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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Mathematics of transformation of 2-D to 1-D coordination.

Let's see an example In cartesian coordinate system: ...
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Transformation of the gradient

For a function $f\in C^2$, $f:\mathbb{R}^n\to\mathbb{R}$ and a point $x\in\mathbb{R}^n$ with $\nabla^2f(x)$ positive definit one can calculate the new point $x^+=x+s$ as follows: Change the ...
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Multivariate to univariate distribution

Say one has a Student's t-copula (where all the margins and the copula can have different degrees of freedom). If you had a matrix of data (f.e. financial returns) and you know that the portfolio ...
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How can I calculate the angle of a line/vector if the center of the image is not (0,0)?

Simple image about the problem How can I calculate the alpha? My center of the image is (320,240) because it is a 640x480 image and the upper left corner is the (0,0). I tried to calculate it with ...
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Finding the function of a sine graph that has both translation and transformation

I can't quite find a problem similar enough to this yet, and I need some serious help. Here is a photo of the graph of the function I am trying to find out: Sorry, but I don't have enough ...
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56 views

2D Fourier transform of characteristic function of stripe on xy plane

Given a stripe $X$ on the xy-plane, namely $X\subset\mathbb{R}^2$, with $X=\{(x,y)\,|\; mx-\frac{1}{2}t \le y \le mx + \frac{1}{2}t$} and its "characteristic" function $$ f(x,y) = \begin{cases} 1, ...
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34 views

Find functions $F(\mathbf{x})$ invariant under a map $\mathbf{x} \to \mathbf{x'}$

We introduce a map $\mathbf{x} \to \mathbf{x'}$, defined as (for example on $\mathbb{R}^3$): $$x'=f(x,y,z) \\ y'=g(x,y,z) \\ z'=h(x,y,z)$$ Note that $f,g,h$ are not all linear (or at least, I'm not ...
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How to prove this identity? Transformation theorem

Let $A\in\mathbb R^n$ be a measurable set with finite measure. For a fixed vector $p\in\mathbb R^{n+1}$ define a cone with basis $A$ and peak $p$ as $$K(A,p)=\{tp+(1-t)q \in\mathbb R^{n+1} \,| \, q \...
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cilindrical over-SURFACE coordinates

I have this problem that I think needs to be solved using over-the-surface coordinates. See attached pic 1: I need a coordinate system (C.S.) with the X´ axis in arbitrary point, but it HAS TO run ...
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30 views

Coordinate transform a triangle

There is a triangle with points P1(x1,y1),P2(x2,y2),P3(x3,y3) on an XY plane. The final ...
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Transformation of a sphere and computing an integral by using sphere coordinates

Let $V \subset\mathbb R^3$be the ellipsoid $$9x^2+4y^2+z^2≤36.$$ How can I express $V$ as a transformation of a sphere and how can I compute the sphere $$\int_v x^2\,d\lambda^3(x,y,z)$$ with sphere ...
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48 views

Transformation matrix

For $x \in \mathbb{C}$ define $A,B \in M(3\times3, \mathbb{C})$ as $$ A = \begin{pmatrix} x & 0 & 0 \\ 0 & x & 1 \\ 0 & 0 & x \end{pmatrix}$$ and $$ B = \begin{pmatrix} x & ...
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Bilinear transformation which maps $z=(\infty, i, 0)$ and $w= (-1, -i, 1)$

I have three equations after simplifying this a bit $a+c=0$ $ai+b-c=0$ $b-d=0$ How do I proceed further? If you care to know this is from the chapter Complex Variables
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1answer
42 views

Is there a universal symbol for transformation or operation?

If I'm talking about a bunch of transformations (translation, rotation, scale, skew, etc) and I want to state $A$ [some transformation] $= B$, what would be the symbol for [some transformation] or ...
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What is the difference between and projection and a reflection, in vector transformation?

In my text book I have the problems of finding the standard matrix of the given linear transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$; Projection onto the line $y = -x$. Reflection in the line ...
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1answer
16 views

Derive the length of the longest line segment that can be enclosed inside the region A.

Q. Let A be the region in the xy-plane given by A={(x,y): x=u+v, y=v, u^2+v^2≤1}. Derive the length of the longest line segment that can be enclosed inside region A. My attempt: I found the equation ...
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Converting coordinates

I am having huge trouble converting between cartesian to polar and to spherical. I need these methods for my limits of integration in most cases but using the formulas never seem to work, I just ...
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1answer
28 views

Distribution of a transform of bivariate to univariate random variable?

