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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43
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3answers
976 views

Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z…: unified treatment of transforms?

I understand "transform methods" as recipes, but beyond this they are a big mystery to me. There are two aspects of them I find bewildering. One is the sheer multiplicity of such transforms. Is ...
18
votes
1answer
283 views

Show that $\phi: \mathbb{R}_3[x]\rightarrow\mathbb{R}^3, \phi(p):=[p(-1), p(0), p(1)] $ is a linear transformation

Let $\mathbb{R}_3[x]$ be a vector space of polynomials p with degree $\leq3$ and show that $\phi: \mathbb{R}_3[x]\rightarrow\mathbb{R}^3, \phi(p):=[p(-1), p(0), p(1)] $ is a linear transformation. ...
14
votes
4answers
1k views

How does multiplying by trigonometric functions in a matrix transform the matrix?

I found this comic: But I can't understand the humor because I can't understand how trig functions affect matrix multiplication. Can someone please explain?
11
votes
2answers
1k 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 ...
10
votes
1answer
197 views

An Unexpected Circle…

I played around with $$z=\frac{-1+e^{it}}{\phantom{-}2+e^{it}}$$ and found that, when I draw the real against the imaginary of $z$, it pretty much looks like a circle. But neither ${\frak{R}} z ...
10
votes
3answers
5k views

Get Transformation Matrix from Points

I have built a little C# application that allows visualization of perpective transformations with a matrix, in 2D XYW space. Now I would like to be able to calculate the matrix from the four corners ...
8
votes
2answers
712 views

Rotations by degrees other than $90, 180,$ and $270$.

Say I have a triangle with vertices $(0,0), (2,4), (4,0)$ that I want to rotate along the origin. Rotation by multiples of $90^{\circ}$ is simple. However, I want to rotate by something a bit more ...
8
votes
2answers
6k 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 ...
8
votes
1answer
235 views

Mirror anamorphosis for Escher's Circle Limit engravings?

You are probably familiar with "mirror anamorphosis," the rendering in a painting of a distorted figure that can be undistorted by viewing in an appropriately tilted or curved mirror. The skull in the ...
7
votes
6answers
2k views

Find eigenvalues of a projection and explain what they mean

Suppose B represents the matrix of orthogonal (perpendicular) projection of $\mathbb{R}^{3}$ onto the plane $x_{2} = x_{1}$. Compute the eigenvalues and eigenvectors of B and explain their geometric ...
7
votes
2answers
10k views

extracting rotation, scale values from 2d transformation matrix

How can I extract rotation and scale values from a 2D transformation matrix? ...
7
votes
5answers
3k 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 ...
7
votes
4answers
943 views

Converting between two frames of reference given two points

I have two points ($P1$ & $P2$) with their coordinates given in two different frames of reference ($A$ & $B$). Given these, what I'd like to do is derive the transformation to be able to ...
7
votes
0answers
217 views

Way to Tietze's Transformation Theorem

during our knot-theory lecture we have talking about the following theorem: Given two finite presentations of the same group, one can be obtained from the other by a finite sequence of Tietze ...
6
votes
3answers
2k views

Is it true that any matrix can be decomposed into product of rotation,reflection,shear,scaling and projection matrices?

It seems to me that any linear transformation in $R^{n\times m}$ is just a series of applications of rotation(actually i think any rotation can be achieved by applying two reflections, but not sure), ...
6
votes
1answer
1k 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 ...
6
votes
6answers
10k 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, ...
6
votes
1answer
508 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 ...
6
votes
1answer
177 views

Definition of the Hamiltonian via Legendre transform.

In my book of classical mechanics (Mathematical methods of classical mechanics by V.I. Arnold), the Hamiltonian is introduced in this way (my translation): Let us consider the system of equations ...
5
votes
3answers
321 views

Fraction of two binomial coefficients

In an exercise I was asked to simplify a term containing the following fraction: $${\binom{m}{k}\over\binom{n}{k}}$$ The solution does assume the following is true in the first step, without ...
5
votes
3answers
189 views

non linear transformation that satisfies $T(cx) = cT(x)$

I am just curious if there is a transformation that does not satisfy $\;T(x+y) = T(x) + T(y),\;$ but satisfies $\;T(cx)=cT(x).\;$ I cannot think of any. Thanks for any help people.
5
votes
2answers
168 views

Prove that the following matrices cannot represent the linear transformation $T$ in ANY basis

$T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ defined as $T(x,y,z) = (2x,z,y)$ is a linear transformation. I need to prove that the following matrices cannot represent $T$ in ANY basis: ...
5
votes
3answers
171 views

Find the necessary and sufficient conditions on $A$ such that $\|T(\vec{x})\|=|\det A|\cdot\|\vec{x}\|$ for all $\vec{x}$.

Consider the mapping $T:\mathbb{R}^n\mapsto\mathbb{R}^n$ defined by $T(\vec{x})=A\vec{x}$ where $A$ is a $n\times n$ matrix. Find the necessary and sufficient conditions on $A$ such that ...
5
votes
2answers
598 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?
5
votes
2answers
759 views

Why is the absolute value needed with the scaling property of fourier tranforms?

