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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Linear transformation formula

How to find formula for linear transformation $\varphi : \mathbb{R}^2 \rightarrow \mathbb{R}^4$ when the following is given: $$\varphi ((5,1))=(2,5,1,1)$$ $$\varphi((1,0))=(3,4,2,2)$$ What is the ...
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1answer
28 views

How to apply coordinate transformations

Lets say I want to rotate a parabola by $\pi/4$ degrees counterclockwise. Wikipedia tells me a counterclockwise transformation would mean: $$ x'=x\cos t-y\sin t \\ y'=x\sin t+y\cos t $$ however ...
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23 views

Multiplication order for coordinate frame transformations

Suppose I have three coordinate frames: $A$, $B$ and $C$. If $T_{AB}$ is the transformation from $A$ to $B$, then which of the following is correct? $T_{AC} = T_{AB} \cdot T_{BC}$ $T_{AC} = T_{BC} ...
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1answer
29 views

Minimum amount of points required to find a transformation matrix

Given a set of point $P$ in $\mathbb R^n$ and the same set of points $P'$ which have been transformed by a transformation matrix: $$L: \mathbb R^n\mapsto \mathbb R^n$$ $$L(p_1) = p_1',\;\; p_1\in ...
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0answers
23 views

Non-linear transformation of symmetric distribution to get non-negative skewness

Say you have a variable $x \sim D(\mu,\sigma^2) $, where $D$ is a symmetric known distribution. I'm looking for two linear or non-linear transformations of $x$ that give one negative and one positive ...
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1answer
31 views

Show that $F$ is not a one-to-one transformation

Given $$F(x,y)=(x-y,y^2-x-2)=(u,v),$$ how to show that this transformation is not one-to-one? And at which points $F$ is locally one to one? While I was drawing this transformation I found that ...
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19 views

When creating conformal images, how do you change the basis of the input lattice such that spirals result in the transformed image?

I am trying to emulate the results shown in the Wikipedia page on Conformal Images in an attempt to better visualize complex functions (and stare at some trippy images, man). The script I wrote ...
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1answer
39 views

Apply Cayley transformation on vector x

If I have $Q = (I + S)(I - S)^{-1}$ ($Q$ is the Cayley transformation of skew-symmetric matrix $S$) then how do I construct a rank-2 $S$ such that $Qx$ has all zeros except the first component?
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1answer
52 views

Cayley transformation of a skew-symmetric matrix is orthogonal?

If $S$ is skew-symmetric ($S^{T} = -S$), how do I show that $Q$ is orthogonal where $$Q = (I + S)(I - S)^{-1}$$ which is the Cayley transformation of $S$.
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Need help with linear transformations (with projection and reflection)?

Let $L$ be the line given by the equation $4x − 3y = 0$. Let $S : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be reflection through that line, and let $P : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be ...
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1answer
41 views

Solving a transformation equation involving vectors and quaternions

I'd like to solve the following equation for $c$, where $a$, $c$, and $d$ are position vectors represented by quaternions with $w$ (the real component) set to $0$ and $b$ is a unit quaternion: ...
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18 views

Finding the Transformation given the domain and the codomain in $\Bbb R^3$

So I am given the domain and the codomain of three matrices such that the $F: \Bbb R^3 \to \Bbb R^3$, $T(1,0,-1) = (2,2,1)$ and $T(1,1,0) = (1,1,0)$, the point here and the question rather is not to ...
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0answers
21 views

How do I compute uncertainties in the positions transformed using Helmert transformation?

I am familiar with Python, MATLAB and Mathematica. My data is given by Cartesian coordinates (positions in X,Y,Z (meters), standard deviations XX, YY, ZZ (meters) and a correlation matrix (XY, YZ, ...
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1answer
90 views

Difference between transform and transformation.

