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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Equations transformations with roots

How does the following transformation works (do not write that it is easy i want the answer): $$\ln \sqrt[n]{\frac{n!}{n^n}}=\frac{\ln \frac{n!}{n^n}}{n}$$
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25 views

Finding the relative pose of a robot gripper

I have a robot arm with a gripper. I know the gripper pose (relative to the robot base coordinate system) at any moment. At startup, I record the pose of the gripper and set this as the original pose ...
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2answers
49 views

Formula for the sum of fractions [duplicate]

How to find the formula for the sum of fractions like this? $$\frac{1}{1\times 2}+\frac{1}{2\times 3}+\ldots+\frac{1}{n\times (n+1)}=$$
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1answer
53 views

Making sense of polar coordinates transformation on the derivatives

I would like to make sense of the transformation of the differentials in polar coordinates (to fix the ideas). To be more precise, the "right" way to find the transform for the differential and the ...
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2answers
53 views

Cayley Transform and Eigenvalues

I have a particular operator, namely $A=-i\frac{d}{dx}$ that I would like to Cayley transform. $A$ is defined on the Hilbert space $L^{2}[0,1]$ and has domain $\mathcal{D}_{\alpha}=\{g:g \in ...
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1answer
50 views

Rotate a vector about a given axis by the use of a quaternion

I encountered a problem in programming where I need to rotate a given vector about a given angle. To be precise, I need to change it to a quaternion so that I can later change it to a 4x4 matrix to ...
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0answers
81 views

Rotational matrix problem?

In the problem yo-yo is made of two identical cylinders of radius $R$, thickness $h$ and mass $M$, and the yo-yo is let go. In order to define the position of the yo-yo, I need as position vector and ...
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1answer
33 views

Normal distribution

X follows a regular normal distribution on V with center $\xi$ and inner product $<\cdot,\cdot>$, and let $\eta \neq 0$ be a vector in V. Show that the reel stochastic variable ...
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1answer
48 views

Changing $[0,2\pi)$ with $S^1$ such that a map defined on $[0,2\pi)$ stays unchanged

* Consider the following procedure of changing the domain of a map, but the map remaining essentially the same - illustrated, for concreteness, in case of the polar-coordinates map.* Let ...
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1answer
45 views

Fourier Transform - Laplace Transform - Which variable transform?

I need to know when do I have to transform $x$ and when $y$ in a PDE in Fourier Transform and Laplace Transform. In an exercise of Fourier Transform, I have to solve a Laplace Equation, where ...
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1answer
37 views

Lotka-Volterra coordinates transformation

I would like to ask the following: Given a Lotka-Volterra predator-prey system, \begin{align} & \frac{dx}{dt}={\alpha}x-{\beta}xy \\ & \frac{dy}{dt}=-{\gamma}y+{\delta}xy \end{align} , with ...
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3answers
130 views

Dimension of Hom(U,V)

I know this has been asked before - I am really struggling to understand what people have said though, so I want to ask for myself. If U,V are vector spaces over field K, with dimensions n,m ...
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0answers
39 views

Transfer function: steady-state solution of equation

Place the transfer function in the form $$H(i\omega) = \frac {1}{R}e^{-i\phi}$$ and use this result to find the steady-state solution of the equation$$x'' + x'+4x = 3*cos(2t)$$ I don't really ...
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1answer
162 views

Fourier COSINE Transform (solving PDE - Laplace Equation)

I'm trying to solve Laplace equation using Fourier Cosine Transform (I have to use that), but I don't know if I'm doing everything OK (if I'm doing everything OK, the exercise is wrong and I don't ...
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0answers
41 views

How can I show that the AR process is nonstationay if x(n) has nonzero mean?

This is a first-order-real-valued autoregressive (AR) process $y(n)$ that satisfies the real-valued difference equation $y(n)+a_1y(n-1)=x(n)$ where $a_1$ is a constant and x(n) is a white noise ...
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0answers
38 views

How to apply the chain rule for partial derivatives to transformations?

I'm currently working to solve the Black-Scholes model partial differential equation (it's a model for a.o. stock option prices). The Black-Scholes equation for a calloption C(S,t) is given by $ ...
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0answers
17 views

How to understand all types of transforms as linear operators on function spaces?

I would really appreciate if someone can point to a simple general framework that can help me to understand integral transform from a generalized point of view so that there is no longer a fourier ...
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27 views

Prove a formula after change of variable?

