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-...

learn more… | top users | synonyms

1
vote
1answer
35 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 ...
1
vote
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 :)
1
vote
0answers
19 views

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 ...
1
vote
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: ...
0
votes
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} ...
1
vote
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 ...
4
votes
1answer
101 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 ...
0
votes
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 p}...
2
votes
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 ...
5
votes
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?
2
votes
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 ...
1
vote
1answer
19 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 ...
0
votes
0answers
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 ...
2
votes
1answer
31 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 ...
0
votes
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 ...
0
votes
0answers
16 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!!!
0
votes
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 $|v| < v_{max}, |\omega| < \omega_{max}$,...
0
votes
1answer
49 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 <...
1
vote
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?
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 \...
0
votes
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 ...
0
votes
0answers
18 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 $20$, ...
1
vote
0answers
31 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, ...
2
votes
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)=\...
2
votes
3answers
66 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 $z$...
1
vote
1answer
48 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. ...
0
votes
1answer
23 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 ...
1
vote
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 ...
0
votes
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=\...
5
votes
2answers
716 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) = \...
0
votes
0answers
20 views

affecting the final result of function depending on external factor

Suppose I have a function $f(x) = \frac{x}{x+y}$ whose range is in the interval $[0,1]$ and there is an external factor say $a$, such that $a$ is in the interval $[0,1]$, moreover, a predefined ...
0
votes
0answers
13 views

Solving a transformation problem in complex (Argand) planes

Below is an example in a text-book section on using complex-number arithmetic to represent transformations in Argand planes Example A transformation $T$ of the $z$-plane to the $w$-plane is ...
0
votes
1answer
39 views

What's the formulation of N-point radix-N for NTT

We can write the formulation for the buttlerfly function applied in FFT as \begin{align*}y_0 &= x_0 + x_1,\\ y_1 &= x_0 - x_1. \end{align*} As seen here. For FFT (Fast Fourier Transform) we ...
0
votes
1answer
30 views

How to determine if an affine transformation would cause reflection?

I have a list of affine transformation matrices and I want to write a code to delete the transformation matrices that applying them on an image would cause reflection. after seeing this image in ...
0
votes
0answers
23 views

Distribution of discrete function of continuous random variable?

It has been quite some time that I did statistics, and I am not sure how to figure out the distribution of a function of a random variable if the function itself discretizes (if that is a word) the ...
1
vote
1answer
58 views

Bring system in normal form up to the second order

Bring the system $$ x'=y+xz,\quad y'=x^2+y^2+z^2,\quad z'=-2z+xy $$ to a normal form up to the second order (kill all non-resonant quadratic terms). The equilibrium is $(0,0,0)$ and the ...
0
votes
1answer
24 views

Invariance properties of transformations

In Gentle's Matrix Algebra (2007, p. 175), he presents a table of what features of vectors various transformations preserve. What does it mean to say a transformation T preserves some property of a ...
-1
votes
1answer
31 views

How to approximate linear relationship between two timeseries? [closed]

I have two time series A and B I would like to solve for the equation in the form $$y_t = m x_t + b$$ that transforms a point $y$ at time $t$ from series $B$ to the corresponding point $x$ at time $...
0
votes
0answers
120 views

PDFs of Piecewise Transformations: why doesn't it apply in this case?

This is from Casella and Berger's Statistical Inference, although it is more of a probability question than a stats question. Theorem 2.1.8 Let $X$ have pdf $f_{X}$, let $Y = g(X)$. Define the ...
0
votes
1answer
25 views

The correct order for applying transformations?

I had a line, y=-1/2 x I wanted to reflect everything onto the line so as you can see by my steps in the picture below(sorry, I dont know the coding) I first rotated the line theta degrees then ...
1
vote
1answer
34 views

rational integral with a quartic function in the denominator

Let's say I've got the integral: $\int [ 2 Q^4 - 5 Q^2 + 3 ]^{-1} dQ$ This integral evaluates to: $\int [ 2 Q^4 - 5 Q^2 + 3 ]^{-1} dQ = \tanh^{-1}\left( Q \right) + \sqrt{ \frac{2}{3} } \tanh^{-1}\...
1
vote
1answer
36 views

Transform PDE to ODE (3 variable case) with given boundaries

How can I transform the following PDE into an ODE? I tried using three different functions $H(x),G(y)$ and $F(t)$ but that didn't help hence I did not post it here. I really hope someone can help me ...
0
votes
1answer
13 views

Three Variable Transformation and Independence

If $X_1$, $X_2$, and $X_3$ are independent, identically distributed random variables and $Y_1$, $Y_2$, and $Y_3$ are functions of them, how do I show that the $Y$ variables are mutually independent? ...
1
vote
1answer
20 views

Integral with more-dimensional substitution variables

Good day, In the lecture of partial differential equations we had the following transformation: $$\int_{||\nu||=1} h(x+\nu c t, \tau) d\nu = \frac{1}{c^2 t^2} \int_{||y-x||=ct} h(y,\tau) dy$$ for $...
0
votes
0answers
41 views

Is there a product integral that preserves zeroes?

The integral essentially takes the arithmetic mean of the range of a function multiplied by the domain, adding together each possible output weighted by the amount of the domain accounted for by that ...
3
votes
2answers
109 views

Cayley's Theorem for Semigroups

I've read and fully understand Cayley's theorem for groups, however when I get to the theorem for semigroups I come to a complete stop. I've figured that the identity and cancellative properties are ...