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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Transforming Vectors

Let $T$ be the linear transformation from $\mathbb{R}^3$ to $\mathbb R^3$ that reflects every vector about the $xy$-plane and then triples its length. How do I find the matrix for $T$?
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Given two linear transformations, find the preimage of a given point for the composite transformation

If someone could run quickly through the theory and methods on this it would be hugely appreciated. Thank you. Let $f: \Bbb R^2 → \Bbb R^2$ be reflection in the line $y = x$ and let $g: \Bbb R^...
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Tricky change-of-basis transformation problem

I have absolutely no idea what to do here because of the $\sin(x).$ Let $V = \text{Span}\left\{x, x^3, \sin(x) \right\}$, and consider the basis for $V$ given by $\beta = \left\{x-2x^3, x^3+\sin(x), -...
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How to find the Fourier Transform of the form $\frac{cos(2\pi t)}{t^2}$?

I'm having trouble on figuring out how find the Fourier Transform of the following function, and I'm not allowed to use the straight up definition of the Fourier Transform but rather use it's ...
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Does the Fourier Transform exist for f(t) = 1/t?

My professor says that the following function has a Fourier Transform: $$f(t) = \frac{1}{\pi t}$$ He said that all I have to do is apply some of the Fourier Transform properties and not the direct ...
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Linear Transformation and Matrices

I have been studying linear algebra for a while now, and I still can't understand the basic concept of linear transformation and the easy ''translation'' of them the matrices. I understand that every ...
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Clarification of a transformation step (probably very simple).

A simple and quick question. Have been sitting over it for a while now but i can't get it right: Could someone just clarify how this transformation has been done? $$\frac{(v+1)^2 2^{v^2}}{2^{v^2+2v+1}...
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50 views

Null Space of Transformation

I am given that $V$ is n-dimensional vector space over $\mathbb{C}$ and $T \in L(V)$. And $T$ has least $m$ distinct nonzero eigenvalues. How do I show that $\text{null}(T^{n-m}) = \text{null}(T^{n-m+...
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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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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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903 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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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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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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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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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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52 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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226 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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276 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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336 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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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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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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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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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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131 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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103 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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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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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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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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295 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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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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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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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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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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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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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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180 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
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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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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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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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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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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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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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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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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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221 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, $\{\...