If I have a change of variable $(x,u)\to (X,U)$ given by $$X=x+\epsilon u,U=u-\epsilon u.$$ How to prove the formula $$\frac{\partial U(X,0)}{\partial \epsilon}=\phi(X,u(X))-u'(X)\xi(X,u(X)),$$ where ...
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1answer
26 views

Cross-sectional function for a surface of revolution

If I take a one-to-one function $f(x)$ and rotate it about the x-axis, how can I describe a function resulting from a cross-section of the solid of revolution? I'm not talking about the circular ...
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0answers
23 views

Shear matrix simple explanation

I can understand translation, dilation and rotation matrices, but the shear one is still obscure to me (despite understanding what shearing means graphically). This is the matrix: ...
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3answers
37 views

How does $(x+3)^2 - 2^2$ become $(x+1)(x+5)$?

I don't understand how $(x+3)^2 - 2^2$ can be transformed to equal $(x+1)(x+5)$. A short demonstration and/or reference to math rules would be very kind. Thanks.
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1answer
51 views

Transformation Matrix for cube in 2D

My task is to transform the cube from the left corner to the big cube in the middle: What I did was: First i scale the cube: $$ \begin{pmatrix} 4 & 0 & 0 \\ 0 & ...
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17 views

Transform Confocal Ellipsodal to Spherical Coordinates

I heard that someone published a paper showing that the confocal ellipsoidal coordinate system can transform into the spherical coordinates under special limit evaluations, however I was unable to ...
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2answers
75 views

Sum of uniform random variables $U(0,1)$ and $U(0, a)$

The problem I have is: $X \sim U(0,1), Y \sim U(0,a)$ are independent random variables. Find the pdf of $X + Y$. I've got stuck in an integral-problem, and will show you what I've tried. Skip to the ...
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1answer
59 views

Hyperbolic Geometry: Question about the Transitivity of Möbius transformations

I was confronted with this exercise in the book Hyperbolic Geometry by Anderson which states: In each case, find $m \in Möb(\mathbb{H})$ such that the property holds, or prove that no such $m$ ...
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2answers
32 views

Linear transformation with special properties

how should I do that please (I had this in my test yesterday)? Linear transformation $f:\mathbf{R}^{10} \to \mathbf{R}^7$ has an attribute that every vector $\mathbf{v}$ for which is true that ...
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21 views

Can Magnification/Scaling (transformation) be prepresented by a vector?

Vectors represent three bits of information: Magnitude, Line of Action, and Direction. A Translation (transformation) can be represented by a vector: object is moved By so much (magnitude) Along a ...
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1answer
26 views

Transformation theorem: calculate picture of a set

I have this function: $T:(0,\infty)^2 \rightarrow T((0, \infty)^2), \quad T(x,y)=\left( \frac{y^2}{x},\frac{x^2}{y} \right)$ Now I try to estimate $T(M)$ with: $0<p<q, \quad 0<a<b$ ...
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0answers
70 views

Transforming the cubic Pell-type equation for the tribonacci numbers

The Lucas and Fibonacci numbers solve the Pell equation, $$L_n^2-5F_n^2=4(-1)^n\tag1$$ The tribonacci numbers $z = T_n$ are positive integer solutions to the cubic Pell-type equation, $$27 x^3 - 36 ...
2
votes
1answer
38 views

Tietze transformation

I have a question in which I have to transform $\textbf{I.}$ $\langle a,b,c \mid b^2, (bc)^2\rangle$ to $\textbf{II.}$ $\langle x,y,z\mid y^2, z^2\rangle$ using Tietze transformations. My ...
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74 views

Decomposition of 4x4 or larger affine transformation matrix to individual variables per degree of freedom.

There are a couple of problems and solutions where affine matrices are decomposed into their seperate tranformations. However they are all for the 2D case and I`m finding it difficult to generalise it ...
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1answer
77 views

Variants of the change-of-variables formula

Consider the following change of variables formula for $f:X\rightarrow Y$, that holds for any "reasonable" $g:B\subseteq Y \rightarrow \mathbb{R}$ and $A\subseteq X$ $$ \int_B g(x)\ {\rm ...
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32 views

Linear transformation of vector

I have computer graphics class and i had something like that on lecture: $$ \begin{bmatrix} \overrightarrow{b1} & \overrightarrow{b2} & \overrightarrow{b3} \end{bmatrix} \begin{bmatrix} c1\\ ...
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1answer
32 views

What is a transformation that can't have shearing called?

