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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Determine mutual location of two coordinate systems, given two sets of points

My problem is: we've got tracking device and a robot. Tracking device provides set of $n$ points in cartesian coordinates(taken from markers on that table) and robot driver returns position of ...
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2 views

Transformation from unknown orientation representation to DCM

I'm working with some really strange software which has some sort of custom orientation representation, and I'm trying to get it into a standardized format (direction cosine matrix). However, that's ...
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77 views

Is there a space in which the $\vec a$ in $\sin(a_1\cdot x)+\sin(a_2\cdot x)$ is linear?

I have equations of the form $\sin(a_1\cdot x)+\sin(a_2\cdot x)=y$ (actually more complicated, but that's the general essence). I want to solve for $\vec a$ using linear regression instead of ...
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17 views

What is transform of $\cos(1/t)$?

Let $T\{f(t)\} = TF(m)$. That is let $f(t)$ be a function of $t$. $TF(m)$ be some resultant (non-trigonometric) transformation in $m$ obtained by applying the transformation operator $T\{\}$. Are ...
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1answer
61 views

Transformation theorem

Given $X_1$ is $\Gamma(\alpha,1)$ distributed and $X_2$ is $\Gamma(\beta,1)$ distributed and set $$Y=\frac{X_1}{X_1+X_2}.$$ The task is to show that $Y$ is $\operatorname{Beta}(\alpha,\beta)$ ...
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1answer
48 views

Transforming the probability distribution that have unknown form [closed]

I have the following expression, which is difficult to compute as the explicit form of the probability distribution is unknown. $\int_{0 \leq y < t} y^2 Q(y)dy$. The density $Q(y)$ is for $y ...
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1answer
22 views

Significance of homothety mapping incircle to circumcircle

Are there any special properties of the homothety mapping the incircle of a triangle to the circumcircle? For example are the centers of this homothety triangle centers?
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1answer
27 views

How to construct center of homothety for two circles which overlap

In general any two circles have two centers of homothety. They have only one center when the circles have the same radius or when the circles have the same center. Given two circles of different ...
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1answer
36 views

Converting a matrix from one base base to another.

I have this basis $B = ((1,0,1),(0,1,-1),(1,-1,0))$ That is represented by: $$[T]_B = \begin{pmatrix} 1 & 0 & 1 \\ 1 & 1 & 2 \\ 1 & 1 & 2 \end{pmatrix}$$ I want to convert ...
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1answer
37 views

Prove/disprove: if $X$ is an eigenvector of $T$ then X is a singular matrix

I have this question: Let $A$ be a non-scalar matrix of an order $(N\times N)$. And let $T:M_{n\times n}^R \rightarrow M_{n \times n}^R$ such that: $T(X) = AX$ for every $ X \in M_{n\times n}^R$ ...
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14 views

Prove that the medians of a triangle concurr using homothety

We are given a triangle ABC with medial triangle DEF (D on BC, E on AC, F on AB). It is easy to prove, assuming that the medians of a triangle concurr at the centroid G, that there is a homothety with ...
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1answer
21 views

Prove that two circles are homothetic

I am trying to prove that any two circles are homothetic. In general there are two centers of homothety, one at the intersection of the external tangents and one at the intersection of the internal ...
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20 views

Prove that similar triangles with the same orientation are homothetic

I am trying to rederive some basic results about homothety. I am trying to prove that two similar triangles with the same orientation are homothetic. I have proved that homothety maps a line AB to a ...
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1answer
87 views

Prove that $T$ is not diagonizable

I'm having difficulties with this exercise, can anyone give me a hand? Let $T:R^3 \rightarrow R^3$ be a linear transformation. It's know that $(1,1,0), (1,1,1)$ are eigenvectors of $T$ and: ...
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1answer
51 views

Proving that $T:\mathbb R^N \rightarrow \mathbb R^N$ is not surjective

Let $T:\Bbb R^n \rightarrow \Bbb R^n$ be a linear transformation and let $u_1,u_2$ different vectors in $R^n$ such that for every $v \in \Bbb R^n$ $Tu_1 \cdot v = Tu_2 \cdot v$. Prove that $T$ is ...
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17 views

How to compute or describe the geometric distance between two 3*3 homography matrices?

The problem is similar to this, are there any geometric methods that can measure the distance between two homographies and tell whether these homographies can describe the same or similar ...
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0answers
12 views

3D analogue to following matrix transformation template?

