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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1answer
135 views

The Matrix of a Transformation from P2 to P1

I don't understand how T(1), T(x), and T(x^2) were found in the picture so I did it using another method I saw on StackExchange. (a + b)x - c => -c + (a + b)x + 0x^2 so the first row would be {0, 0, ...
0
votes
1answer
68 views

Line plane intersection

I have two planes in $\mathbb R^3$ as shown below: axes representation corrected after MvG's comment Each plane is a finite area, a rectangle with length and width $H_l, H_w$. Each plane has its ...
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0answers
37 views

inverse laplace tranform

I have a simple question, There are some functions $f(t)$, $g(t)$ and lets say $F(s)$ and $G(s)$ for the form of Laplace transform of $f(t)$ and $g(t)$, respectively. While I am solving ...
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0answers
108 views

How do I calculate 3D movement based on yaw, pitch and roll?

I'm creating a 3D game demo and I need to calculate the position of the player in the space (i.e. the player's x, y and z coordinates). I understand that this would be affected based on the camera ...
0
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1answer
21 views

Is a series that contains the index term a function of the same series without the index term?

Can it be shown that $U_{2} = \sum_{i=1}^{n} [i*g(Y_{i})]$ is a function of $U_{1}=\sum_{i=1}^{n} g(Y_{i})$ ? My intuition tells me that this is not true because of the changing (for lack of a ...
4
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2answers
109 views

Fractional Linear Transformation: Region between two circles to strip

I'm trying to find Fractional Linear Transformation (if one exists) that maps the region between the circles $\|z+1| = 1\}$ and $\{|z|=2\}$ to the region between the horizontal lines $Im(z) = 1$ and ...
1
vote
1answer
51 views

Prove the result of multiplying a complex number by (1 + i)

I know that if I multiply a complex number that has the form a + bi by the number (1 + i), this will rotate the vector that corresponds to the complex number by 45 degrees in the counterclockwise ...
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0answers
57 views

Reducing heat equation into nondimensional form

I want to get nondimensional form of heat equation $u_t=a(x,t)u_{xx}$. For the case of $a(x,t)=a(t)$, by setting $A(t)=\int_0^ta(\eta)d\eta$ and $t=\phi(\tau)$, where $\phi$ is the inverse mapping ...
0
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0answers
19 views

Make a conform map g that sends the unit disk to $A = \{w: \operatorname{arg}(w) \in(\frac{\pi}{4},\frac{3\pi}{4})\}$ such that $g(2i) = 0$

make a conform transformation g that sends the unit disk to $A = \{w: \operatorname{arg}(w) \in(\frac{\pi}{4},\frac{3\pi}{4})\}$ such that $g(2i)$. I actually solved it by taking the inverse of $f(x) ...
1
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0answers
171 views

Mapping A point from one 3D Coordinate System to Another 3D coordinate System with Euler Angles between the two systems given

Suppose I have a point in the green coordinate system, and I wish to describe it in reference to the orange coordinate system. I know the roll, pitch, and yaw of the green system with respect to the ...
2
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2answers
54 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
votes
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} ...
0
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1answer
56 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
53 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
116 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
votes
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
37 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
77 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} ...
1
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1answer
51 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
76 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
89 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
votes
1answer
61 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
38 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}) = ...
0
votes
1answer
161 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
58 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 ...
0
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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
110 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 ...
0
votes
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 ...
1
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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 $x=u-v$
0
votes
1answer
27 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
51 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
35 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
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 ...
0
votes
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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0answers
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 ...
1
vote
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?
1
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1answer
116 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
199 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
113 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
287 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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0answers
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 ...
1
vote
1answer
85 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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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
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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vote
2answers
114 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
68 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
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2answers
86 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 ...