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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13 views

Name for affine transform which consists of scale and translate

Is there a concise name for this type of transform? Viewed as an affine transform, the matrix would have the form: k 0 dx 0 k dy 0 0 1 The best I can come ...
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0answers
24 views

Is there a similarity solution for this PDE? (with discussion, kindly check)

I have a PDE for $h(x,t)$ of this form $$h_t+Ah^{-1}+(h^3h_x)_x+Bh_{xx}+(h^3h_{xxx})_x=0,$$ where the subscripts denote the partial derivatives, and $A$ and $B$ are all constants. I'm wondering ...
4
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3answers
211 views

Find the distribution of the series $Z = X_1+X_2+…+X_N$

"Let $0<p=1-q<1$. Suppose that $X_1,X_2,...$ are independent Ge(q)-distributed R.V.'s and that $N \in Ge(p)$ is independent of $X_1,X_2,...$. Find the distribution of $Z=X_1+X_2+...+X_N$." I ...
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1answer
14 views

Basic inverse $z$ transform

I have trouble finding a (probably) pretty easy inverse of a $z$ transform. $$H(z) = \frac{z-0,5}{z+0,5}$$ I used the polynomial division on it to get a proper fraction and got $$H(z) = 1 - ...
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2answers
22 views

Range of continuous transformation on closes set

let $f$ be a continuous transformation and $F$ closed set. Prove that the range $f(F)$ does not have to be closed.
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21 views

Invariance of stationary wavelet transform

Suppose we are given 64 points $x_1,\ldots ,x_{64}$ and divide them into two groups $x_1,\ldots, x_{32}$ and $x_{33},\ldots , x_{64}$. Then we apply stationary wavelet transform to both these groups ...
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1answer
29 views

What is Fourier transform of $g(y)=e^{-\pi y^2 +2\pi yx}$ for all $y\in \mathbb R.$?

Let $x\in \mathbb R.$ Define $g:\mathbb R\to \mathbb R$ as $g(y)=e^{-\pi y^2 +2\pi yx}$ for all $y\in \mathbb R.$ My Question is: What is the Fourier transform of $g$? In other word, how to ...
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39 views

Change from Fourier Space to Real Space

I have a function in 3D fourier Space $$g(\textbf {k})=\frac{\hat{k}_i}{\hat{k_j}}f(\textbf {k}),$$ where $\hat{\alpha}$ is a fixed vector and $i$ and $j$ are the components of the relevant vector, ...
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1answer
12 views

Clarification on computing the Moment Generating Function of a Gamma-distribution

In my text book the MGF of a Gamma distributed R.V., $X \in \Gamma(p,a)$, is computed as follows: $$ \psi_X(t)=\int_0^{\infty}e^{tx}\frac 1 {\Gamma(p)}x^{p-1}\frac 1 {a^p}e^{-x/a}dx= $$ $$ =\frac 1 ...
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1answer
12 views

How to compute the Moment Generating Function of R.V.'s $X_1 - X_2$

I want to show that $$Y_1 + Y_2 \overset{d}= X_1 - X_2$$ where $Y_1, Y_2 \sim \text{Laplace}(1)$ and $X_1,X_2 \sim \Gamma(2,1)$, by checking their Moment Generating Functions. For the left-hand ...
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16 views

Z transform to difference equation?

For a z transform to fully describe an equation, you need the z transform itself and the ROC. You can convert the z transform to a difference equation easily if it's rational. How can I covert the ...
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1answer
34 views

Linear transformation understanding

Let $S: R^2 → R^2$ be the function defined by $S(x, y) = (x − y, y)$ for all $(x, y)$ I've found the matrix for this which was question 1. These are the two questions I am struggling to understand. ...
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1answer
62 views

Calculate the determinant of the matrices $a_{ij}=\frac{1}{i+j-1}$ and $b_{ij}=\frac{1}{i+j}$?

I would like to know if there is any formula for calculating determinants of the following symmetric matrices: $$ A=[a_{ij}]_{n\times n},\qquad a_{ij}=\frac{1}{i+j-1}, $$ and $$ B=[b_{ij}]_{n\times ...
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3answers
44 views

Log transformation

Supose I have a series of numbers from 1 to 10. Their mean value is 5.5. Now supose I apply some transformation like $y=2x+1$. Now their mean value is 12. Now, if I want to get back the original mean, ...
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1answer
155 views

Prove that every triangle is the orthogonal projection of an equilateral one

Prove that every triangle is the orthogonal projection of some equilateral triangle. This problem appears in a book I'm working through in the chapter on transformations in space. There is a rather ...
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2answers
39 views

