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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the Fourier transform of a constant

How to calculate the Fourier transform of a constant without the aid of duality property? In other words, how do I calculate $$ \int_{-\infty}^{\infty}e^{-j\omega t}dt? $$
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44 views

What is the image of this mobius transformation

Consider the standard mapping $w=\frac{1}{z}$. What is the image of the "half" plane above the line whose imaginary part is $c$, for the three cases of $c\gt 0 , c=0 , c\lt 0$? For $c=0$ obviously ...
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34 views

Transformation for two different boundary functions in Stefan problem

Peace be upon on all of you, I have one-dimensional Stefan problem. Let say we have two boundary conditions of $u(t,s_{1}(t))=g_{1}(t)$ and $u(t,s_{2}(t))=g_{2}(t)$, where $u$ is temperature, $t$ is ...
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2answers
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 ...
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105 views

Dimension of image of a skew symmetric map is even

If $A$ is a skew-symmetric linear transformation on a finite-dimensional Euclidean space, then rank $\rho(A)$ of $A$ i.e., the dimension of image of $A$ is even. I am trying for a geometric proof of ...
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28 views

How to create own transformation change of variables

Evaluate the following integral using change of variables. Draw the original and new regions of integration. $$\int\int_{R} \frac{1}{x^2-y^2} dA$$ where R is bounded by the lines $x +...
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34 views

Composition of linear transformations that preserve angles

Given two invertible linear transformations T1,T2 in L(V) that preserve angles i.e. $\frac{(T(u), T(v))} {∥T(u)∥∥T(v)∥} = \frac{(u, v)} {∥u∥∥v∥} $. How can I show that T1T2 and T-1 also preserve ...
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83 views

Matrix with linear transformation with reflection

Find the matrix of the linear transformation A which is the reflection in the line $y = \sqrt{2}x$ with respect to the standard basis in $\mathbb{R^2}$. I Have no idea how to approach this problem...
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61 views

Help me generalize what this divisor transform does.

I have an algorithm which takes as input the series expansion of: $$\frac{-(1 + ax(-2 + x + ax))}{-1 + ax} \tag 1$$ or expressed differently: $$\left\{a^0,(-a)^1,a^1,a^2,a^3,a^4,a^5,a^6,a^7\right\}$...
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49 views

Dot product significance in vector transformation?

Suppose we multiply a 3 component vector by some 3x3 transformation matrix. Is it correct, then, to say the following about the transformed vector? Each component of the transformed vector is equal ...
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218 views

Conformal mapping of part of an annulus

I have a question about conformal mapping. I am wanting to map annuli to some other simple domain (probably rectangular). I have an image of my problem below In image A. we see a standard annulus ...
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268 views

Determine if the following function is one-to-one and/or onto

$T(x,y,z) = (xy,yz,xz)$ For one to one, I made $(x,y,z)=(u,v,w)$ and solved. $$xy=uv\to y=\frac{uv}{x}$$ $$\frac{uz}{x}=w$$ $$xz = uw \to x = u$$ $$uy = uv \to y = v$$ $$vz = vw \to z = w$$ So ...
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1answer
321 views

What is the correct change of variables to yield convexity in this nonlinear optimization problem?

$$ \text{min. } x/y \\ \text{s.t. } 2\leq x \leq 3 \\ x^2+y/z\leq \sqrt{y} \\ x/y=z^2 \\ x,y,z\geq 0 $$ To transform this problem into a nonlinear convex optimization problem, both the objective ...
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1answer
24 views

How can I find the density function of Z?

I am trying to find the density function for Z, this is what I am doing but I am not getting an appropiate function, I don´t know if there is something wrong with limits of the intregral. Or if this ...
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1answer
29 views

What is $f(T)$?