Suppose we have two random variables $$R\sim U[1-\varepsilon,1]\;\;\;\;\; \Theta\sim U[0,2\pi],$$ and a third random variable $$X=g(R,\Theta)=R\cos\Theta.$$ What is the density $p_X(x)$? The ...
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1answer
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inverse Mapping in Transformation of a random variable

I have a question concerning the the inverse mapping in the image . text extracted from Casella Statistical inference $g^{-1}(A) = \{ x \in \chi : g(x) \in A\}$ I know the idea that they want to ...
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22 views

Determinant in the transformation theorem

Where does the $|det|$ come from in the transformation theorem? It is pretty much the first time I saw a $|det|$ in analysis.
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Rotation of a coordinate system

Suppose that I rotate the (traditional) coordinate system $(x,y)$ by an angle $\theta$ to obtain a new coordinate system $(s,n)$. Consider a velocity vector $$v = (v_x,v_y) = v_xe_x + v_y e_y,$$ where ...
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1answer
27 views

Cone under similarity transformation

Suppose we have a cone passing through the origin of $xyz$ coordinate system. Now, the question is that whether we can find an invertible transformation on this coordinate system that turns the cone ...
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Is there a mathematical term for the “de-peaking” of a dataset

I have several sets of data that exhibit similar "peaks" when viewed as a graph, e.g.: A more useful representation of this data (for my purposes) is ${y = abs(x - 70)}$. The representation of the ...
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Probability - Finding the Support of a Joint Transformation

$$ f(x,y) = \left\{ \begin{array}{ll} 12xy(1-y) & \quad 0< x < 1, 0<y<1 \\ 0 & \quad \text{elsewhere} \end{array} \right. $$ ...
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Find the basis $\mathcal{V}$ of $\mathbb{R}^4$ and $\mathcal{W}$ of $\mathbb{R}^3$.

Let $T:\mathbb{R}^4\to\mathbb{R}^3$ be a linear function with the transormation matrix given as: $$A=\begin{pmatrix} -3 & 2 & 3 & -3 \\ 4 & 0 & -4 & 4 \\ 2 & 0 & -2 &...
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46 views

If $f_X(x) = x/8$, find the pdf of $Z = \log (x/4)$.

Given the function $f_{X}(x) = \left\{ \begin{array}{lr} \ \frac{x}{8} \quad & : 0 <x <4 \\ 0 & :\text{Otherwise} \end{array} \right. $ Find the PDF of $Z = \log_{e}(\...
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25 views

Lineaire transformation of matrices, how to tackle

I've been learning linear algebra but can't understand the concepts of linear transformation. Correct me where i'm wrong: Say i'm given $T:R^2 \longrightarrow R^2$ is my transformation. This tells ...
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29 views

Co-ordinate transformation of metric

In a past exam paper that I am using to prepare for my upcoming finals, I have encountered the following question (paraphrased): Given the metric: $$\mathrm{d}s^{2} = -c^{2}\:\mathrm{d}t^{2}+\left(...
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How to graph $x^2 -4x$?

I know about transformations and how to graph a function like $f(x) = x^2 - 2$. We just shift the graph 2 units down. But in this case, there's an $-4x$ in which the $x$ complicated everything for me. ...
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How can I find the Linear Map given the Image? [closed]

Find the linear map $F : \mathbb{R}^3 → \mathbb{R}^3$ whose image is a subspace with the basis: $\{(1,2,3),(4,5,6)\}$.
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Sine Curve Circular Transform - Parametric Equations

Is there a way to transform a sine curve so that the x-axis of the sine curve would become a circle, with the sine wave oscillating around the now-circular x-axis? What would be the parametric ...
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1answer
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How to calculate rotation rates of a rotating body relative to another rotating body?

I have two 3D bodies A and B, each of them is rotating around its own Z-axes with an angular velocity (e.i. yaw rate) of $\dot{\alpha}_A$ and $\dot{\alpha}_B$, respectively, relative to an absolute ...
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42 views

Joint pdf of two transformed variables ($W$ and $Z$) from joint pdf of $X$ and $Y$.