Rotation matrix construction

I'm reading a book on how to construct transformation matrices and I'm stuck in a certain point. From the book: Now here's the figure that I don't understand: How come the opposite edge in the ...
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1answer
17 views

Measuring Image of a Set Using Jacobian Integral

Assume $T:\mathbb{R}^d \to \mathbb{R}^d$ is a differentiable mapping and $E$ be a measurable set. Show that $m(T(E))=\int_E |det(DT(x))|dx$. I am thinking I might use the following Theorem, but ...
2
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1answer
24 views

Convolution Problem

while working on a signal processing problem i've reached to the following: So my aproach was: Am I doing something wrong? Is it valid Y(f)=[X(f) x H(f)]*W(f)=X(f) x [H(f)*W(f)] If you could ...
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1answer
23 views

Construction of a Transformation of a Random Vector that Preserves Independence

Let $X_1, \dots, X_n$ be $n$ independent random variables, not necessarily normal. Let $Y_1 = \sum_{i=1}^{n}\alpha_i X_i$ a given linear combination of the random variables. Is there a known, ...
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1answer
20 views

How do I find the Probability Function of this Transform?

Given the Probability Generating Function for a non-negative, integer-valued, R.V. $X$ as: $$ g_X(t)=\log\left(\frac 1 {1-qt}\right). $$ How do I compute its Probability Function, $P(X=k)$? A ...
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1answer
46 views

Biliniear form to inner product

Let $f:V\times V\rightarrow F$ be a bilinear form in a finite inner product space V. If $F=R$, how can I prove that there exists a single linear transformation $T:V \rightarrow V$ so that for each ...
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2answers
81 views

Closed form solution to rotation in arbitrary many planes in arbitrary dimensions.

In each coordinate space $V$ with dimension $\dim(V)$, we can describe any rotation operator $R : V \to V$ as a product of rotations in as few as $\dim(V) - 1$ orthogonal planes in the space. Let's ...
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12 views

Transformation Matrix for particular problem

I have a question regarding transformation matrices. I have two images both showing a table. I have coordinates of the corners of the tables, and now I want to apply a transform to 1 of the images so ...
2
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1answer
16 views

Finding Equality around Axis of Symmetry

I have a particular function that for even numbers $m$ obeys the following equation: $$f_{m,n}\left(\frac{2}{m}-x\right)=(-1)^nf_{m,n}(x)$$ Now when I put in odd values for $m$ and plot the ...
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1answer
41 views

Probability Theory - Transformation (of two variables) of continuous random variables

Let $X_1$ and $X_2$ be independent and identically distributed continuous random variables, with probability density function $$p(x)=\begin{cases} \exp(-x), & \text{if}\ x>0 \\ ...
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1answer
60 views

Transformation of continuous, independent random variables

Let $X_1$ and $X_2$ be independent and identically distributed continuous random variables, with probability density function $$p(x)=\begin{cases} \exp(-x), & \text{if}\ x>0 \\ ...
1
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2answers
59 views

Probability Theory - Transformation of independent continuous random variables

Let $X_1$ and $X_2$ be independent and identically distributed continuous random variables, with probability density function $$p(x)=\begin{cases} \exp(-x), & \text{if}\ x>0 \\ ...
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0answers
46 views

How do I find the matrix with respect to a different basis?

I tried to solve this question but the answer is totally different, can you explain how to solve it
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2answers
32 views

Transformation Matrix for Derivative

I have figured out how to show Part A by using properties of derivatives. For Part B, we know that $T(f)$ is $\begin{pmatrix} 4a+2b+c\\4a+b\\2a\end{pmatrix}$, so when asked to find a matrix $A$, ...
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2answers
29 views

How do I scale a triangle given its cartesian cooordinates?

Given the cartesian $(x,y)$ coordinates of three points $a, b$ and $c$ that form an equilateral triangle $ABC$, how do I scale them using its center point so that its position on the cartesian plane ...
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1answer
45 views

If T is a normal transformation, does that mean that $||v||^2=||Tv||^2$?

If T is a normal transformation, does that mean that $||v||^2=||T||^2$ for all $v\in V$ ? Where $V$ is a vector space And if yes, how to prove it? EDIT: To be clear I mean that $||v||^2=(v,v)$
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1answer
29 views

Proving a property for a normal transforation $T$ for which $T^{-1}=-T$

Let $V$ be a unitary space. Given a normal transforation $T$ for which $T^{-1}=-T$. Let $v \in V$ and $u=Tv$ I need to prove that $Tu=-v$ (which I managed to do easily, so we can consider it as ...
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30 views

What is the name of this transformation's property?

