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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Gaussian function at a rotated and translated coordinate system

I'm reading a paper and this coordinate transformation came along. In the $z_{i}=0$ plane the electric field is writen as $E=\exp[-x_{i}^2]$. The author says it's more convenient to work with the ...
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1answer
25 views
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Transformation of a plane

I have the $(x,y)$-plane $$\left\{(x,y,z)\in \mathbb{R}^3 | x,y\in \mathbb{R}, z = 0 \right\}.$$ I need a transformation matrix to transform this to the plane $$ \left\{ (x,y,z) \in \mathbb{R}^3 | x+...
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19 views

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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3answers
2k views

How to find the equation of the graph reflected about a line?

Consider the graph of $y = e^x$ (a) Find the equation of the graph that results from reflecting about the line $y = 4$. (b) Find the equation of the graph that results from reflecting about ...
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1answer
25 views

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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1answer
6k views

Matrix for rotation around a vector

I'm trying to figure out the general form for the matrix (let's say in $\mathbb R^3$ for simplicity) of a rotation of $\theta$ around an arbitrary vector $v$ passing through the origin (look towards ...
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2answers
39 views

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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1answer
503 views

Variance stabilization for Poisson data

Intro Let $Z > 0$ be a random variable with the mean and variance defined as $\mathbb{E}\{ Z \}$ and $\operatorname{Var}\{ Z \}$, respectively. The variance stabilization transform (VST) $f(z)$ ...
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3k views

Negative & Positive Shear Factor

My question relates to constructional geometry & matrices aren't to be involved in the solution because stated Math level is up to O Levels... The figure below shows shear with y=3 as invariant ...
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1answer
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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1answer
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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2answers
32 views

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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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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1answer
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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390 views

What is the sense of bottom row of affine transform matrix?

Usually affine transform matrix (in 2D) is represented like where block A is responsible for linear transformation (no translation) and block ...
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1answer
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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2answers
43 views

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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2answers
39 views

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
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Fourier Transform of a Polynomial

Lets say you are given \begin{equation} f(x)=1+x^3 \end{equation} and the definition of Fourier transform: \begin{equation} \hat{f}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ikx}f(x)dx, k\...
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1answer
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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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1answer
1k views

Finding reflection of a matrix

To 2 decimal places, what is the value of the lower-right entry in the reflection matrix $Q_a $if a = 1.05? Not even sure where to begin, is there a formula? This is what I could find in my textbook
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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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3answers
30k views

Image and Kernel of a Matrix Transformation

So I had a couple of questions about a matrix problem. What I'm given is... Consider a linear transformation $T: \mathbb R^5 \to \mathbb R^4$ defined by $T( \overrightarrow{x} )=A\overrightarrow{x}$, ...
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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
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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1answer
25 views

Image under $T_j$ of the basis vectors $e_1$ and $e_2$.

Define the linear transformation. Decide which of the mappings of $\mathbb R^2$ to itself given below are linear. $$\begin{align}T_1(x,y)&=(x+2y,y-2x)&T_2(x,y)&=(x,2x+y)\\T_3(x,y)&...
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1answer
15 views

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

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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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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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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Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z…: unified treatment of transforms?

I understand "transform methods" as recipes, but beyond this they are a big mystery to me. There are two aspects of them I find bewildering. One is the sheer number of them. Is there a unified ...
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1answer
161 views

What is this subclass of the class of monotonic transformations?

Let $u$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$. Then $v$ is called a positive monotonic transformation of $u$ if $u(x) < u(y)$ if and only if $v(x)<v(y)$ and similarly for ...
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1answer
38 views

Probability function (p.f) of a random variable

If we have a Bernoulli random variable $W$ that is derived from a Variable $T$ (Poisson $\lambda$), by the following rules $W =$ (if $T=0$ then $W=1$ and if $T>0$ then $W=0$), I am having trouble ...
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1answer
31 views

Probability function and random variables

Given a Bernoulli r.v., $W$, which is derived from r.v. $T$ (Poisson) (a) if $T=0$ then $W=1$ and (b) if $T>0$ then $W=0$. One has to show that the sample mean (the proportion of $0$s in the ...
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1answer
155 views

Find a matrix such that the image is equal to the solution space of a linear system of equations

$x_1 + 2x_2 + x_3 − x_4 = 0$ $−x_1 + 2x_2 + x_3 + x_4 = 0$ $x_1 + x_3 = 0$ Consider the following matrix $A$ from the system of equations: $$A = \left(\begin{array}{crc} 1 & 2 & 1 & ...
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649 views

Finding the eigenvalues of a $3N \times 3N$ block matrix

I have a block matrix of size $3N \times 3N$ of the form $$B = \begin{bmatrix} A & C & \ldots & C\\ C & A & \ldots & C\\ \vdots & \vdots & \ddots & \vdots\\ C &...
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1answer
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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1answer
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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1answer
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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3answers
47 views

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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1answer
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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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855 views

Inverse rotation euler angles

I have three angles representing a rotation (Pitch, roll and yaw). I need the inverse rotation (working on coordinate system transforms). What I do now is transforming these angle to a rotation matrix ...