I was told that there is a difference between a transform and a transformation. Can anyone point out clearly. For example : Is Laplace Transform not a transformation ?
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How to formulate a coordinate transformation

Thank you in advance for taking the time to consider this. I'm trying to figure out how to formulate a coordinate transformation problem (at least that is what I think it is). Background: I have an ...
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1answer
27 views

Perspective transformation matrix application

I need to transform an angled photographed pice of paper to a "flat" image. I found this question & solution here on Mathematics and tried it out for the image given in the solution: The values ...
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19 views

Syncronize positions of 2 rectangles with different origin point while rotation

Suppose we have 2 rectangles in Cartesian coordinate system with (0,0) at the top left corner of the screen. Both of rectangles (a and ...
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1answer
12 views

Equations transformations with roots

How does the following transformation works (do not write that it is easy i want the answer): $$\ln \sqrt[n]{\frac{n!}{n^n}}=\frac{\ln \frac{n!}{n^n}}{n}$$
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20 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
40 views

Formula for the sum of fractions [duplicate]

How to find the formula for the sum of fractions like this? $$\frac{1}{1\times 2}+\frac{1}{2\times 3}+\ldots+\frac{1}{n\times (n+1)}=$$
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1answer
43 views

Making sense of polar coordinates transformation on the derivatives

I would like to make sense of the transformation of the differentials in polar coordinates (to fix the ideas). To be more precise, the "right" way to find the transform for the differential and the ...
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2answers
45 views

Cayley Transform and Eigenvalues

I have a particular operator, namely $A=-i\frac{d}{dx}$ that I would like to Cayley transform. $A$ is defined on the Hilbert space $L^{2}[0,1]$ and has domain $\mathcal{D}_{\alpha}=\{g:g \in ...
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1answer
37 views

Rotate a vector about a given axis by the use of a quaternion

I encountered a problem in programming where I need to rotate a given vector about a given angle. To be precise, I need to change it to a quaternion so that I can later change it to a 4x4 matrix to ...
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0answers
52 views

Rotational matrix problem?

In the problem yo-yo is made of two identical cylinders of radius $R$, thickness $h$ and mass $M$, and the yo-yo is let go. In order to define the position of the yo-yo, I need as position vector and ...
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1answer
28 views

Normal distribution

X follows a regular normal distribution on V with center $\xi$ and inner product $<\cdot,\cdot>$, and let $\eta \neq 0$ be a vector in V. Show that the reel stochastic variable ...
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1answer
47 views

Changing $[0,2\pi)$ with $S^1$ such that a map defined on $[0,2\pi)$ stays unchanged

* Consider the following procedure of changing the domain of a map, but the map remaining essentially the same - illustrated, for concreteness, in case of the polar-coordinates map.* Let ...
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1answer
33 views

Fourier Transform - Laplace Transform - Which variable transform?

I need to know when do I have to transform $x$ and when $y$ in a PDE in Fourier Transform and Laplace Transform. In an exercise of Fourier Transform, I have to solve a Laplace Equation, where ...
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1answer
33 views

Lotka-Volterra coordinates transformation

I would like to ask the following: Given a Lotka-Volterra predator-prey system, \begin{align} & \frac{dx}{dt}={\alpha}x-{\beta}xy \\ & \frac{dy}{dt}=-{\gamma}y+{\delta}xy \end{align} , with ...
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3answers
99 views

Dimension of Hom(U,V)

I know this has been asked before - I am really struggling to understand what people have said though, so I want to ask for myself. If U,V are vector spaces over field K, with dimensions n,m ...
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35 views

Transfer function: steady-state solution of equation

Place the transfer function in the form $$H(i\omega) = \frac {1}{R}e^{-i\phi}$$ and use this result to find the steady-state solution of the equation$$x'' + x'+4x = 3*cos(2t)$$ I don't really ...
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1answer
134 views

Fourier COSINE Transform (solving PDE - Laplace Equation)

I'm trying to solve Laplace equation using Fourier Cosine Transform (I have to use that), but I don't know if I'm doing everything OK (if I'm doing everything OK, the exercise is wrong and I don't ...
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How can I show that the AR process is nonstationay if x(n) has nonzero mean?

This is a first-order-real-valued autoregressive (AR) process $y(n)$ that satisfies the real-valued difference equation $y(n)+a_1y(n-1)=x(n)$ where $a_1$ is a constant and x(n) is a white noise ...
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0answers
32 views

How to apply the chain rule for partial derivatives to transformations?

I'm currently working to solve the Black-Scholes model partial differential equation (it's a model for a.o. stock option prices). The Black-Scholes equation for a calloption C(S,t) is given by $ ...
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How to understand all types of transforms as linear operators on function spaces?

I would really appreciate if someone can point to a simple general framework that can help me to understand integral transform from a generalized point of view so that there is no longer a fourier ...
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27 views

Prove a formula after change of variable?