What would be the analogous form for the transformation matrix of a 3-dimensional shape as opposed to the 2-dimensional shape form presented below? My impression is that the analogue would be a 4*4 ...
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25 views

Finding transformation with respect to a basis

Let $T:R^3 \rightarrow R^3$ be a non-invertible linear transformation that's represented with respect to the base: $ B = ((1,0,1),(0,1,-1),(1,-1,0))$ By the matrix: $$[T]_B=\begin{pmatrix} 1 & 0 ...
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9 views

lat/lon spherical coordinates to equidistant spherical coordinates

How to transform spherical data expressed in latitude/longitude pairs (parallels/meridians) in a new set of pair expressed just in parallels pairs? In other words, I need to transform data expressed ...
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2answers
28 views

Given two basis, find the transformation matrix from one to another

I have these two basis of $M^R_{2x2}$: $C= (\begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}, \begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}, \begin{pmatrix}0 & 0 \\ 1 & ...
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33 views

Is there a Fourier invariant basis?

There are some functions which are invariant under Fourier transformation up to scaling factors, eg. sech(pi*x), Gaussian function etc.. Is there a set of basis functions, which form an invariant ...
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24 views

Multi-Variable Calculus: Change variables in order to use a specific region for integration

After reviewing change of variables, I realized that every text I read provided the change of variable equations in the problem descriptions. From what I understand, the main use of changing variables ...
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3answers
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Composition of Rotation and Translation in the Complex Plane — Finding Angle of Rotation and Point

A rotation about the point 1-4i is -30 degrees followed by a translation by the vector 5+i. The result is a rotation about a point by some angle. Find them. Using the formula for a rotation in the ...
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21 views

extract rotation, scale values from 2d transformation matrix (more clarification)

I have read the questions related to my query this and this. Based on the answer I have calculated the scale, rotation and translation. Some mistake certainly I have done that's why I could not get ...
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2answers
68 views

What could be the mathematical equation of the given signal?

We know that Fourier series for periodic signal $y(t)$ is given by $$ y(t) = \sum\limits_{m=0}^{+\infty} a_m \cos(w_m t) + \sum\limits_{m=0}^{+\infty}b_m \sin(w_m t). \quad (2)$$ Now,I want to find ...
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1answer
30 views

How to compute the normal form of this geometric object?

Given this quadric: $x_1^2+5x_2^2+9x_3^2+4x_1x_2+2x_1x_3+10x_2x_3-2x_3=2$ Maple screenshots: How to put it into the normal form $\Large\frac{x_1^2}{a^2}+\frac{x_2^2}{b^2}-\frac{x_3^2}{c^2}=1$ ...
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1answer
30 views

Rewriting matrix transformation as standard matrix

I'm following a textbook chapter on matrix transformations, and one of the examples seems off. Would this not actually be: $$T\begin{pmatrix}\begin{bmatrix}x_1 \\ x_2\end{bmatrix}\end{pmatrix} = ...
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1answer
18 views

Change from one cartesian co-ordinate system to another by translation and rotation.

There are two reasons for me to ask this question: I want to know if my understanding on this issue is correct. To clarify a doubt I have. I want to change the co-ordinate system of a set of ...
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17 views

are there a transformation for f(1/t)

I have been learning Transformation of functions lately, and found that transformation(some like laplace) make the function linear which inturn makes easy for computation and mostly analysis of ...
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2answers
40 views

Right coordinates of a slanting line when slope is zero and left coordinates never changed after transformation

I have a line in a program I am developing that I want to remove the slant (slope to zero) then get the new coordinates after transformation that removes the slope. This is how the line with the ...
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9 views

Calculate camera view and projection matricies from projected points

I’m stuck on a project for a client.. I need to find the answer to this to proceed: Given (n) coordinates in 3D space and (n) corresponding coordinates in 2D space as projected onto a camera’s image ...
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2answers
30 views

Finite double sum: Improve index transformation

In order to prove a rather complicated binomial identity a small part of it implies a transformation of a double sum. The double sum and its transformation have the following shape: ...
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24 views

transofrmations (a,b,c) to (x,y,z)

I'm not 100% sure linear algebra will crunch this problem, but hopefully so. This may just be a case of matrices, which would be good cause I like those. Imagine we have a robot with a camera ...
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1answer
16 views

Finding the eigenvalues and eigen vectors of linear transformation

$$T:Z_5^{2x2}\rightarrow Z_5^{2x2}$$ $$T \left( \begin{array}{ccc} a & b\\ c & d \\ \end{array} \right) = \left( \begin{array}{ccc} a & a+b\\ b+c & c+d \\ \end{array} \right)$$ This ...
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26 views

Log-like transform function which is close to 1 for small numbers

I am regular user of stackoverflow.com, but I think that my question can be better answered here. Please let me know if my inputs are not clear or if should move this question to some other forum. I ...
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1answer
30 views

Why does this hyperboloid change into a surface? [duplicate]

Given this equation $x^2+y^2+z^2+2xy+2xz+2yz-x-y-z=6$ and the corresponding quadric: If I rearrange the equation to $(x+y+z-3)(x+y+z+2)=0$ (which is equivalent), I get: So, which is the right ...
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40 views

How to transform (rotate) this hyperbola?

