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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7
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2answers
9k views

extracting rotation, scale values from 2d transformation matrix

How can I extract rotation and scale values from a 2D transformation matrix? ...
0
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1answer
29 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
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2answers
346 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
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2answers
722 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
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2answers
39 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
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0answers
82 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
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1answer
47 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 ...
0
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1answer
60 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 ...
0
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1answer
35 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
53 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 ...
0
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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
42 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
22 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
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0answers
15 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
52 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
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0answers
16 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
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0answers
16 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 ...
0
votes
1answer
59 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} ...
0
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0answers
46 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
1
vote
1answer
104 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 ...
18
votes
1answer
282 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
76 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
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2answers
32 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
70 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
31 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
52 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
42 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
22 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
365 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
20 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
48 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
46 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
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3answers
77 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
86 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 + ...
1
vote
2answers
232 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
30 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
26 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
87 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 ...
0
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0answers
56 views

More on transformations and convolution on continuous random variables

This question is related to my last question but I've done some more exploring and then got stuck again. I decided to modify the problem a little bit and use a transformation of a random variable that ...
1
vote
1answer
75 views

Linear Algebra - Question about transformation and characteristic polynomial

I have some trouble with this question, I tried to solve it but I'm not sure that my solution is correct. I'll be glad if somebody could take a look. Data : T : R^4 --> R^4 (linear transformation) ...
0
votes
0answers
38 views

Reversible smoothing of a two dimensional function (or an image)

Smoothing of an image, or a two dimensional function is quite easy, there are many methods to achieve it, using average of near elements. But how to make it reversible? Maybe DCT (discrete cosine ...
0
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1answer
24 views

Getting linear combinations in linear algebra?

I failed a homework problem a few days ago. I can't figure out how they got the answers, which have been given in green as corrections. Help me figure how they got them;
0
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1answer
34 views

Finding the image of a region transformed by a mapping

The only examples I've found are either very complicated, or state the transformation like y=g(u,v) x=f(u,v), whereas this question states u and v in terms of x and y. I'm not sure how to get ...
3
votes
3answers
252 views

Basis in the vector space of all polynomials

Let $V$ vector space of all polynomials $p(t) = a_0 + a_1t + \cdots + a_nt^n$,$\forall n \in\mathbb{N}$ and $a_0,\ldots,a_n \in\mathbb{R}$. How can I prove that $ \gamma = \{1,t,t^2,\ldots\}$ is a ...
1
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1answer
60 views

Looking for peculiar vector transformation

I have a vector of numbers from 0 to 1. For example: [0.5, 0.5, 0.1]. I need to find a transformation which increases sum of the vector to asked number and: -keeps the order of elements (if element1 ...
0
votes
0answers
28 views

What kind of a matrix transformation is this?

Just playing around with some simple 2x2 linear transformations got me thinking about another type of transformation I havent heard of before, and cant seem to find any info about. Say you have a ...
1
vote
1answer
101 views

Consistency of a ratio between positive and negative numbers

I want to model the inverse relationship between two sets of numbers $A, B$ both in the domain $[-5, 5]$. That is, for the same value $A$ I need a number that decreases linearly as $B$ increases, ...
1
vote
1answer
16 views

Will statistical analysis of transformed data hold for the original one?

I have a data with distribution like chisq-squared one. But ANOVA and t-test need the data to be normal distributed. So I want to do the Box-cox transformation to the data, but my concern is will the ...
1
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1answer
42 views

Constructing regular integer matrices with distinct integer eigenvalues

How can I construct matrices with positive integer values and distinct integer eigenvalues (not necessarily positive, but 0 should not be an eigenvalue). The standard-method to construct matrices ...