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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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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16 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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14 views

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
23 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 a similarity transformation on this coordinate system that turns the cone ...
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21 views

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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1answer
25 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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25 views

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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1answer
44 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 = ...
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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
28 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} = ...
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3answers
41 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
28 views

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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22 views

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
17 views

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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1answer
41 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} ...
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2answers
29 views

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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1answer
134 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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1answer
40 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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3answers
56 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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2answers
62 views

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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1answer
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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2answers
29 views

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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1answer
14 views

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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58 views

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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2answers
20 views

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 ...
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1answer
15 views

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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1answer
22 views

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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0answers
12 views

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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19 views

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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1answer
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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1answer
22 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 ...
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1answer
29 views

Finding the area between two curves using a set of transforms and their Jacobian

I have the following transforms: $\begin{align} x &= u^2 - v^2 \\ y &= 2uv \end{align}$ and am tasked with finding the area between the following curves: $\begin{align} x &= 4 - ...
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4answers
50 views

Negation with De Morgan’s law

I'm having a hard time getting my head around transformation proofs. There is one particular example demonstration in the material I'm studying which I can't make sense of From this ¬ (¬ (¬ p) ∨ ¬ ...
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1answer
36 views

What is known about the space of measure-preserving transformations?

I started reading about measure-preserving transformations, the ergodic theorems and mixing, but I was also wondering what is known about the space of measure-preserving transformations. The books ...
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1answer
21 views

Transforming an integral to a different domain

For a given $v(x)$ with $x\in[0,1]$, use the variable transformation $x=g(\eta)=\frac{1}{2}\eta+\frac{1}{2}$ to transform the integral $I=\int_0^1v(x)dx$ to an integral over $[-1,1]$. My doubts: ...
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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. ...
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2answers
70 views

Calculate Rotation Matrix to align k n dimensional vectors

I have a $k$ number of $n$-dimensional vectors written with respect to two rotated frames: $X= \{\vec{x}_1,\vec{x}_2,...,\vec{x}_k\}$ and the same rotated vectors: $X'= ...
2
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1answer
38 views

Finding a Transformation for a Sum of Exponentials

I am looking to see if it is possible to find a transformation $T_i(f(x))$ such that $$T_1\left(e^x+e^{ix}+e^{-x}+e^{-ix}\right)=e^x-ie^{ix}-e^{-x}+ie^{-ix}$$ ...
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1answer
37 views

What is wavelet tranform in simple words?

I have read wiki and other sources and have still problem understanding the wavelet transform. What is the basic idea in simple words? Does the Fourier uncertainty hold for wavelet transform?
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Derivate formula for Radon-transformation

For the Radon-transformation $\mathcal{R}f(r,\omega)=\int_{\{x:x\cdot\omega=r\}}f(x)\mathrm{d}\sigma(x)$ with $r\in\mathbb{R},\omega\in\mathbb{S}^{n-1}$ I want to prove the following derivative ...
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2answers
50 views

Laplace transform of $\sin(\sqrt t)$

How can I use this differential equation $$4tf''(t) +2 f'(t) + a^2 f(t)=0$$ to show that $$L(\sin(\sqrt{t}))=\frac{1}{2}\sqrt{\pi}\,\frac{1}{s^{\frac{3}{2}}}\,e^{\frac{-1}{4s}}$$
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11 views

Abel transformation - sources

I'd like to study the Abel transformation, that is, $$Af(x) = \int\limits_x^\infty\frac{f(t)t}{\sqrt{t^2 - x^2}}\ \mathrm{dt},\quad x\in(0,\infty).$$ I'm especially interested in ...
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1answer
25 views

Area of Region using Transformation

Let R be the region bounded by the curves x = 0, y = sin(x)+1, y = sin(x), and y = 2 − x. Find the area of R. I need to use a transformation to find this, but I could not solve it using a ...
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1answer
29 views

A Deeper Understanding / Interpretation of Homographies

I currently understand that a homography matrix, which allows for a mapping between planes in 3-dimensions, is a $3\times3$ matrix of the following general form: $$\begin{bmatrix} \vert & \vert ...
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20 views

Zero Order Laguerre Transform of Sin(at)

Zero Order Laguerre Transform is given by $$L\{f(t)\}=\int_0^\infty e^{-t} L_n(t)f(t)dt $$ I've to prove ...
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0answers
19 views

Transform matrix into constant diagonal matrix (or hollow matrix)

Does there exist a (possibly unique) orthogonal transformation, $U$, which will create a hollow matrix (or matrix with constant diagonal entries) from an arbitrary symmetric matrix, $A$? ...
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0answers
10 views

Conformal transformation of an annulus

I am given two circles in the complex plane, one of radius a, the other of radius b such that $a<b$. Their centres are separated by a distance h such that $a+h<b$. I need a conformal ...
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26 views

what is a ordinally quadratic function?

A function is ordinal equivalent to another means there exist a (unique) monotonic transformation between wiki definition of ordinal utility. I am a little confused, a function is ordinally quadratic ...
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17 views

What kind of transformation matrix should i use?

I am trying perform inverse kinematics on a 6 jointed robot, but is having a hard time determining how my transformation matrix should look like. I am using a piece of software to which you feed an ...
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1answer
32 views

Map an ellipsoid to a sphere

If I have a ellipsoid described by: $(\boldsymbol{x} - c)^T \boldsymbol{A} (\boldsymbol{x} - c) = 1$ How do I get the transformation to an unit sphere centered at the origin? From the principal ...