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Questions tagged [transformation]

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-transformations).

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extracting rotation, scale values from 2d transformation matrix

How can I extract rotation and scale values from a 2D transformation matrix? ...
3
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1answer
2k views

Causal Inverse Z-Transform of Fibonacci

Say the Fibonacci sequence is defined by: $y(n) = y(n-1) + y(n-2)$ initial conditions: $y(0)=0, y(1)=1$ I incorporate those initial conditions as: $y(n) = y(n-1) + y(n-2) + \delta(n-1)$ I ...
5
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1answer
2k views

Solving a cubic polynomial equation.

Overview I have tried finding a solution to this problem myself and I have flailed. Its just a challenge for me. could you please tell me how far am I in solving this question? My approach for ...
8
votes
1answer
12k views

Matrix for rotation around a vector

I'm trying to figure out the general form for the matrix (let's say in $\mathbb R^3$ for simplicity) of a rotation of $\theta$ around an arbitrary vector $v$ passing through the origin (look towards ...
3
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1answer
3k views

How do I find the matrix with respect to a different basis?

I tried to solve this question but the answer is totally different, can you explain how to solve it
3
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1answer
415 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)\\ \mathcal{F}[-ixf(x)...
13
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3answers
126k views

Image and kernel of a matrix transformation

I had a couple of questions about a matrix problem. What I'm given is: Consider a linear transformation $T: \mathbb R^5 \to \mathbb R^4$ defined by $T( \vec{x} )=A\vec{x}$, where $$A = \left(\...
17
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1answer
30k views

“Well defined” function - What does it mean?

What does it mean for a function to be well-defined? I encountered with this term in an excersice asking to check if a linear transformation is well-defined.
2
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1answer
932 views

Active and passive transformations in Linear Algebra

I am trying to understand what each transformation means and what their differences are but many books that don't state which transformation they are referring to make it a bit confusing to understand ...
16
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2answers
33k views

How to transform a set of 3D vectors into a 2D plane, from a view point of another 3D vector?

I googled around a bit, but usually I found overly-technical explanations, or other, more specific Stackoverflow questions on how 3D computer graphics work. I'm sure I can find enough resources for ...
4
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2answers
18k views

How to multiply vector 3 with 4by4 matrix, more precisely position * transformation matrix

All geometry in computer graphics are transformed by position * transform matrix; The issue is the fact that position is a vector with 3 components (x,y,z); And transform matrix is a 4 by 4 with one ...
10
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1answer
877 views

A limit and a coordinate trigonometric transformation of the interior points of a square into the interior points of a triangle

The coordinate transformation (due to Beukers, Calabi and Kolk) $$x=\frac{\sin u}{\cos v}$$ $$y=\frac{\sin v}{\cos u}$$ transforms the square domain $0\lt x\lt 1$ and $0\lt y\lt 1$ into the ...
3
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2answers
1k views

Why can any affine transformaton be constructed from a sequence of rotations, translations, and scalings?

A book on CG says: ... we can construct any affine transformation from a sequence of rotations, translations, and scalings. But I don't know how to prove it. Even in a particular case, I found it ...
1
vote
1answer
346 views

Matrix in canonical form of an orthogonal transformation

Let: $$A = \frac 12 \begin{pmatrix} 1 & -1 & -1 &-1 \\ 1 & 1 & 1 &-1 \\ 1 & -1 & 1 & 1\\ 1 &1 &-1 & 1\end{pmatrix} $$ Prove there exists an ...
2
votes
1answer
167 views

Change of coordinate codomain from $[-1,1]$ to $[0,1]$

How does one translate coordinates from $[-1,1]$ to $[0,1]$? That is, suppose we have an ordered pair $(x,y)$ which lies between $[-1,1]$ and want to push into the range delimited by $[0,1]$. A lot ...
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vote
2answers
142 views

What amplification can I apply to $y=\sin x$ for it to be a perfect oscillating arc?

A perfect arc is $y=\sqrt{|1-(x-1)^2|}$. A sin wave is $y=\sin({\pi x\over2})$ I am curious how I can amplify the sin wave so that it's a perfect alternating arc. In the link below you can see ...
1
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1answer
132 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 ...
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2answers
5k views

Getting a transformation matrix from a normal vector

I'm trying to randomly generate coordinate transformations for a fitting routine I'm writing in python. I want to rotate my data (a bunch of $(x,y,z)$ coordinates) about the origin, ideally using a ...
18
votes
3answers
5k views

Is a Fourier transform a change of basis, or is it a linear transformation?

