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37
votes
2answers
689 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 ...
17
votes
1answer
268 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
925 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
191 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 ...
8
votes
1answer
153 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 ...
8
votes
3answers
4k 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 ...
7
votes
6answers
1k 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
4k 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 ...
7
votes
4answers
523 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
149 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
2answers
660 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 is simple. However, I want to rotate by ...
6
votes
3answers
1k 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
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 ...
5
votes
3answers
288 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
163 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
1answer
791 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 ...
5
votes
3answers
160 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
363 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
6answers
7k 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, ...
5
votes
2answers
485 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
236 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
733 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
1answer
466 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 ...
5
votes
1answer
63 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
3answers
60 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
119 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
1answer
1k 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
131 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
0answers
121 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
281 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
263 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
4answers
6k 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
5answers
433 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
3answers
7k views

extracting rotation, scale values from 2d transformation matrix

how can i extract rotation, scale values from 2d transformation matrix? i mean a have a 2d transformation ...
4
votes
2answers
142 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: ...
4
votes
2answers
736 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
1k 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
108 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
99 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
300 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 ...
4
votes
1answer
116 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 ...
4
votes
2answers
567 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 ...
4
votes
2answers
97 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 ...
4
votes
2answers
50 views

Characteristic polynomial of a mapping from matrices space to matrices space

Let $T$ be the linear map from $M_n \to M_n$ given by TX=AX, while A is as well a matrix $n \times n$ (a) Write out the characteristic polynomials for $T$ (b) Show that if A is ...
4
votes
2answers
106 views

Fourier, Laplace, … and other Integral-transformations

I know Laplace, Fourier and Mellin-Transformation. Is there a general theory of transformations? My main interest is about classification of transformations satisfying specified properties like ...
4
votes
1answer
130 views

Is there a geometric argument that the Legendre transform of a convex function is convex?

I am trying to build intuition on Legendre transforms. Arnold's Mathematical Methods of Classical Mechanics has some nice geometric interpretations, but he does not provide a proof that the Legendre ...
4
votes
2answers
279 views

Transforming Differential Equation to a Kummer's Equation

I'm trying to transform an equation of the form $$ yw''(y) - [b - ay] w'(y) - [d + ey]w(y) = 0 $$ into the form of a Kummer's or confluent hypergeometric differential equation: $$ y w''(y) + [f - ...
4
votes
2answers
2k views

Why composite transformations are multiplied to the right side?

I have seen that many composite transformations have the later transformation multiplied to the right side of the matrix. Say I have matrix an existing transformation matrix $\mathbf{M}$ and then ...
4
votes
4answers
77 views

Showing that a transformation $T:\mathbb R^3 \to \mathbb R^2$ is linear

OK, I am trying to prove the following transformation is linear, and find the basis for $\ker(T)$ and Im$(T)$ (also denoted in our textbook by $N(T)$ and $R(T)$ ). Then we're suposed to find the ...