Let the joint distribution of $X$ and $Y$ be given by: $f(x,y) = e^{-x}$ if $0 < y \leq x < \infty$ Define $Z = X+Y$ and $W = X-Y$ Find the joint pdf of $Z$ and $W$ Calculate $f_{ZW} (0.1,...
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Double integral over region $R$ using change of variables

Can someone help step by step on the following? Evaluate $\iint_R x^2y^4\,dxdy$, where $R$ is the region bounded by $xy = 4$, $xy = 10$, $y = x$, and $y = 6x$ by using the change of variables ...
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138 views

Linear Algebra - Preservation of inner product

Consider the vector space $\mathbb{R}^2$ with the standard inner product given by $ \langle(a, b), (c, d)\rangle = ac + bd$. (This is just the dot product.) (a) Let $\theta \in [0,2\pi)$ and let T : $...
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42 views

A probability transformation

Let $X,Y,Z$ be continuous random variables; $Z$ are independent of $X,Y$. Is the following transformation right ? \begin{align} P(X,Y \in (a,b),Y+Z \notin (a,b))&=\int_a^bP(X \in (a,b),y+Z \notin (...
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59 views

Distribution of $R^2 = X^2 +Y^2$ where (X,Y) is a point on the unit circle

So I have a point $(X,Y)$ chosen from the unit disk with uniform distribution. And I'm attempting to find the distribution of $R^2$, where $R$ is the distance from the point to the origin. Now ...
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Can a linear transformation $\mathbb{R}^n \rightarrow \mathbb{R}^n$ have a complex eigenvector?

Can a transformation $\mathbb{R}^n \rightarrow \mathbb{R}^n$ have a complex eigenvector? Similarly, can a transformation $\mathbb{C}^n \rightarrow \mathbb{C}^n$ have a real eigenvector?
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22 views

Defining a region in $\mathbb{R}^2$

I was trying to do this exercise but my answer doesn't match with the solution and I'm wondering why: Consider the coordinates transformation defined by $x=2u+v$ and $y=u^2-v$. Being $T$ the ...
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Transformations of points in the plane

Hopefully somebody understands what I mean here, If take a polynomial with complex numbers as input, then I will get a complex number as an output. If the input and output are plotted on an Argand ...
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Transformation of a real number 4 by 4 matrix to a 3 by 3 matrix

Are there any transformation methods that could be used to represent a $4\times 4$ matrix of real constants as a finite product/sum of $3\times 3$ (numeric: real or complex) matrices?
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True or false: Every transformation T: Cn --> Cn (n ≥ 2) has n distinct eigenvectors

True or false: 1) Every transformation T: Cn --> Cn (n ≥ 2) has n distinct eigenvectors 2) Every transformation T: Cn --> Cn (n ≥ 2) has at least 1 eigenvector 3) Every transformation T: Rn --> Rn (...
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Polar Coordinate Conversion (Integration)

I want to convert some integrals to use polar coordinates as my differentials, my problem is getting the limits. So this is the first concept I am not understanding: If I have a circle in the xy-...
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Matrix transformations on objects

I am trying to solve the following question: I have created the scaling, translation and rotation matrices that I feel will transform the left figure to the figure on the right: Scaling $$ \...
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Aligning 2 Coordinate Systems

I have a camera and a table and I want to align the camera to co-exist in the same coordinate system as the table. Here is an image of the setting. What type of mathematical transformations I need to ...
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Prove that $l$ and $l_{AB}$ are parallel if and only if $\sigma_B \sigma_l \sigma_B \sigma_A \sigma_l \sigma_A = id$

Prove that $l$ and $l_{AB}$ are parallel if and only if $\sigma_B \sigma_l \sigma_B \sigma_A \sigma_l \sigma_A = id$ I imagine that this proof has to be along the lines of a proof by contradiction, ...
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Linear transformations and possible dimension mismatch

The problem: Let $L: R_4 \to R_3$ be defined by $$L([u_1, u_2 ,u_3 ,u_4]) = [u_1 ,(u_2+u_3), (u_3 + u_4)]$$ Let S and T be the natural bases for $R_4$ and $R_3$, respectively. Find the ...
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47 views

How to you find out what a matrix does to an equation.

Lets say I have an equation of a plane, $$x-3y+2z=0 $$ and I get matrix to transform it with say a 3x3 matrix with just a-i as place holders for the values in the matrix. How would I find what the ...
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25 views

Get vertex points of transformed rectangle knowing bounding box and transform matrices

(I'm not a mathematician so talk down to me). I have a rectangle that has been transformed by a series of matrix transforms. I can recover the transform matrices and get the x,y coordinates of each ...