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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13 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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18 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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0answers
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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1answer
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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0answers
11 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
40 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
26 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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0answers
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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1answer
44 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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9 views

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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18 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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0answers
16 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 ...
2
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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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33 views

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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2answers
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
33 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
25 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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3answers
73 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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2answers
43 views

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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2answers
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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0answers
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
39 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
25 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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2answers
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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0answers
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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 ...
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2answers
36 views

Split series to alternating series

Let $a_n$ be sequence of positive numbers. Is that true that: $\sum_{n=1}^{\infty} (-1)^n\cdot a_n$ converges $\implies$ $\sum_{n=1}^{\infty} a_{2n}-a_{2n+1} $ converges $\sum_{n=1}^{\infty} ...
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0answers
21 views

Inverse Fourier transform of $\frac{\alpha}{\alpha+\|w\|_2^d}$

I want to calculate the inverse Fourier transform of $\frac{\alpha}{\alpha+\|w\|_2^d}$ where, $w \in R^D$ and $d$ is some positive integer. $\| \|_2$ is a 2 norm of a vector and $ \alpha $ is some ...
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0answers
21 views

Change of variables of quantities

I am just trying to see if I can rewrite one set of quantities in terms of the others given the following transformation rules: $$p^2=0,\,\, q^2=-Q^2, \,\,(p+q)^2=0,\,\, \frac{2 m \cdot p}{2 q \cdot ...
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0answers
24 views

Angle by which tangents to curves at $z_{0}$ are rotated under the mapping $w = z^{2}$

I have to find an angle by which tangents to curves at $z_{0}$ are rotated under the mapping $w = z^{2}$ if (a) $z_{0} = i$, (b) $z_{0} = -1/4$, (c) $z_{0} = 1+i$, and also find the corresponding ...
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0answers
24 views

How would I solve these problems?

(a) According to a theorem, if A is a real 2 x 2 matrix with complex Eigenvalue λ=a-bi (b is not equal to 0) and associated eigenvector w =u +iv in C2 , then choosing P = vectors [u, v] will result in ...
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1answer
16 views

Quaternion for transforming one frame to other?

I am new to quaternions and learning how they can replace rotation matrices. I know that we can use rotation matrices to describe a transformation from one frame to other. Where one may be a rotated ...
2
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1answer
28 views

Change of variables, integration

In a finite element analysis, I am evaluating the following integral: $$\int_{0}^{h}\left ( 1-\frac{x}{h} \right )*\left ( x \right )dx$$ but I want to apply a transformation from x to integrate ...
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12 views

Why does “up to scale” make homograph matrix lose one freedom?

Can anyone explain "if H is up to scale, then dof(H)=8" in the following discussion? degree of freedom of Homography matrix Thank you!!!
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14 views

How to go from Affine to a Non-linear transformation

If you were able to take two sets of matrices and transform one to attempt to match the other using an affine method such as Least Squares, how could you replicate this process better using a ...
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1answer
22 views

State transformation for non-holonomic differential equation.

Given a non-holonomic dynamical system, \begin{align*} \dot x = v\cos\theta \\ \dot y = v\sin\theta \\ \dot \theta = \omega \end{align*} with constraints $|v| < v_{max}, |\omega| < ...
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0answers
14 views

Translate and Rotate mesh

I have a mesh constituted of some vertices in 3d space, let's call them $(x_1,y_1,z_1),(x_2,y_2,z_2),\cdots,(x_n,y_n,z_n)$. The mesh's central point is $(0,0,0)$. How to find out the new coordinates ...
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3answers
44 views

How can you enlarge a shape about a point other than (0,0), using matrices?

If I want to enlarge a shape, $A$, by scale factor $k$ about $\left(0,0\right) $ I multiply each point (in the form $\begin{bmatrix}x\\y\end{bmatrix}$) by $kI$. However, I can't work out a general ...
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1answer
23 views

Inverse Z-transform of $\frac{10}{z-5}-\frac{2}{z-1}$

I must calculate the inverse z-transform of $\frac{10}{z-5}-\frac{2}{z-1}$. I decided to use the known formula $H(n-1)a^n\rightarrow \frac{a}{z-a}$, where $H(n)$ is the heaviside signal. I finally get ...
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0answers
24 views

Finding transform matrix from resulting multiplypoint function

Two matrix transformation functions exist within the Unity3D API: 1) MultiplyPoint 2)MultiplyPoint3X4 3X4 matrix (2) preforms a standard transform against a vector (And ofc is easily replicated ...
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0answers
26 views

Fourier transform of $H(-t)e^{5t}$

i have to calculate the Fourier transform in the title. My professor says the result is $\frac{1}{5-2\pi i f}$. I start from $H(t)e^{\alpha t}$, and i calculate the transform $H(t)e^{5t}\rightarrow ...
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1answer
16 views

Can you kindly explain me in detail this Fourier transform?

I've this function to transform not using the general formula, but just substituting the known transform (i.e. $\text{rect}(t)\rightarrow \text{sinc}(f)$): $\frac{\sin(6\pi t)}{t}$ I know the ...
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17 views

Transforming parts of functions

I have a function in the form: $$ \mathrm{e}^{-t\lambda} \cdot \left[t\lambda - {(t\lambda)^2 \over 2}\right] $$ If one were to plot this for say $\lambda = \frac{2}{3}$ and $t$ from $0$ to $20$, ...
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1answer
32 views

Logarithmic function transformations

The standard log function form is $a \log[k(x-d)] + c$ Where $a$ vertically stretches or compresses $k$ horizontally stretches or compresses $d$ translates left or right $c$ translates up or ...
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25 views

Perspective correction from 3 points and foreshortening factor

I'm working on creating a homography 3x3 matrix to do a perspective correction of a photograph 2D piece of paper. The paper contains 3 markers (like the 3 corner markers of a QR code) in its corners, ...
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3answers
60 views

Image of a family of circles under $w = 1/z$

Given the family of circles $x^{2}+y^{2} = ax$, where $a \in \mathbb{R}$, I need to find the image under the transformation $w = 1/z$. I was given the hint to rewrite the equation first in terms of ...