I have a transformation $P$ with the following property: $P^n = \mathbb{I}$ (the identity) for some specific $n>1$, and all $P^m \neq 1$ for $m \neq n$. What is the name of the property of $P$? ...
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1answer
33 views

Jacobian of the Transformation Problem, Multivariable Calculus

I have the following Jacobian problem: I'm having trouble working through it because the double integral in terms of u and v is throwing me off. Could someone walk me through it? Thanks!
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1answer
41 views

Linear Algebra, Linear Transformation problem [closed]

My task is this I am wondering how to go about doing this. Anyone have ant idea? Thanks!
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1answer
157 views

Prove that $\det(M) = \det(N)$

Let $T:V \to V$ be a linear transformation, and let $B$ and $C$ be two bases for $V$. Let $M$ be the matrix of $T$ with respect to the basis $B$, and let $N$ be the matrix of $T$ with respect to the ...
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1answer
87 views

Rotate points from one plane to another

I'm trying to create a algorithm that will rotate points given on plane 1 to plane 2. I have found two different ways of doing this. My question is ... Why are the transformation matrices different ...
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25 views

Rotating points from one plane to another plane

I'm trying to create a function that will rotate points given on plane 1 to plane 2. I have found two different ways of doing this. The attached spreadsheet shows the two different ways as Solution ...
4
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3answers
576 views

Shift numbers into a different range

I was wondering how can I shift my data that fall between a range lets say [0, 125] to another range like [-128, 128]. Thanks for any help
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1answer
39 views

differential equation problem with laplace - calculators cant solve

I am trying to solve a third order differential equation problem with laplace transform. But I am stuck since 3 days... Could someone tell me what I did incorrectly? I transformed my equation in the ...
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0answers
11 views

aligning a matrix to reference matrix

Assuming X$_0$ as a matrix which represent some sort of transformation between TWO different coordinate system. Now, as a function of time the matrix which has three column vectors evolves in to X'. ...
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1answer
28 views

How do I find the limits of a joint density function and calculate the inequalities?

The r.v.'s ${X_1}$ and ${X_2}$ are independent and equidistributed with density function $$ f_X(x)=4x^3, 0 \le x \le 1, $$ and equal to zero otherwise. Set ${Y_1=X_1\sqrt(X_2)}$ and ...
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10 views

Eliminating correlation by change of variables (on a sphere)

I am trying to understand the derivation of the mean of stationary points $\mathbb{E}[\mathcal N_s^+]$ of a Gaussian random field $V(x_1,\dots,x_N)$ on the upper half of a sphere with radius $R$ as ...
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1answer
43 views

What is the name of this matrix?

I have a vector $a=[a_1 \space a_2 \space a_3 \space a_4 \space a_5 \space \cdots a_n]$ and I want to generate following matrix 'A' from it. $$A=\begin{bmatrix}a_1 & a_2 &a_3\\a_2 & a_3 ...
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12 views

How to transform a set of scores into a normal distribution score?

I have a set of people with scores. Let's say: Person1: 18 Person2: 1,879 Person3: 873 Person4: 1M ... Person2M: 9,387 I would like to give each of these, a 0-100 Score, that is distribution in a ...
2
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0answers
59 views

Find transform matrix that transforms one line segment to another

I have two line segments, one with points $P_1=(x_1,y_1,1), P_2=(x_2,y_2,1)$ and other with points $P_3=(x_3,y_3,1), P_4=(x_4,y_4,1)$. I need to find transform matrix $$M=\left(\begin{array}{ccc} a ...
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1answer
17 views

about Fourier transform. graph of frequency over amplitude

I want to compute the Fourier transform for the function $f(x)=\sin{x}$. The first question is: the Fourier transform is $\pi$? $$a_n=\frac{1}{\pi}\int^{\pi}_{-\pi}{\sin{x}\cos{nx}}=0$$ ...
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1answer
81 views

Change of Basis Matrix: Cartesian to Spherical Laplacian

I was looking at how a change of basis matrix, $[P_{\beta\leftarrow\alpha}]$, is made. While this is a bit more advanced that than what was taught at the course, I wonder what would be the change of ...
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15 views

2D Homogeneous Transformation : Reflection vs Mirroring

I have two questions: (1) Is there any difference between the terms Reflection and Mirroring in 2D Transformation? (2) What are their Transformation matrices with reference to an arbitrary line?
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24 views

Unitary similarity transformations

A complex matrix is said to be "normal" if it commutes with its conjugate transpose. Can you show that a matrix is normal if and only if it is diagonalized by a unitary similarity transformation? ...