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5
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2answers
8k views

extracting rotation, scale values from 2d transformation matrix

How can I extract rotation and scale values from a 2D transformation matrix? ...
0
votes
1answer
21 views

Formula to convert time to pixels

I have a list of times represented as 000000 to 240000. For a web application, I convert those times to pixels by simply ...
0
votes
2answers
275 views

How to enlarge a circle?

if you are given a circle with equation $(x-a)^2 + (y-b^2) = r^2$ and it is enlarged by a factor of $3$ what would the new equation be? Would you put $2x$ an $2y$ in the place of $y$?
1
vote
2answers
642 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 ...
4
votes
2answers
33 views

What is a transformation?

I am not a native English speaker and I have been pointed out that the word "transformation" as a synonym of "function" is grammatically incorrect. However, I even found a wikipedia and a mathworld ...
0
votes
0answers
76 views

Transformation of variables

Let variables $U$ and $V$ be uniformly distributed on $[-\pi, \pi]$, and independent. Let: $$(x,y) = (\cos(U+V),\sin(U-V))$$ What is the probability distribution function of $f_{x,y}(x,y)$ My ...
0
votes
1answer
30 views

Finding image and the null space of a linear transformation

Take $x \in \mathbb{R}^n\backslash\{0\}$ and let $L = \text{span}\{x\}$. Now we consider the linear transformation $$T \colon \mathbb{R}^n \to \mathbb{R}^n,$$ to be given by $T(y) = \text{proj}_{L}y ...
1
vote
0answers
28 views

Halmos Measure Theory section 39 Theorem D

I have trouble explaining the remark "The function $\phi$ plays the role of Jacobian (or, rather, the absolute value of the Jacobian) in the theory of transformation of multiple integrals". I know ...
0
votes
1answer
41 views

Why would the discrete fourier transform “see” signals like this? What is the origin of spectral leakage?

The discrete fourier transform of $x = (x_{0},\dots,x_{N-1})$ is defined as $\displaystyle X_{k} = \sum_{n=0}^{N-1} x_{n}\omega^{kn}_{N}$ where $\omega^{kn}_{N} = e^{-2\pi ik/N}$ and ...
1
vote
1answer
1k views

Negative & Positive Shear Factor

My question relates to constructional geometry & matrices aren't to be involved in the solution because stated Math level is up to O Levels... The figure below shows shear with y=3 as invariant ...
0
votes
1answer
25 views

Mobius transformations are bijections proof

I don't understand the last line of this proof. To show a function is bijective we need to show it is one-to-one and onto. The proof shows that $f$ is one-to-one only. For some reason $f^{-1}$ ...
2
votes
0answers
27 views

Given the minimal polynomial, find the largest invariant subspace

Let the linear transformation T on the vector space $V$ over $\mathbb{Q}$ have minimal polynomial $(x^{7} - x^{3})$. a) What is the largest invariant subspace W of V for which $T (W) = W$? b) Find a ...
1
vote
1answer
19 views

Changing the length scale of the system of coordinates

Change the length scale on the axes of original system of coordinates, in which the equation $$y=x^3-px\qquad\text{(1)}$$ is plotted, i.e. introduce new coordinates $x_1$ and $y_1$ instead of ...
0
votes
1answer
19 views

Transform gradient to reference element

Minimal example of the problem How can you transform the gradient to the reference element?
2
votes
2answers
35 views

Do rotations of one point around all arbitrary axes form a sphere?

Correct me if I am wrong but assume I have a point in 3D which I would like to rotate around all arbitrary axes fixed at common origin. Then this is true that all orbits circled by rotated point will ...
0
votes
2answers
21 views

Strongest 'average' for a diverse set of numbers?

I have a set of numbers consisting of two general size numbers: size 'a', and size 'b' which are about three times bigger in size than size 'a'. There is some variation and the list might look like ...
0
votes
0answers
11 views

Extract translation vector from two homogenous transformation matrices

Given two homogenous transformation matrices $$ A = \begin{pmatrix} a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ ...
1
vote
2answers
43 views

Find the matrix that represents a rotation clockwise around the origin by$ 30∘$ followed by a magnification by a factor of 4.

Find the matrix that represents a rotation clockwise around the origin by 30∘ followed by a magnification by a factor of 4. My attempt: I multiplied the magnification matrix $\left[ ...
0
votes
0answers
6 views

Stable equilibrium position of 3d models.