Given this hyperbola $x_1^2-x_2^2=1$, how do I transform it into $y_1y_2=1$? When I factor the first equation I get $(x_1+x_2)(x_1-x_2)=1$, so I thought $y_1=(x_1+x_2)$ and $y_2=(x_1-x_2)$. ...
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1answer
36 views

Fourier transform of a $H(x)$ product distribution

So I am given this simple example, where $T \in \mathcal{S}(\mathbb{R})$: \begin{equation} T=(\mu +\lambda x+\beta x^2)H(x) \end{equation} where $H(x)\in \mathcal{S}(\mathbb{R})$ (also notated as the ...
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1answer
67 views

Fourier Transform of a Polynomial

Lets say you are given \begin{equation} f(x)=1+x^3 \end{equation} and the definition of Fourier transform: \begin{equation} \hat{f}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ikx}f(x)dx, ...
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0answers
51 views

On the Fourier transform of $f(x)=\ln(x^2+a^2)$

I would like to derive the Fourier transform of $f(x)=\ln(x^2+a^2)$, where $a\in \mathbb{R}^+$ by making use of the properties: \begin{equation} \mathcal{F}[f'(x)]=(ik)\hat{f}(k)\\ ...
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1answer
20 views

How to see transformations on polytopes?

I have a polytope in six dimension with extreme points $(1,0,0,0,0,0)$ $(0,1,0,0,0,0)$ $(0,0,1,0,0,0)$ $(1,1,0,1,0,0)$ $(1,0,1,0,1,0)$ $(0,1,1,0,0,1)$ $(1,1,1,1,1,1)$ $(0,0,0,0,0,0)$ Each of the ...
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On the convolution of $f(x)=\sin x/x$ and $g(x)=1-|x|$

I am having trouble with computing the convolution of $f(x)=\sin x/x$ and: \begin{equation} g(x)=\begin{cases} 1-|x|,& -1 \leq x \leq 1 \\ 0, & x \notin [-1,1] \end{cases} \end{equation} I ...
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15 views

What does the s-Transform (exponential transform) mean conceptually? What does it show us?

I don't understand the conceptual idea. If I have PDF, and I calculate its s-transform for some s, what do I know that I did not know before?
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2answers
62 views

On the Fourier transform of $f(x)=e^{-x^2+2x}$

So, I have the $f(x)=e^{-x^2+2x}$ and to take the FT of it, I complete the square: \begin{equation} f(x)=e^{-x^2+2x \pm1}=e^{-(x-1)^2}e \end{equation} Then, by knowing that the FT of $g(x)=e^{-x^2}$ ...
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608 views

Integral becomes improper after a substitution

I'm suprised about the following phenomenon which I would like to discuss with you. Consider the proper integral $$\int_{\pi/4}^{\pi/2}\frac{1}{\sin(x)}dx.$$ Since $\sin(x)$ is a diffeomorphism on ...
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1answer
17 views

Transformation of two i.i.d. uniform random variables

G'day folks, I'm trying to work through a problem in preparation for an exam and it's got me stumped. The question is: Let $X_{1}$ and $X_{2}$ be i.i.d. $U(0,1)$ random variables. Let ...
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1answer
21 views

Interpolate between 3D plane and 3D hemisphere

I have a simple 3D plane whose points (different $x, y$ values, but all $z = 0$) need to be mapped to 3D Cartesian coordinates in order to form a hemisphere. However, I also would like to be able to ...
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8 views

Theory that studies varous transformations

It is obvious that Fourier transform, Laplace transform and integration itself are similar things. So, which kind of mathematics generalizes such transformations?
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20 views

Fourier cosine transformation

Good day! I'm studying right now some transformation and I encountered the following equation: $$(2\pi)^{-n/2} \int_{-\infty}^\infty\cdots\int_{-\infty}^\infty \exp\left(-\frac{1}{2} ...
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32 views

Is my proof true or not ? $rk(A) + rk(B) \ge rk(A+B)$

I know that my question has already an answer here, but I have proved it another way and I want to see whether my proof is true or not? If we assume $A$ , $B$ and $A+B$ are respectively the matrices ...