Let a linear transformation $T:\mathbb{R}^3\to \mathbb{R}^3$ defined as $T(v_1, v_2, v_3) = (v_1, v_3 - 2v_2, -v_3)$. Calculate $f(T)$ where $f(X) = -X^2 + 2 \in \mathbb{R}[X]$ I'm not so sure how ...
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1answer
74 views

Transformation of a Random Variable

We have a random variable $x$ with p.d.f. $\sqrt{\dfrac{\theta}{\pi x}}\exp(-x\theta)$, $x>0$ and $\theta$ a positive parameter. We are required to show that $2\theta x$ has a $\chi^2$ ...
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70 views

Complex analysis question, maximum principle application

Let $\Omega=\{z, \text{Re}z>0\}$ Suppose that $f$ is continuous in the closure of $\Omega$ and $f$ is holomoprhic on $\Omega$ and there are constants $A<\infty $ and $\alpha<1$ such that $$|f(...
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129 views

Equation of smooth spline curve

This is a homework question a)Assume an equilateral triangle ABC of a side AB = a = 10.The coordinate of A is (5, 3).The slope of the segment AB is 2.This triangle controls a curve.This smooth ...
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1answer
101 views

transformation of conic section

Given is the conic section $x^2 +xy + y^2 +2x +3y -3 = 0$. The following tasks: 1.) What is the coordinate matrix $A_1 = M_{\beta} (\sigma) $ of the bilinearform? 2.) do the transformation and ...
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118 views

How to write this term compactly?

How can I write this term in a compact form where $a$ only appears once on the RHS (in particular without cases)? $T(a) = \begin{cases} a^2 &,\text{ if $a \leq 0$}\\ 2a^2 &,\text{ ...
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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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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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58 views

probability: transformation of a random variable $Y = X^4 + 1$

Find the PDF of $Y = X^4 + 1$ if $X\sim\exp(\lambda)$. When a transformation is not one-to-one, we have multiple solutions for $X$. Take for example $Y = X^2$. Then \begin{align*} x_1 &= \...
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294 views

Box Muller Transform - Proving that Z is Normal Distribution

I'm studying the Box Muller transform and I cannot see how Z0 and Z1 represent standard normal distributions. I've looked at the wikipedia page for the box-muller function but they don't seem to have ...
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1answer
124 views

how to obtain transformation matrix A in y = Ax + b notation?

I'm trying to obtain original transform matrix A and its translation vector b From y=Ax+b equation. I have original values of vectors before transform and translation (x) and vectors after transform ...
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1answer
52 views

Finding the image of a linear transformation given other images

Suppose there is a linear translation $T: \mathbb{R^3} \rightarrow \mathbb{R^3}$ such that $$T(\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}) = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}, \ T(\begin{...
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1answer
56 views

Matrix representation, linear transformation , general question on linear algebra

Let $X$ be a finite dimensional vector space over $K$ and define $T:X\rightarrow X$ to be a linear transformation on $X$. If $\alpha, \beta$ are two different basis for $X$ then we know that the ...
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0answers
37 views

Integration matrix

I want to do integration(summation) of a signal(x) using matrix multiplication. I am looking for a transformation matrix, I corresponding to integration such that F = I * x , where x is the signal ...
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35 views

One question on Matrix Equation

Assume $\hat{M}_1, \hat{M}_2, \hat{T}_{11}, \hat{T}_{12}, \hat{T}_{21}, \hat{T}_{22}$ are $2\times 2$ matrix. And $a, b, A, B, C, D$ are all numbers, satisfying the following relation: \begin{align} ...
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1answer
30 views

Homeomorphism from $[0,1]\times[0,1]$ to $\overline{D}(0,1)$?

I'm trying to construct a homeomorphism from $[0,1]\times[0,1]$ to $\overline{D}(0,1)$. I'm pretty sure there is one. I've been trying to work geometrically : mapping $[0,1]\times[0,1]$ to $[-1/2,1/2]...
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176 views

Comparing function to parent function without graphing

How can I compare this function to the parent function without graphing? Where did the 5/4 come from and what steps do I need to take to solve this?
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36 views

Prove the linear transformation that takes all linear maps T: V → W to their respective matrix representations is an isomorphism.

Let V, W be finite dimensional vector spaces. Prove the linear transformation that takes all linear maps T: V → W to their respective matrix representations is an isomorphism. Thanks in advance! I ...
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3answers
118 views

Rewrite formula

I have the formula: $$ R(T)=R(T_0)e^{-B(\frac{1}{T_0} - \frac{1}{T})} $$ How can I write this to T=...? I came this far: $$ \ln(\frac{R(T)}{R(T_0)})= -B(\frac{1}{T_0} - \frac{1}{T})$$ $$ \frac{ln(\...
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24 views

Transformation as function of time, Solve for time

I'm trying to create a flawless a priori collision solver. I have two local coordinate systems which map to global coordinates using $[translate][rotate][scale]$, and map to eachother using $[TR1][...
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1answer
113 views

What's the inverse of the Weierstrass-Mittag-Leffler-Transform $\exp\left[g(z) + \int_\mathbb C f(y)\ln(z-y)\,dy\right]$?