I've frequently heard that a Fourier transform is "just a change of basis". However, I'm not sure whether that's correct, in terms of the terminology of "change of basis" versus "transformation" in ...
9
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3answers
7k views

Shift numbers into a different range

I was wondering how can I shift my data that fall between a range lets say [0, 125] to another range like [-128, 128]. Thanks for any help
16
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6answers
14k views

Can non-linear transformations be represented as Transformation Matrices?

I just came back from an intense linear algebra lecture which showed that linear transformations could be represented by transformation matrices; with more generalization, it was later shown that ...
13
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6answers
30k views

Finding a Rotation Transformation from two Coordinate Frames in 3-Space

The question I'm trying to figure out states that I have 3 points P1, P2 and P3 in space. In one frame (Frame A I called it) those points are: Pa1, Pa2 and Pa3, same story for Frame B (namely: Pb1, ...
13
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4answers
33k views

How do I convert the distance between two lat/long points into feet/meters?

I've been reading around the net and everything I find is really confusing. I just need a formula that will get me 95% there. I have a tool that outputs the distance between two lat/long points. <...
6
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2answers
4k views

Proof of the affine property of normal distribution for a landscape matrix

The widely used/mentioned/assumed affine property of multivariate normal distributions says that: Given a random vector $x \in R^N$ with a multivariate normal distribution -- $x \sim N_x(\mu_x, \...
7
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1answer
547 views

Prove that every triangle is the orthogonal projection of an equilateral one

Prove that every triangle is the orthogonal projection of some equilateral triangle. This problem appears in a book I'm working through in the chapter on transformations in space. There is a rather ...
7
votes
2answers
32k views

Del operator in Cylindrical coordinates (problem in partial differentiation)

I am currently reviewing basic vector analysis and trying to understand every single detail, however, I got stuck in some derivation. What I want to show is the following: Given the del operator (...
5
votes
1answer
19k 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 ...
9
votes
2answers
8k views

How to figure of the Laplace transform for $\log x$?

I was looking at a table of common Laplace transforms of functions when I came across the transform for $\log x$. Apparently, the transform is as follows: $$\mathcal{L} \left\{ \log x\right\}=-\frac{...
3
votes
1answer
2k views

A linear transform of a closed set is closed

A linear transform of a closed set $E\subset \mathbb{R}^d \to \mathbb{R}^d$ is closed. I have seen a lot of similar questions here, but none of them exactly addresses the issue. Please if you find it ...
5
votes
4answers
6k views

Finding a matrix representation of the transpose transformation

Define $T : M_{n×n}(\mathbb{R}) → M_{n×n}(\mathbb{R})$ by $T(A) := A^t$. I know this transformation is linear and just takes a matrix and spits out it's transpose. I also know that the transpose is ...
4
votes
1answer
15k views

Fourier transform of the Cosine function with Phase Shift?

How can i calculate the Fourier transform of a delayed cosine? I haven't found anywhere how to do that. This is my attempt in hoping for a way to find it without using the definition: $$ x(t) = cos(...
3
votes
1answer
6k views

How to find the orthonormal transformation that will rotate a vector to the x axis?

I am having trouble remembering linear algebra. I need to find the orthonormal transformation that will rotate a 3-dimensional vector to the x axis. I could not find any similar question on the net. ...
6
votes
2answers
789 views

How to solve an overdetermined system of point mappings via rotation and translation

I have a set of points in one coordinate system $P_1, \ldots, P_n$ and their corresponding points in another coordinate system $Q_1, \ldots , Q_n$. All points are in $\mathbb{R}^3$. I'm looking for a ...
2
votes
2answers
1k views

Prove two commutative linear transformations on a vector space over an algebraically closed field can be simultaneously triangularized

Prove two commutative linear transformations on a finite-dimensional vector space $V$ over an algebraically closed field can be simultaneously triangularized. It is equivalent to show if $AB=BA$, ...
6
votes
1answer
5k views

Find linear transformation given kernel

Find linear transformation $F$ in canonical bases given $ F: \Bbb R^4 \to \Bbb R^3 $ $ \ker F=\operatorname{span}\left\{\begin{bmatrix}1\\2\\3\\4\end{bmatrix}, \begin{bmatrix}0\\1\\1\\1\end{bmatrix} ...
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5answers
2k views

Transformation T is… “onto”?

