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), ...

learn more… | top users | synonyms

1
vote
1answer
116 views

Know if a 4x4 matrix is a composition of rotations and translations (quaternions)

I am using quaternions to describe 3D transformations. A position in space is representated by a (x,y,z,1) vector, and a transformation by a 4x4 matrix, following quaternions logics as far as I could ...
0
votes
1answer
47 views

what is the rank of the multiplication transformation $A$ if $AX=PX$ and $P$ has rank $m$?

Consider the vector space consisting of all linear transformations on a vector space $V$, and let $A$ be the (left) multiplication transformation that sends each transformation $X$ on $V$ onto $PX$, ...
1
vote
1answer
58 views

Under what conditions is the operator $Ax = [x, y]x'$ a projection?

Suppose that $V$ is a vector space, $x'$ is a vector in $V$, and $y$ is a linear functional on $V$; write $Ax = [x, y]x'$ for every $x \in V$. Under what conditions on $x'$ and $y$ is $A$ a ...
0
votes
2answers
46 views

Prove that a subset of a linearly independent set is a linearly independent set

Let $S$ be a linearly independent subset of a finite dimensional space $V$. Let $S_1 \subset S$, then prove that $S_1$ is linearly independent. I have looked all through my textbook, but I have ...
0
votes
1answer
62 views

When doing 3D rotations my angle flips 180 degrees

I'm implementing 3D rotations for a set of 3D circles. To do that I'm using the parametric equation as described in http://demonstrations.wolfram.com/ParametricEquationOfACircleIn3D/. It works as ...
5
votes
1answer
187 views

Prove spatial velocity identity - screw theory

This question involves a proof regarding coordinate transformations of velocities of screw motions. This comes from "A Mathematical Introduction to Robotic Manipulation" (the text is available for ...
0
votes
1answer
27 views

What kind of a matrix has a unitary diagonalizing matrix?

Suppose $D = P^{-1} A P$. When is $P$ unitary? In other words, what kind of a matrix $A$ should be, such that $D=P^{\dagger}AP$? i.e. what are the conditions a matrix must have to be able to ...
2
votes
1answer
39 views

If $D=P^{-1}AP$, then $f(D)=P^{-1}f(A)P$?

Suppose I have a diagonalizable matrix $A$, such that $D = P^{-1}AP$ Can I apply an element-wise function $f$ and expect that $f(D)=P^{-1}f(A)P$, assuming $f$ is not a linear transofrmation? Or in ...
0
votes
1answer
33 views

going from one unit vector basis to another unit vector basis

Somehow I am confused about this. Say I start with Spherical coordinate $(r,\theta,\phi)$, and I want to find expression $\hat{\phi}$ in terms of $\{\hat{x}, \hat{y}, \hat{z}\}$. At first I thought ...
1
vote
1answer
35 views

Transformation Definition

Let $A$ and $B$ be two $n\times n$ matrices that have no eigenvalues in common. Let $T$ be the transformation $$ T(S):=AS-SB $$ that maps the $n\times n$ matrices, $M_n$, to the $M_n$. Can we ...
1
vote
1answer
58 views

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 $T$?
0
votes
1answer
56 views

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 ...
0
votes
1answer
59 views

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), ...
0
votes
1answer
41 views

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 ...
2
votes
1answer
3k views

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 ...
1
vote
1answer
96 views

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 ...
0
votes
1answer
17 views

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 ...
0
votes
1answer
49 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}) = ...
1
vote
1answer
52 views

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 $$ \int_{-\infty}^{\infty}e^{-j\omega t}dt? $$
0
votes
1answer
43 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 ...
1
vote
0answers
32 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 ...
0
votes
1answer
764 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 ...
1
vote
0answers
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 ...
0
votes
1answer
27 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 ...
1
vote
1answer
31 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 ...
0
votes
3answers
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... ...
1
vote
0answers
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: ...
0
votes
1answer
46 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 ...
1
vote
0answers
211 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 ...
2
votes
1answer
254 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 ...
0
votes
1answer
265 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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
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 $\chi^2$ ...
1
vote
0answers
69 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 ...
1
vote
0answers
123 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 ...
0
votes
1answer
95 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 ...
3
votes
2answers
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{ ...
0
votes
1answer
27 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 0s in the sample), is an ...
1
vote
1answer
36 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 λ), by the following rules W = (if T=0 then W=1 and if T>0 then W=0), I am having trouble finding the pf for W. Any ...
0
votes
1answer
55 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 &= ...
0
votes
0answers
288 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 ...
0
votes
1answer
112 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 ...
0
votes
1answer
51 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}, \ ...
0
votes
1answer
52 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 ...
0
votes
0answers
36 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 ...
1
vote
0answers
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} ...
0
votes
1answer
29 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
vote
0answers
159 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?
0
votes
2answers
34 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 ...