As mentioned in another post, as a consequence of Mittag-Leffler's theorem combined with the Weierstrass factorization theorem, after reducing to the common denominator, any meromorphic function can ...
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Applying PCA on covariance matrix in order to generate a new random variable.

Let $\mathbf{x}$ be a random $n\times1$ real vector, $\mathbf{x}\in\Bbb{R}^n$, which is distributed normally with mean $\bar{\mathbf{x}}$ and covariance matrix $\Sigma_x\in\Bbb{R}^{n\times n}$, i.e. $\...
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1answer
39 views

Calculate projection of a line in a square

Said that we have two points (P1, P2) that form a line, and 3 points (S1,S2,S3) that form a square, how would we calculate the position X and Y of the point resulting from the intersection of the line ...
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1answer
37 views

What is special about a transformation if the matrix of that transformation is symmetric?

If the matrix of a linear transformation T$\colon \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$ with respect to some basis is symmetric, what does it say about the transformation? Is there a way to ...
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19 views

Questions about a special tensor transformation

Suppose tensor $U_{i\alpha\beta}$ with dimension $M*N*N$ satisfy following condition: $$U_{i\beta\alpha}=W^1_{\alpha\alpha'}W^2_{\beta\beta'}U_{i\alpha'\beta'}$$ where $W^1$ and $W^2$ are $N*N$ ...
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1answer
46 views

Finding the transformation matrix R

Please help me in solving this problem, I am not sure what a transformation matrix R is and how to proceed.. Any help is appreciated. Find the transformation matrix R that relates the (orthonormal ) ...
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1answer
91 views

finding if a linear transformation exists, and proving it.

We just started the topic of linear transformations and I have this hw question that I just don't understand. Does there exist a non-trivial linear transformation, represented by some 2x2 matrix, ...
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1answer
216 views

Jacobian determinant of unitary transformation

Is the Jacobian determinant of a unitary transformation equal to one? I ask because I get that impression from the appendix of this paper. They have spherical coordinates for two particles, $\{\...
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49 views

A conformal mapping from a sector to a strip

What is the simplest function that maps the sector $r < 1$, $0 < \theta < \pi$ conformally onto the strip $0 < u < \pi/2$, $v > 0$? Here, $r$, $\theta$, $u$, $v$ have their usual ...
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1answer
39 views

Inverse Laplace transformation of (s^2-4s-2)/((s^2+2)^2)

I approached this problem as follow: $1.$ rewrote $(s^2-4s-2)$ into $(s-2)^2-6$ $2.$ Now break the function into 2 parts: $\frac{(s-2)^2}{(s^2+2)^2} + \frac{6}{(s^2+2)^2}$ the Laplace inverse ...
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3answers
28 views

“compression” transform

Is there a mathematical transform that cuts off a signal at two extreme values? Here is code to do what I want: ...
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1answer
173 views

Legendre transform concave function

Let $f$ be a concave function and define $f^*(y) := \inf_{x}(yx-f(x))$. Is this in any sense related to the Legendre transformation? -If yes, is $f^*$ also concave? Is this transformation invertible ...
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31 views

If S and T are transformattion mappings, what is [ST]?

S and T are transformation mappings, what does [ST] and [TS] mean? Does it mean transform via S and then apply T to the result and vice versa?
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45 views

Conformal mapping of nonsimply connected domains

The question asks: Map the complement of the arc $|z|=1$, $y\geq 0$ on the outside of the unit circle so that the points at $\infty$ correspond to each other. How would you construct such conformal ...
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1answer
48 views

Linear Algebra Dimension

Let $L(U,V)$ = $\{T:U\rightarrow V\ :\ T\ \text{linear}\},$ and dim $(U)=n$, dim $(V)=m$. Then show that $$ \dim L(U,V) = mn. $$ I don't know how to begin and I already searched the internet to find ...
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114 views

Linear Algebra - - Linear transformation

The matrix $$ A=\left[\begin{array}{ccc} 1 & -a & a \\ -1 & a & a+2 \\ 1 & 2a+3 & -3a-4 \end{array}\right], $$ where $a \in \mathbb{R}$, represents a linear transformation $T: \...