I thought you have to say a mapping is onto something... like, you don't say, "the book is on the top of"... Our book starts out by saying "a mapping is said to be onto R^m", but thereafter, it just ...
3
votes
2answers
344 views

suppose $|a|<1$, show that $\frac{z-a}{1-\overline{a}z}$ is a mobius transformation that sends $B(0,1)$ to itself.

Suppose $|a|<1$, show that $f(x) = \frac{z-a}{1-\overline{a}z}$ is a mobius transformation that sends $B(0,1)$ to itself. To make such a mobius transformation i tried to send 3 points on the edge ...
2
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1answer
4k views

Transformation of ellipsoid to sphere

So I need to find an volume-preserving mapping from an ellipsoid to a ball (solid sphere). (Specifically: $\dfrac{x^2}9 + y^2 + z^2 \le 3$, but I'd rather understand the general case than just get ...
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0answers
337 views

Algorithm to determine matrix equivalence

I'm a physicist who's not particularly good at linear algebra so please accept my apologies if this is standard textbook stuff that I'm just unaware of. I have two real rectangular matrices $A_{mxn} ...
0
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1answer
658 views

Elliptic Coordinates - Inverting the transformation

The standard way to transform elliptic coordinates $(\mu, \nu)$ $\ to$ Cartesian coordinates $(x,y)$: $x = a \cosh(\mu) \cos(\nu)$ $y = a \sinh(\mu) \sin(\nu)$ Is there any way to get the ...
0
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1answer
2k views

How to find the rotation matrix that will align an arbitrary vector to an axis

If I have a vector that starts at the origin, how can I find the transformation matrix that will align it with the positive y-axis. So it basically turns into a positive-y axis? EDIT: I also forgot ...
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2answers
2k views

3d transformation two triangles

I have two triangles in 3d. I need to calculate transformation matrix(3X3) between two triangles in 3D. 1)How can I calculate the transformation matrix(rigid) while fixing one of the points to the ...
3
votes
0answers
194 views

Proof for a summation-procedure using the matrix of Eulerian numbers?

I've discussed a procedure for divergent summation using the matrix of Eulerian numbers occasionally in the last years (initially here, and here in MSE and MO but not in that generality and thus(?) ...
1
vote
1answer
354 views

Linear Transformations finding matrix in respect to a basis and coordinate change matrix.

Define $T: Poly_2 \ to\ Poly_2$ by $$T(at^2+bt+c)=3ct^2 +2at-b$$ 1) Show that T is a linear transformation and give a matrix A for T with respect to the basis $B=\{t^2,t,1\}$. 2) Give a ...
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2answers
4k views

How do I map the torus to a plane?

Please see my answer on Perlin noise first. A bit of background. Imagine a solid texture, like an actual block of sky and cloud. If you "cut a sheet" of sky and display it as an image, you'd get ...
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2answers
172 views

Find and sketch the image of the straight line $z = (1+ia)t+ib$ under the map $w=e^{z}$

I need to find and sketch the image of the straight line $z = (1+ia)t +aib$, where $-\infty < t < + \infty$, $a,b\in \mathbb{R}$, and $a \neq 0$, under the map $w = e^{z}$. In order to ...
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vote
4answers
189 views

Why do we define a linear transformation to have the property that $f(cW)=c f(W)$?

Why we define a lin tranfs to have the property that $f(cW)=c f(W)$ ? let $V,T$ be any two vector spaces and let $f:V\rightarrow T$ be a linear transformation between $V $and $T $ why do we ...
1
vote
1answer
166 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]...
0
votes
1answer
66 views

Biliniear form to inner product

Let $f:V\times V\rightarrow F$ be a bilinear form in a finite inner product space V. If $F=R$, how can I prove that there exists a single linear transformation $T:V \rightarrow V$ so that for each $v,...
0
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1answer
56 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+...