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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2answers
58 views

suppose $|a|<1$, show that $\frac{z-a}{1-\overline{a}z}$ is a mobius transformation that sends $B(0,1)$ to itself.

Suppose $|a|<1$, show that $f(x) = \frac{z-a}{1-\overline{a}z}$ is a mobius transformation that sends $B(0,1)$ to itself. To make such a mobius transformation i tried to send 3 points on the edge ...
2
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1answer
89 views

Which linear transformations preserve this?

Let $a,b,x,y\in \mathbb{Z}$ (with $a,b$ given) and consider the equation $a(x^2+y^2)=bxy$. Consider transformations taking $x$ to $px+p'y$ and $y$ to $qx+q'y$. For which integers $p,q,p',q'$ is it ...
0
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1answer
26 views

Matrix of $L(A)=A^{T}$ From $R^{2 \times 2} \rightarrow R^{2 \times 2}$

A bit of trouble with this question: Find the matrix of the linear transformation $L(A)=A^{T}$ From $R^{2 \times 2} \rightarrow R^{2 \times 2}$ with respect to the basis $\begin{bmatrix} 1&0\\0&...
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1answer
59 views

Inverse Fourier Transform Proof

I am aware of how Fourier Transformation and Fast Fourier Transformation works, however I do not understand the logic of the inverse of FFT. Could someone explain why the inverse fourier ...
1
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1answer
55 views

Is monotony preserved under expectation?

let $X_1 \sim f_1(x)$ and $X_2 \sim f_2(x)$. Suppose we know that $\mu_1=E(X_1)<E(X_2)=\mu_2$ and let $\nu_1=E(\log(X_1))$ and $\nu_2=E(\log(X_2))$. Since $\log$ is monotonically increasing, my ...
0
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1answer
128 views

How to rotate an orientation (Euler angles)

If I have an orientation defined by Euler angles and I want to simulate a rotation of the coordinate system about the origin (doesn't matter to me how the rotation is specified), how would I get the ...
1
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1answer
42 views

Slice, projection, contour: A terminology question.

Consider a multivariate function, say $y=f(x_1,x_2,\dots,x_n)$, and suppose that $z=f(x_1,x_2,\dots,x_{n-1},g(x_1,x_2,\dots,x_{n-1}))$. What do we call $z$ with respect to $y$? Projection, level set, ...
0
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1answer
63 views

Computing range, null space, and matrix of a linear transformation

Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be defined by $(a_1, a_2, a_3) \mapsto (a_1, a_2, -a_1-a_2)$. I have to find $R(T), N(T)$ and a matrix that represents $T$. I know for my matrix that represents $...
0
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1answer
39 views

Find a surjective linear map with the following conditions

Find a surjective Linear map $ T: \mathbb{C}^{3} \rightarrow \mathbb{C}^{3} $ such that $ T(1,0,0) = (0,i,0) \space $ and $ T(0,i,0) = (0,0,1) $ We must also verify that $T$ is injective. I suppose ...
0
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2answers
79 views

Is this transformation surjective?

Consider the transformation $T:C_{\mathbb R} [0,1] \to \mathbb R$ defined by $T(f(t)) = \int_0^1 f(t)dt$. Is this transformation surjective? It would be enough to show that $$\mathbb{R} \subseteq\...
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1answer
53 views

What does it mean for a matrix to change basis?

I understand what it means for vectors, i.e. $$ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} ...
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0answers
85 views

DPE problem invlolving Fourier transforms / partial eq.

Don't even know where to start with this question! would really appreciate some guidance.
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2answers
95 views

How does the Jacobian relate to sketch of x,y coordinates with u,v constant?

