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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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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18 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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2answers
77 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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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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1answer
30 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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30 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
51 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 $T^{*}...
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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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2answers
1k 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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1answer
20 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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1answer
40 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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2answers
50 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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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
46 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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17 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
42 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
36 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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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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1answer
1k views

How can I transform coordinate systems with quaternions?

I have a coordinate system $0$ which I'd first like to rotate about its $z$-axis which gives me system $1$, and afterwards rotate system $1$ about its $y$-axis which gives me system $2$. See picture: ...
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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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1answer
102 views

optimal monotonic transform: $\min_f (f(x)-y)^2$

Given two vectors of length $N$ denoted by $x_i$ and $y_i$, $1\leq i\leq N$, what is the monotonic transformation $f(x)$ that minimizes the overall distance $D=\sum_{i=1}^{N}{(f(x_i) - y_i)^2}$. Does ...
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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 p}...
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0answers
33 views

How to determine changing scale factors when performing coordinate transfomations?

To explain: I have two coordinate systems. One $(x,y)$ and the other $(x',y')$ as seen in this diagram. Coordinate systems I am trying to convert the coordinate in the $(x,y)$ system to the rotated ...
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2answers
2k views

Transformation matrix to go from one vector to another

I've two vectors $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$. How to find transformation matrix for transform from a to b?
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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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1answer
24 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 ...
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25 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
34 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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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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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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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 $|v| < v_{max}, |\omega| < \omega_{max}$,...
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1answer
50 views

Finding the relative pose of a robot gripper

I have a robot arm with a gripper. I know the gripper pose (relative to the robot base coordinate system) at any moment. At startup, I record the pose of the gripper and set this as the original pose <...
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2answers
7k views

How can I determine the scale factor of a pantograph from the ratio of the arms?

I know this is probably simple but I just can't see it. How can I determine the scale factor of a pantograph from the ratio of the arms?
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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
47 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
24 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
17 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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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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35 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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1answer
50 views

Fourier transform of $\frac{\sin(6\pi t)}{t}$

I have to calculate the fourier transform of this function in time domain: $\frac{\sin(6\pi t)}{t}$. First I tough to use the definition of $\operatorname{sinc}$ function as $\operatorname{sinc}(t)=\...
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3answers
67 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 $z$...
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1answer
51 views

$Z$ coordinates disappear in the general rotation transformation matrix.

I wanted to generate the general rotation transformation matrix ($3D$). But when I did the multiplication the result didn't include the original $Z$ coordinates,I don't know why the $Z$ disappeared. ...
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1answer
30 views

Prove a transformation is injective if its restrictions are injective.

Suppose that $V$ is a vector space, and let $V → W$ be a linear map. $$V_0 ⊆ V_1 ⊆ · · · ⊆ V_i ⊆ V_{i+1} ⊆ · · · ⊆ V$$ are subspaces of $V$ (one for each $i = 0, 1, 2, \ldots$) and inclusions ...
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1answer
12 views

Can you stretch a function with a zero or undefined gradient?

If $y=f(x)$ is either $y=3$ (zero gradient) or $x=2$ (undefined gradient), is it possible to stretch $y=f(x)$ by graphing $y=af(x)$ or $y=f(ax)$? If it is possible to stretch them, can you only ...
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1answer
47 views

Find the density of $Z=\frac{X}{Y}$ for an exponential distribution?

We have the iid random variables $(X,Y)$ where $f_x(x)=\lambda e^{-\lambda x}$, $x>0$. We are given $Z=\frac{X}{Y}$ and asked to find the cdf and the density function. Here's my attempt. $Z=\...
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717 views

Should I trust Mathematica or numerous other sources on this Fourier transform

Assume $a>0$ So Mathematica claims $$F\{e^{-a|t|}\}(\omega) = \frac{a\sqrt{\frac{2}{\pi}}}{a^2+\omega^2}$$ However, I've read about another transform pair (page 3): $$F\{e^{-a|t|}\}(\omega) = \...