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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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 ...
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98 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 ...
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62 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 $f^{-1}$ ...
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89 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 ...
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38 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 ...
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2answers
56 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 ...
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29 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 ...
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19 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}\\ ...
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23 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 ...
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2answers
257 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[ ...
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119 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} ...
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114 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 ...
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2answers
48 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 ...
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68 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 ...
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1answer
105 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, ...
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50 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: ...
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23 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 - ...
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28 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 ...
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26 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 ...
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3answers
95 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} ...
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146 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 + ...
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68 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 ...
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29 views

Transform gradient to reference element

Minimal example of the problem How can you transform the gradient to the reference element?
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81 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 ...
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91 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 ...
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1answer
98 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) ...
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28 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;
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50 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 ...
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1answer
66 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 ...
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469 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 ...
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450 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, ...
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1answer
23 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 ...
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1answer
48 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 ...
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158 views

What kind of transformation an upper triangular matrix represents

Every matrix represents a linear transformation, but depending on characteristics of the matrix, the linear transformation it represents can be limited to a specific type. For example, an orthogonal ...
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36 views

Rational quadratic forms

The quadratic form $$10x^2+20y^2+2z^2+4xy-6xz+8yz$$ can be written as $x^TAx$, where A = [ [10,2,-3] , [2,20,4] , [-3,4,2] ] Using diagonalization, this can be written in the form ...
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1answer
351 views

Decompose distortion affected homography matrix

I am working on a system that finds homography between images taken by moving (shaking) camera with rolling shutter and map. The map is orthogonal image of flat 2D plane and the camera images are ...
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39 views

Linear Probability Density Transformations

Suppose that $\mathbf{y=Ax}$ and that a probability density function over $\mathbf{x}$ is defined as $p(\mathbf{x})$. If $\mathbf{A}$ has an inverse then the PDF over $\mathbf{y}$ is given by ...
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306 views

How can I transform a 3D triangle to xy plane

Suppose I am given a triangle ABC and its corresponding vertex coordinates in 3D. I want to transform ABC in such a way so that vertex A lies on global (0,0,0) coordinate, B lies on (dist, 0, 0) ...
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1answer
73 views

Transformation to avoid division by 0

I want to model the relationship between a discrete variable $A$ with values in the range $[-5,5]$ and a continuous variable $B$ as $\dfrac A B$. How do I transform the data to avoid dividing by ...
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1answer
66 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 ...
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1answer
83 views

Using Kolmogorov's 0-1 law in proof of shift map being ergodic

Why should ${\cal E}_\theta$ be trivial?. I dont see how Kolmogorov's 0-1 law says that in this case we should take the 0 option. This is only mention of ${\cal E}_\theta$ in my notes I can find. ...
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1answer
42 views

What is the difference between a bijection and a reversible transformation?

I was reading http://arxiv.org/abs/quant-ph/0101012v4 and one of the axioms is that there needs to be a continuous reversible transformation between states. What is the difference between that and a ...
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169 views

Show $rk(A) + rk(B) \ge rk(A+B)$

Show $rk(A) + rk(B) \ge rk(A+B)$, where $A,B \in M_{m\times n}(\mathbb{F})$ I'm trying to think in terms of linear transformations. We can define $T_a, T_b:\mathbb{R}^n\rightarrow \mathbb{R}^m$ I ...
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45 views

Find that the given linear transform is a isomorphism

I'm studying Linear Algebra and I'm having trouble demonstrating that a function is a isomorphism, that is: "Given the linear transform $T: V \rightarrow W$, $T$ is a isomorphism if and only if it is ...
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Understanding a definition for vector-spaces

Let $V$, a finite dimensional vector space, and $L$, a subspace of $V$. Let $T:V^*\rightarrow L^*$ defined as: $T(\varphi)(x)=\varphi(x)$ for all $\varphi \in V^*$. Prove $T$ is onto. Well, I'm ...
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$V = \operatorname{Im} T + \ker T $ then $ \operatorname{Im} T \cap \ker T = \{0\}$

Let $F$ be a field, let $V$ be a vector space with finite dimension over $F$ and let $T$ be a linear operator on $V$. Prove that: a) If $V = \operatorname{Im} T + \ker T $ then $\operatorname{Im} T ...
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41 views

$\ker S$ is not contained in $\ker T$ implies $\dim \Im T \ge 1$

Let $T,S:V\rightarrow W$.where $V$ is a finite vector space above $F$ and $W$ is one-dimensional vector-space above $F$ ($\dim W = 1$). It is given that $\ker S$ isn't contained in $\ker T$. Why is ...
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47 views

what's a homogeneous transformation?

Minkowski writes in his paper on Time and Space: If, for simplicity, we retain the same zero point of space and time, the first-mentioned group signifies in mechanics that we may subject the axes ...
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402 views

Isomorphism implies direct sum of Kernel and Image

If $f: U \rightarrow V$ and $g: V \rightarrow W$ are linear transformations between vector spaces over a field $K$ such that $ g \circ f$ is an isomorphism, then $V = \operatorname{Im}f \oplus ...
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43 views

Finding this integra(Radon Transform)

I need to find the radon transform of the following function. But I got stuck in finding this integral. Assume that $\delta$ is the Dirac delta distribution. Let $\chi$ be given by $$\chi(t) = ...