I have 2 models, described by vertices arrays. The aim is to find stable equilibrium position of one of the models upon the other. The algorithm should consider the possibilities of transformation of ...
1
vote
2answers
282 views

Understanding perspective transform matrix elements interpretation

I am representing 3D points (vectors) in the following way: ...
1
vote
0answers
14 views

A question on the procedure of finding the matrix of a linear transformation of a polynomial and a combination of its derivatives

I'm trying to self-study Linear Algebra from Linear Algebra Done Wrong, but the book doesn't have solution manual so my question might be extremely easy, apologize in advance: The question is for the ...
1
vote
2answers
259 views

Approximate a polynomial function using a sum of sine waves

I have a polynomial function which I need to approximate by a sum of sine waves with constant amplitude along a given domain. From what I hear, this might be a good time to make use of Fourier ...
0
votes
1answer
45 views

Use the transformation $x=u^2-v^2$, $y=2uv$ to evaluate the integral

$$\int_0^1 \int_0^{2\sqrt{1-x}} \! \sqrt{x^2+y^2} \, \mathrm{d}y\,\mathrm{d}x$$ Here's where I'm at: $J(x,y)=4u^2+4v^2$ Substituting $x$ and $y$ into the integral: $\sqrt{(u^2-v^2)^2+4u^2v^2} ...
2
votes
1answer
295 views

Exponential Function Shifts

I have some confusion about shifts concerning exponential functions. I can best describe my question with an example. Take y = e^-(x-3). This graph has a reflection over the y-axis and is shifted ...
-1
votes
0answers
19 views

Transformations of variables

I don't know how apply the second transformation of variables in this integral. The integrated function can be assumed to be $N_{MJ}(\bar\xi)*D^{-1}(\bar\xi)*\bar\xi_i*\bar\xi_n=\bar\xi_1* \bar\xi ...
0
votes
0answers
41 views

How do you solve a linear transformation with no transformation matrix given?

I am stuck, I can't see how Tff was found with no transformation matrix. And now am being asked to find Tgg, help me http://oi60.tinypic.com/33yrplv.jpg
0
votes
1answer
43 views

I need help with linear transforms? Linear Algebra [closed]

In the question below, how was [T]ff found? I have tried but I can't understand how because I usually start from a given matrix with variables, but non is given here. website is here; ...
1
vote
1answer
90 views

Relation between two distributions expressed in terms of their CDFs

Not great at stats, and having trouble wrapping my mind around this. Would love an explanation, not overly detailed, in plain english of what these transformations mean. The bias correction ...
1
vote
1answer
219 views

Subspaces, transformation matrices exercise

I have trouble understanding the following exercise so I would really appreciate any help you could give me: Let $k$ be a non zero vector in $\mathbb R^n$, written in standard basis. Let $H$ be ...
18
votes
1answer
276 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. ...
0
votes
1answer
74 views

Matrix representation of transformation in ordered bases

An example question asks me to determine $[T]_{\beta}^\gamma$ where $\beta,\ \gamma$ are standard ordered bases of $\mathbb{R}^n$ and $\mathbb{R}^m$ respectively, of $$T_1: \mathbb{R}^n \rightarrow ...
1
vote
2answers
18 views

Linear transformation - linear matrix & kernel

I have a problem understanding getting the KERNEL and IMAGE of a linear transformation. We have the following transformation given: $$ \mathbb{R}_{2}[ x ] \rightarrow \mathbb{R}_{2}[ x ] $$ $$ (\phi ...
-1
votes
1answer
60 views

Find conjugate transpose (adjoint) of linear transform

A difficult question I've been trying to tackle but I seem to hit a dead end. Let $V$ be an inner product space over $\mathbb R$. We are required to find $T^{*}$ such that $$\left<T(u),v\right> ...
0
votes
1answer
22 views

Translation of basis for a vector space on the specified distance

In the Euclidean space $XYZ$ is a basis $X_1Y_1Z_1$ defined that is specified by the vectors $\overrightarrow {O_1X_1}$, $\overrightarrow {O_1Y_1}$ and $\overrightarrow {O_1Z_1}$. How to calculate ...
1
vote
1answer
27 views

Verify Result of a Calculation

In the journal: "A Closed Form Solution for the Similarity Transformation Parameters of Two Planar Point Sets", I cannot get same value for scaling factor for the same problem in the journal. Here is ...
0
votes
1answer
37 views

Transform square region to triangular region

How do you express x and y in terms of u and v so that the region $\{(u,v): 0\le u, v\le 1\}$ is mapped to the triangular region in the $xy$-plane with vertices $(0,0)$, $(1,0)$, and $(0,1)$? Now, ...
0
votes
1answer
31 views

Determine whether the following map is a linear transformation.