If I have a change of variable $(x,u)\to (X,U)$ given by $$X=x+\epsilon u,U=u-\epsilon u.$$ How to prove the formula $$\frac{\partial U(X,0)}{\partial \epsilon}=\phi(X,u(X))-u'(X)\xi(X,u(X)),$$ where ...
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1answer
24 views

Cross-sectional function for a surface of revolution

If I take a one-to-one function $f(x)$ and rotate it about the x-axis, how can I describe a function resulting from a cross-section of the solid of revolution? I'm not talking about the circular ...
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15 views

Shear matrix simple explanation

I can understand translation, dilation and rotation matrices, but the shear one is still obscure to me (despite understanding what shearing means graphically). This is the matrix: ...
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3answers
37 views

How does $(x+3)^2 - 2^2$ become $(x+1)(x+5)$?

I don't understand how $(x+3)^2 - 2^2$ can be transformed to equal $(x+1)(x+5)$. A short demonstration and/or reference to math rules would be very kind. Thanks.
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1answer
41 views

Transformation Matrix for cube in 2D

My task is to transform the cube from the left corner to the big cube in the middle: What I did was: First i scale the cube: $$ \begin{pmatrix} 4 & 0 & 0 \\ 0 & ...
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0answers
16 views

Transform Confocal Ellipsodal to Spherical Coordinates

I heard that someone published a paper showing that the confocal ellipsoidal coordinate system can transform into the spherical coordinates under special limit evaluations, however I was unable to ...
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2answers
72 views

Sum of uniform random variables $U(0,1)$ and $U(0, a)$

The problem I have is: $X \sim U(0,1), Y \sim U(0,a)$ are independent random variables. Find the pdf of $X + Y$. I've got stuck in an integral-problem, and will show you what I've tried. Skip to the ...
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1answer
53 views

Hyperbolic Geometry: Question about the Transitivity of Möbius transformations

I was confronted with this exercise in the book Hyperbolic Geometry by Anderson which states: In each case, find $m \in Möb(\mathbb{H})$ such that the property holds, or prove that no such $m$ ...
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2answers
32 views

Linear transformation with special properties

how should I do that please (I had this in my test yesterday)? Linear transformation $f:\mathbf{R}^{10} \to \mathbf{R}^7$ has an attribute that every vector $\mathbf{v}$ for which is true that ...
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Can Magnification/Scaling (transformation) be prepresented by a vector?

Vectors represent three bits of information: Magnitude, Line of Action, and Direction. A Translation (transformation) can be represented by a vector: object is moved By so much (magnitude) Along a ...
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1answer
26 views

Transformation theorem: calculate picture of a set

I have this function: $T:(0,\infty)^2 \rightarrow T((0, \infty)^2), \quad T(x,y)=\left( \frac{y^2}{x},\frac{x^2}{y} \right)$ Now I try to estimate $T(M)$ with: $0<p<q, \quad 0<a<b$ ...
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0answers
66 views

Transforming the cubic Pell-type equation for the tribonacci numbers

The Lucas and Fibonacci numbers solve the Pell equation, $$L_n^2-5F_n^2=4(-1)^n\tag1$$ The tribonacci numbers $z = T_n$ are positive integer solutions to the cubic Pell-type equation, $$27 x^3 - 36 ...
2
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1answer
26 views

Tietze transformation

I have a question in which I have to transform $\textbf{I.}$ $\langle a,b,c \mid b^2, (bc)^2\rangle$ to $\textbf{II.}$ $\langle x,y,z\mid y^2, z^2\rangle$ using Tietze transformations. My ...
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47 views

Decomposition of 4x4 or larger affine transformation matrix to individual variables per degree of freedom.

There are a couple of problems and solutions where affine matrices are decomposed into their seperate tranformations. However they are all for the 2D case and I`m finding it difficult to generalise it ...
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1answer
71 views

Variants of the change-of-variables formula

Consider the following change of variables formula for $f:X\rightarrow Y$, that holds for any "reasonable" $g:B\subseteq Y \rightarrow \mathbb{R}$ and $A\subseteq X$ $$ \int_B g(x)\ {\rm ...
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Linear transformation of vector

I have computer graphics class and i had something like that on lecture: $$ \begin{bmatrix} \overrightarrow{b1} & \overrightarrow{b2} & \overrightarrow{b3} \end{bmatrix} \begin{bmatrix} c1\\ ...