0
votes
0answers
29 views

Determine if the following linear transformation is surjective or injective

Let $S \left(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}\right) = $ $\begin{pmatrix} x_1 & -2x_2 & x_3 & x_4\\ 2x_1 & - 4x_2 & -3x_3 & ...
0
votes
3answers
54 views

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation. Find $T(x)$

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation with $T \left(\begin{bmatrix} 1 \\ -2 \\ -1 \\ \end{bmatrix}\right) = \begin{bmatrix} 1 \\ -1 \\ 2 \\ ...
1
vote
0answers
12 views

scale transformation is invariant for H_1

Consider the subspace $H_1$ of $C_0(0,\infty)$, where $\phi=\int_0^t\dot{\phi}(s)ds$ and $\int_0^{\infty}{\dot{\phi}}^2ds<\infty$. The transformation is $(T\phi)(t)=t\phi(\frac{1}{t})$. How to ...
1
vote
2answers
39 views

Linear operator exists then differentiable?

Let $E_{\text{open}} \subseteq \mathbb{R}^n$, and let $\vec{x_o} \in E$. Let $\vec{f}: E \rightarrow \mathbb{R}^m$. If there exists a linear operator $A: \mathbb{R}^n \rightarrow \mathbb{R}^m$. such ...
0
votes
1answer
30 views

Linear transformation from $R^2$ to $R^2$.

Let $\vec{f}: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, where $\vec{f} (\vec{x}) = (x+y^2, x^3+5y)$ and $\vec{x} = (x,y) \in \mathbb{R}^2$. Let $\vec{h} = (h_1, h_2)$ and $\vec{a} = (1,1) \in ...
0
votes
0answers
24 views

Abel and Radon Transform

I am learning Radon and Abel transforms. As far as I understood, basically both the transforms are projection of a 3D object onto a 2D plane. Then what is the difference between both transforms? Under ...
1
vote
1answer
23 views

Plane transformations

I need help in understanding how plane transformations work: for example, let $$A = \{(x,y) \in \mathbb{R}^2: x^2 + y^2 < 1\}$$ Now let's change coordinates like this: $$x = u^2 - v^2$$ $$y = ...
0
votes
1answer
51 views

Why is this transformation true?

I have just a simple question i think. I tried to implement the $\chi^2$-test.I have this document where it is said on page 41, that I have to implement the test like this:$$\frac{\sum_{0\leq i < ...
1
vote
0answers
41 views

The Fourier Stieltjes transform is uniformly continuous

Let $G$ be a locally compact Abelian group and $\hat{G}$ be its dual group, that is the group of all complex functions $\gamma:G\to\mathbb C$ such that ...
0
votes
3answers
526 views

A Möbius transformation maps circles and lines to circles and lines. What exactly does that mean?

The title pretty much says it all. I am also looking for a concrete example if possible. I have looked at the proof, but I'm not exactly sure what it means because I am kind of confused on what the ...
0
votes
1answer
60 views

How to come up with a formula for converting a set of numbers?

I have a set of values (W1, X1 and T1). Via some magic, the combination of these numbers result in a number X2. The full spreadsheet is here, but the gist of the problem looks like this: ...
1
vote
1answer
109 views

Creating a 3D surface from 2D graphs

So I have two sets of equations: $\mathcal{A}$ = \begin{equation} \{ f(y_{0},x), \, f(y_{1},x) , \;... \;, f(y_{n},x) \} \end{equation} $\mathcal{B}$ = \begin{equation} \{ g(y,x_{0}), ...
5
votes
2answers
444 views

Are Legendre transforms of non-convex functions useful?

Do Legendre transforms have any applications that do not appeal to convexity? What is the intuitive interpretation of the Legendre transform of a non-convex function?
1
vote
2answers
137 views

Can the inverse of this logit-like transformation be stated analytically?

For $\alpha \geq 0$ the transformation $x \mapsto \log(x) - \alpha \log(1-x)$ maps the unit interval to the real line (in fact for $\alpha = 0$ the transformation is not surjective). For $\alpha=1$ ...