What is a transformation called when it can have separate scaling for x and y, rotation, and translation, but it cannot have shearing or scaling AFTER rotation? Basically if this transformation is ...
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1answer
40 views

coordinate transformation and tensor

A 2 dimensional Euclidean space is represented by two different coordinate systems: the Cartesian system $(x_1,x_2)$ and an alternative system $(\xi^1,\xi^2)$ where ...
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32 views

Linear Algebra - verification of my answer, basis for $ImT$

I'd like to verify this answer, because I think that the answer in my book is incorrect. I'll be very glad if someone could tell me, if the basis I found for $ImT$ is correct. Let : $T:R^3 ...
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1answer
35 views

Matrix transformation conserving the “positive semi-definite” aspect

Let's say I have two covariance matrices $A$ and $B$ (so they're both positive semi-definite), What kind of transformations can I apply on either one of them or both without loosing the ...
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0answers
18 views

Composition of a rotation and a homothetic transformation of different centers?

Consider the rotation $r_{\Omega,\alpha}$ of center $\Omega$ and angle $\alpha$. Furthermore let $h_{\lambda,S}$ be the homothetic transformation of center $S\neq \Omega$ and ratio $\lambda$. What ...
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1answer
43 views

Proof : If $S,T:V \rightarrow V$ (V is finite) and $KerS=\{0\}$ then $Im(TS)=Im(T)$

I have this problem : Proof : If $S,T:V \rightarrow V$ (V is finite) and $KerS=\{0\}$ then $Im(TS)=Im(T)$ My solution : Let $v \in ImT$ exist $w \in V$ such that $T(w)=v$. Since $KerS=\{0\}$ ...
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1answer
45 views

Composition of translation and rotation is a rotation, but what is its center?

Consider the rotation $r_{\Omega,\alpha}$ of center $\Omega$ and angle $\alpha$.Furthermore let $t_{\vec{v}}$ be the translation by vector $\vec{v}$. Then $$t_{\vec{v}}\circ ...
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1answer
10 views

Bounds of a Bivariate Function

I am given that $h(x, y) = \frac{x}{(x+y)}$ , $x > 0$ , and $y > 0$. I am supposed to deduce that the bounds for $h(x, y)$ are $0 < h(x, y) < 1$, but I do not understand how to arrive at ...
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2answers
395 views

What is the difference between coordinates transformation and change of coordinates?

In the context on 3D computer graphics, what is the difference between coordinates transformation and change of coordinates? It can just be a matter of notation, but my book makes a clear distinction ...
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0answers
16 views

Show: $s_A\circ s=s\circ s_B \iff$ $d$ is the bisector of $[AB]$

Consider two distinct points $A$ and $B$ of the plane space. Let $d$ be a line different from $(AB)$. Denote $s_A$ (resp. $s_B$) the central symmetry of center $A$ (resp. $B$). Denote $s$ the symmetry ...
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0answers
17 views

Equality with fourier transform

I have problem with the following equality where the Fourier transform appears: Assume that $u_1,u_2:\mathbb{R}^n\to\mathbb{C}$ are Schwartz functions. Prove that for any $\xi\in\mathbb{R}^n$, ...
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0answers
41 views

probability integral transformation and distribution of P= P[ |T| <= |t|] .

The task is to find the distribution of P. where , P=P[ |T| <= |t|]. (T is a continuous random variable with PDF f(t)). now , I tried to make the following two arguments : 1.P= P[ |T| <= |t|] ...
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2answers
59 views

Why do a set of continuous transformations form a manifold?

I am reading Sean Caroll's book on GR, and he defines manifolds to be "a space that may be curved and have a complicated topology, but in local regions looks just like R$^n$. Here by "looks like" we ...
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1answer
25 views

Expressing the inverse of $C(x) = (\langle x,a_i \rangle )$

Assume we have the following linear transformation: $$C(x) = \tilde{x} = \left( \begin{array}{c} \langle x, a_1 \rangle\\ \vdots \\ \langle x, a_k \rangle\\ \vdots \\ \langle x, a_n \rangle ...
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0answers
49 views

Is there an interesting interpretation of the ROWS of an affine transform matrix?

Context: I have a question about affine transform matrices in 3-space. Matrices are 4x4, with the right-most column being the translation, and the bottom row being [0,0,0,1]. In discussions I read ...
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3answers
92 views

Find the transformation matrix that has kernel the span of two vectors

Find a linear map $T : R^4\to R^3$ whose kernel is generated by $v=(1, 2, 3, 4)$, $w=(0, 1, 0, 1)$ This question has been answered but I don't understand the workings. In particular I don't ...
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2answers
42 views

Find all Linear Transformations with specified Image and Kernel [closed]

$T: R^3 -> R^3$ Find all the linear transformations such that: The vectors $$v = \left( \begin{array}{c} 1\\ -1\\ 0\\ \end{array} \right)$$ $$w = \left( \begin{array}{c} 0\\ 1\\ 1\\ ...