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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36 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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1answer
63 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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1answer
42 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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37 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
42 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
125 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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3answers
214 views

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
121 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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1answer
312 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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28 views

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
90 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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34 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
14 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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1answer
48 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
131 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
69 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
88 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
84 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
160 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
62 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
56 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
86 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
48 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
235 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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1answer
318 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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1answer
85 views

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
49 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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1answer
34 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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1answer
78 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
93 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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2answers
99 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
86 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
39 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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1answer
31 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
94 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 ...
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1answer
80 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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0answers
127 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
97 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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1answer
41 views

What is a transformation that can't have shearing called?

What is a transformation called when it can have separate scaling for x and y, rotation, and translation, but it cannot have shearing or scaling AFTER rotation? Basically if this transformation is ...
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1answer
42 views

coordinate transformation and tensor

A 2 dimensional Euclidean space is represented by two different coordinate systems: the Cartesian system $(x_1,x_2)$ and an alternative system $(\xi^1,\xi^2)$ where ...
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1answer
42 views

Matrix transformation conserving the “positive semi-definite” aspect

Let's say I have two covariance matrices $A$ and $B$ (so they're both positive semi-definite), What kind of transformations can I apply on either one of them or both without loosing the ...
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1answer
55 views

Proof : If $S,T:V \rightarrow V$ (V is finite) and $KerS=\{0\}$ then $Im(TS)=Im(T)$

I have this problem : Proof : If $S,T:V \rightarrow V$ (V is finite) and $KerS=\{0\}$ then $Im(TS)=Im(T)$ My solution : Let $v \in ImT$ exist $w \in V$ such that $T(w)=v$. Since $KerS=\{0\}$ ...
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1answer
72 views

Composition of translation and rotation is a rotation, but what is its center?

Consider the rotation $r_{\Omega,\alpha}$ of center $\Omega$ and angle $\alpha$.Furthermore let $t_{\vec{v}}$ be the translation by vector $\vec{v}$. Then $$t_{\vec{v}}\circ ...
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1answer
13 views

Bounds of a Bivariate Function

I am given that $h(x, y) = \frac{x}{(x+y)}$ , $x > 0$ , and $y > 0$. I am supposed to deduce that the bounds for $h(x, y)$ are $0 < h(x, y) < 1$, but I do not understand how to arrive at ...
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2answers
1k views

What is the difference between coordinates transformation and change of coordinates?

In the context on 3D computer graphics, what is the difference between coordinates transformation and change of coordinates? It can just be a matter of notation, but my book makes a clear distinction ...
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0answers
62 views

probability integral transformation and distribution of P= P[ |T| <= |t|] .

The task is to find the distribution of P. where , P=P[ |T| <= |t|]. (T is a continuous random variable with PDF f(t)). now , I tried to make the following two arguments : 1.P= P[ |T| <= |t|] ...
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2answers
67 views

Why do a set of continuous transformations form a manifold?

I am reading Sean Caroll's book on GR, and he defines manifolds to be "a space that may be curved and have a complicated topology, but in local regions looks just like R$^n$. Here by "looks like" we ...
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1answer
27 views

Expressing the inverse of $C(x) = (\langle x,a_i \rangle )$

Assume we have the following linear transformation: $$C(x) = \tilde{x} = \left( \begin{array}{c} \langle x, a_1 \rangle\\ \vdots \\ \langle x, a_k \rangle\\ \vdots \\ \langle x, a_n \rangle ...
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66 views

Is there an interesting interpretation of the ROWS of an affine transform matrix?

Context: I have a question about affine transform matrices in 3-space. Matrices are 4x4, with the right-most column being the translation, and the bottom row being [0,0,0,1]. In discussions I read ...