I understand how to prove the scaling property of Fourier Transforms, except the use of the absolute value: If I transform $f(at)$ then I get $F\{f(at)\}(w) = \int f(at) e^{-jwt} dt$ where I can ...
5
votes
2answers
270 views

Transformations that leave a binomial distribution invariant

The binomial distribution is written as $$p(r|n,\theta )=\binom{n}{r}\theta ^r(1-\theta )^{n-r}$$ where $n$ is a positive integer, $0\leq\theta\leq1$, and $r$ is an integer taking values from $0$ to ...
5
votes
1answer
843 views

Using quaternions instead of 4x4 matrices for transformations

I'm interested in implementing a clean solution providing an alternative to 4x4 matrices for 3D transformation. Quaternions provide the equivalent of rotation, but no translation. Therefore, in ...
5
votes
2answers
124 views

What does the Yoneda lemma say for the identity functor and finite sets?

So I try to plug in the simplest arguments into the Yoneda lemma and see how to interpret it. I'll try it for the identity functor and the category of finite sets, in particular, I use an three ...
5
votes
1answer
95 views

How to compute (and check) this transform matrix?

Background: This is a homework exercise which asks to compute a transform matrix. The answer has been published by our teacher. However, my approach goes a different way and gets a different solution. ...
5
votes
1answer
139 views

Can a transformation matrix be expressed in terms of the vector to be transformed?

I'm currently learning linear algebra with my friend via an online course, and we have a disagreement that we would like settled. Upon learning that vectors can be projected onto lines by a simple ...
5
votes
1answer
69 views

How to transfrom my equation to $Y=KX^2$

In general , $$\vec{C}(u)=\vec{a_0}+\vec{a_1} u+\vec{a_2} u^2$$ is a parabolic arc between the points $\vec{a_0}$ and $\vec{a_0} + \vec{a_1} + \vec{a_2}$. So I'd like to prove it by myself: My trial ...
5
votes
3answers
75 views

I'm looking for the name of a transform that does the following (example images included)

I'm in the usual situation that if I would know what the name of the thing was, then I could find the answer. Since I dont know the name, here is what I'm looking for: Suppose I have the following ...
5
votes
1answer
230 views

Jacobian of Fourier Transformation

I am trying to calculate the Jacobian determinate of the Fourier transform which I stumbled upon when studying the Path Integral in Quantum Field Theory. I know the answer should be $1$ but I don't ...
5
votes
2answers
2k views

Finding the Dual Basis

Define the four vectors in $\mathbb{R}^4$ by $$v_1=\left( \begin{array}{ccc} 1 \\ 0 \\ 0 \\ 0 \end{array} \right), v_2=\left( \begin{array}{ccc} 1 \\ 1 \\ 0 \\ 0 \end{array} \right), v_3=\left( ...
5
votes
1answer
110 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 ...
5
votes
0answers
54 views

Applying PCA on covariance matrix in order to generate a new random variable.

Let $\mathbf{x}$ be a random $n\times1$ real vector, $\mathbf{x}\in\Bbb{R}^n$, which is distributed normally with mean $\bar{\mathbf{x}}$ and covariance matrix $\Sigma_x\in\Bbb{R}^{n\times n}$, i.e. ...
5
votes
0answers
155 views

How to use polynomial or conformal transformation

In my research, I came to a transformation problem. The simple version is an initial circle (or sphere) region is advected by some deformational flow. After some time the circle will be deformed into ...
4
votes
3answers
417 views

Image of function definition notation

In my Linear Algebra and Geometry textbook, it defines the image of a linear transformation $T$ as: $$\operatorname{Im}\, (T) := \{\; w \in W : \; w=Tv \;\;\text{ for some } v \in V \} $$ As far as ...
4
votes
4answers
266 views

Can anybody explain how these logarithms are transformed?

I'm learning for my algorithms exam and I can't derive two logarithm transformations: $ 3^{log_{4}(n)}=n^{log_{4}(3)} $ $ log_{3}(n)=log_{3}(e)*ln(n) $ I'm not very strong in logarithms, anybody ...
4
votes
3answers
844 views

Geometric interpretation of linear transformation

I have a linear transformation, given by the following matrix $$ \begin{pmatrix} x_1\\ x_2 \end{pmatrix} \mapsto \begin{pmatrix} 2 & 2\\ -1 & -1\\ \end{pmatrix} \begin{pmatrix} x_1\\ x_2 ...
4
votes
4answers
8k 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. ...
4
votes
2answers
204 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
4
votes
5answers
490 views

Help a newbie understand Linear Algebraic terms

I am taking a class in Algebra but I am having a problem grasping exactly what it is I am being asked to do -- I think I am having a problem with the vocabulary being used. I have a couple of ...
4
votes
1answer
3k 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 ...
4
votes
2answers
809 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 ...
4
votes
1answer
41 views

Is every symmetric matrix diagonalizable?

I know that Hermitian matrices are always diagonalizable and real symmetric matrices are real Hermitian matrices and therefore diagonalizable. But, it is always not the case that a symmetric matrix ...
4
votes
1answer
4k 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.
4
votes
1answer
119 views

How do you create a “stretch” transformation while keeping volume constant?

A stretch transformation can be represented as: $$ \begin{bmatrix} k & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$ However, this changes the volume of any object which ...
4
votes
1answer
105 views

Beta integral transformation

It's a homework task and I can't get past the last step. Task is to prove that $$ B(x,y)=\int\limits_0^1 \frac{\tau^{x-1}+\tau^{y-1}}{(1+\tau)^{x+y}} \mathrm{d}\tau $$ By substituting ...
4
votes
2answers
7k views

Value range of normalization methods? min-max, z-score, decimal scaling

I am working my way through Normalization (data transformation) of data and was curious about four methods: min-max normalization, 2. z-score, 3. z-score mean absolute deviation, and 4. decimal ...