So I have to determine if the following is a linear transformation: $$T: F(I) \rightarrow F(I)$$ defined by: $$T(f) = 2f$$ I know that if you let $T: V\rightarrow W$ be a linear transformation. Then: ...
0
votes
2answers
18 views

Transformation matrix of a polynomial

I would really appretiate some help about the following transformation matrices. We have to write a tranformation matrix in basis $B = \{ 1 + x, x + x^2, x^2 \}$ with a polynomial $(Ap)(x) = (x^2 - ...
2
votes
2answers
278 views

Find the matrix A of the linear transformation T(M)

I know that if I substitute the first matrix for $T(M)$ I see what T does to each of the basis vectors. I don't understand how that creates a $3\times 3$ matrix though. I was looking at this ...
0
votes
1answer
15 views

Coordinates rotation and function change

In the Cartesian coordinates $(x,y)$, I have a vector function $\bar{f}(x)=\hat{x}A\cos(yk)$, where $A$ and $k$ are constants. I make now a 45 degrees rotation (in the same plane) to the new set of ...
0
votes
1answer
25 views

Randomly generating special affine transformations

I want to generate many random special affine transformations, that is, affine transformations that preserve volume (determinant equal to 1). I need quite a few of them. Is there a better way than ...
0
votes
1answer
39 views

Discrete Fourier Transform Interpretation

Using Mathematica I took the Discrete Fourier Transform (DFT) of a vector whose entries are volumes of a particular stock. The power spectrum is plotted below: There are two questions that I have ...
1
vote
3answers
66 views

Transformation of two independent uniform random variables

Suppose $X,Y \sim \text{Uniform} \left(0,1 \right)$ are independent. Then I need to find the PDF for $W=X/Y$. By the CDF technique this is seen to be : $$F_W( w)=\int_{0}^1 \int_{0}^{wy} ...
1
vote
1answer
50 views

Find the equation of the linear transformation of an orthogonal projection on the line y=mx.

Let $T : \mathbb R^2 → \mathbb R^2$ the orthogonal projection on the line $y = mx$. Prove that for all $a, b \in \mathbb R$, $$\begin{align}T((a,b)) = {\frac{1}{m^2 + 1}}(a+mb, ma + ...
0
votes
1answer
360 views

Finding reflection of a matrix

To 2 decimal places, what is the value of the lower-right entry in the reflection matrix $Q_a $if a = 1.05? Not even sure where to begin, is there a formula? This is what I could find in my textbook ...
0
votes
2answers
108 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, ...
0
votes
1answer
24 views

Endomorphisms and Invariant Subspaces

I have a question or two regarding the following exercise: Let $\alpha$ be the endomorphism of $\Bbb{Q}^4$ defined by: $$\alpha : \left[\begin{matrix}a \\ b \\ c \\ d \end{matrix}\right] \mapsto ...
0
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0answers
27 views

linear algebra question related to basis, kernel and linear transformation [duplicate]

Let V be a 2-dimensional vector space, and let α=e1,e2 be a basis for V. Define a linear transformation T:V→V by declaring that: T(e1+e2)=2e1−e2 T(e2)=4e1−2e2. a. Find [T]α,α. (one alpha is upper ...
0
votes
0answers
18 views

Source for Kontorovich-Lebedev transformation formulas in Erdelyi's “Table of integral transforms”

I am looking for the sources (i.e. papers with the detailed derivations) of the Kontorovich-Lebedev transformation formulas in Erdelyi's "Table of integral transforms, Volume 2" (McGraw-Hill 1954, ...
1
vote
1answer
81 views

Given $A$, find invertible $B$ such that $B^{-1}AB$ is positive

Given $A \in Mat(n,n,\mathbb R)$, is there always an invertible matrix B, such that $B^{-1}AB$ is positive, assuming all eigenvalues of A are positive and simple ? If yes, is it possible to classify ...