given a certain vector $v$ on the surface of a sphere centered at $0$, I'm trying to find another vector $w$ such that $w$ and $v$ are colinear and $w$ is on the surface of an ellipsoid also centered ...
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16 views

How do projections in 3D with homogeneous coordinates work?

Affine 3D transformations can be expressed in homogeneous coordinates by a matrix $M \in \mathbb{R}^{4 \times 4}$. This means we have 16 parameters to calculate. The first thing I asked myself is how ...
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1answer
40 views

Problem finding Jacobian when computing density

I'm having trouble finding the Jacobian when trying to compute a distribution. If $(X,Y)$ is a point on a unit disk with radius $1$, I'd like to find the density of the distance between the point and ...
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1answer
26 views

Explain Similarity Transformation

For two n-by-n matrices $X$ and $Y$, we know that they are similar if the following is true for some invertible n-by-n matrix $M$: $X = M^{-1}YM$. Can anyone explain what the $M$ and $M^{-1}$ are ...
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39 views

How do I graph these curves?

My teacher taught me this in class but I still don't understand it. Could someone please explain how to graph $y= x +\frac1x$ and $y = x - \frac1x$? Thank you :)
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Fourier Transform of triangle function 𝑥(𝑡)=Δ((t-1)/2)

Can you please tell me if my working is right for the fourier transform of this function: 𝑥(𝑡)=Δ((t-1)/2) My workings are: my workings I have used the fourier transform standard results. Please ...