T is a non-linear transformation, with the following component functions: x = u/v, y = v On a sketch of the x-y plane, with u and v constant, how does the Jacobian, J = 1/v, relate to the sketch of y ...
0
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1answer
75 views

Multiplication order of rotation matrices

I have three 3D coordinate frames: O, A and B, as shown below. I want to know the rotation matrix RAB between A and B, that is the rotation that is required, with respect to the frame A, to move ...
0
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1answer
40 views

Change of Bases defined by a Plane

In the plane $V$ defined by the equation $x_1-2x_2+2x_3=0$, consider the basis $\mathcal{A} = (\vec{a_1}, \vec{a_2}) = \Bigg(\begin{bmatrix}2\\1\\0\end{bmatrix},\begin{bmatrix}-2\\0\\1\end{bmatrix}\...
0
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1answer
178 views

Transformations between two coordinate systems on a rigid body

I have two coordinate frames, A and B, which are rigidly attached to each other on a body. This body then translates and rotates, such that A starts at A1, and moves to A2, and B starts at B1, and ...
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0answers
60 views

Fourier Tansform of derivative on Wolfram Alpha

If I'm not mistaken, the Fourier Transform of the $n$th order partial derivative of a function with respect to $x$, using the transform variable $k$ is: $$(i*k)^n * [F(k)]$$ so for the $1$st order ...
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1answer
40 views

$T(f(t))$ from “the space of all polynomials”

Let $T(f(t)) = (f(0), f(1), f(2), f(3),\cdots)$ from $P$ to $V$, where $P$ denotes the space of all polynomials. Is $T$ linear and if so, is $T$ an isomorphism? I feel like a counterexample is ...
4
votes
1answer
124 views

Hamiltonian, symplectic transformation

I am trying to understand symplectic transformations. Assume that $H(q,p)$ is a Hamiltonian and the corresponding Hamiltonian equations are given as, \begin{split} & \dot q = \frac{\partial H}{\...
0
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1answer
30 views

Proving that a transformation of a function gives a positive result

If $x$ is real and: $$p = \frac{3(x^2+1)}{2x-1}$$ Prove that: $$ p^2-3(p+3)\geq 0$$ I think this has something to do with equating the discriminant to $0$, but I'm not entirely sure I'd really ...
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1answer
51 views

$X$ and $Y$ are ind. exponentially dist. ran. variables w/para. $\beta_1$ and $\beta_2$. Let $U=X+Y$, verify that $f_u(u)= \int_0^u f_{xy}(u-v,v)dv$.

I am a little lost with transformations with exponential distributions, any help would be much appreciated! The given hint is $0<x<infty$ and $x=u-v$
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1answer
28 views

Transformation matrix for a 3d->2d projection

We know $\mathbf{\hat{y}} = X\mathbf{w}$ and $A$ is the subspace in which $\mathbf{\hat{y}}$ lies (so the columns of the $X$ matrix define the subspace $A$). $\mathbf{\hat{y}}$ (2-dimensional vector) ...
0
votes
3answers
53 views

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 ...
0
votes
1answer
37 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 ...
1
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1answer
67 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 P\...
0
votes
1answer
44 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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0answers
42 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 (...
1
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1answer
46 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?
1
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1answer
151 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
234 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 ...
1
vote
1answer
139 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: $$a+(b*...
1
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1answer
335 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 ?
0
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0answers
30 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 ...
1
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1answer
98 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 ...
0
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0answers
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 ...
0
votes
1answer
14 views

Equations transformations with roots

How does the following transformation works (do not write that it is easy i want the answer): $$\ln \sqrt[n]{\frac{n!}{n^n}}=\frac{\ln \frac{n!}{n^n}}{n}$$
0
votes
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 <...
1
vote
2answers
180 views

Formula for the sum of fractions [duplicate]

How to find the formula for the sum of fractions like this? $$\frac{1}{1\times 2}+\frac{1}{2\times 3}+\ldots+\frac{1}{n\times (n+1)}=$$
0
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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 ...
1
vote
2answers
93 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 C^{\infty}...
2
votes
1answer
90 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 ...
0
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0answers
169 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 ...
1
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1answer
66 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 $Y=<X-\xi,\eta&...
1
vote
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 \begin{...
0
votes
1answer
89 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 $y>0$...
0
votes
1answer
50 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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votes
3answers
254 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 ...
2
votes
1answer
333 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 ...
3
votes
1answer
88 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
50 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 $ \...