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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27 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
98 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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1answer
151 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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69 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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1answer
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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1answer
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
101 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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1answer
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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0answers
51 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
68 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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3answers
487 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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1answer
505 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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3answers
171 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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2answers
37 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
369 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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0answers
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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1answer
332 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
76 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
67 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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2answers
179 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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3answers
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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1answer
34 views

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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87 views

$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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1answer
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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0answers
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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1answer
410 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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1answer
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) = ...
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1answer
87 views

What are Basis images?

I have read that using Fourier transformation we can decompose any arbitrary image into orthogonal basis images and reconstruct it back. But what are basis images actually?
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51 views

$T (x_1,x_2,x_3,…,x_n) = (-x_3,x_3,x_4,x_5,…) $ then $ W \ne ker T$

Let $V$ the vector space of all sequences of real numbers and $W$ the subspace given by $W = \{(a,a,0,0,...) | a \in R\}$ , and $T : V \rightarrow V$ given by $T (x_1,x_2,x_3,...,x_n) = ...
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2answers
34 views

If $f\in V$ of degree $n$ then for every $g \in P_n(\Bbb R)$ there exist scalars $c_0,c_1,..,c_n$ such that $g = c_0f + c_1f'+ … + c_nf^{(n)}$

Let $V=P(\Bbb R)$ and $1 ≤ i$ be the vector space of the polynomials with real coefficients, on the field of real numbers $\Bbb R$. Let $T_i(f)=f^{(i)}$ the $i$th derivate of $f$. a) I have to show ...
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1answer
43 views

Matrices as linear transformations

I am reading a proof which claims: A matrix of $m\times n$ is a linear transformation from $m$ vector-space to $n$ vector-space, And therefore, by the dimension theorem: $m = \dim\ker A + ...
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1answer
45 views

Matrix transformations

I have to find components of a matrix for 3D transformation. I have a first system in which transformations are made by multiplying: $M_1 = [Translation] \times [Rotation] \times [Scale]$ I want to ...
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1answer
170 views

How does the z transform work in practice?

What I've found I've implemented a PID controller using the equations 7 and 9 of this article, which states that: $$\frac{U(s)}{E(s)}=K_p+\frac{K_i}{s}+K_ds$$ Translates to ...
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0answers
39 views

One double integral elated problem

The bit I am stuck is the limits in the double integral. I tried X from 0 to uy and Y from 0 to infinity, this is obviously incorrect. I just want to know the complete double integral in the order ...
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2answers
29 views

Showing a set is a basis

Let $V, W$ vector spaces and $f, g:V\rightarrow W$, linear transformations. $\ker f \subset \ker g$. Now, let $\{v_1,...,v_n\}$ a basis for $\ker f$ and we'll complete it with $\{u_1,...,u_m\}$ to a ...
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4answers
102 views

$\{ v_1,v_2,…,v_n\}$ is basis of $V$ if and only if $\{ v_1,v_1 + v_2,…,v_1 + v_2+…+v_n,\}$ is a basis of $V$

Let $V$ a vector space over a field $K$. Is it true $\{ v_1,v_2,...,v_n\}$ is basis of $V$ if and only if $\{ v_1,v_1 + v_2,...,v_1 + v_2+...+v_n,\}$ is a basis of $V$ ? I made some examples and ...
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1answer
44 views

$V$ and $W$ finite vector spaces with dimension $n$ and $r$ with $\{ v_1,v_2,…,v_n \} \subset Ker T$ and $\{ u_1,u_2,…,u_s \} \subset V$

Let $V$ and $W$ finite vector spaces with dimension $n$ and $r$ respectively and $T: V \rightarrow W$ linear transformation, $\{ v_1,v_2,...,v_n \} \subset Ker T$ and $\{ u_1,u_2,...,u_s \} \subset ...
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3answers
57 views

Diagonalizable operator of a finite vector space

Let $V$ a vector space of finite dimension, $dim (V) = r$, and $T: V \rightarrow V$ a diagonalizable operator with $ \lambda _1,\lambda_ 2,...,\lambda _r$ distincts eigenvalues of $T$ then $ (T- ...
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1answer
110 views

Linear transformation matrix representation with differentiation answer confirmation

I hope you liked the title. I have a question that is as follows: Consider the linear transformation $T: P_3(\mathbb{R}) \to P_3(\mathbb{R})$ given by $$T(f(x))=f(0)+f'(x)+f''(x)$$ Where the ...
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1answer
110 views

Finding a basis for $\ker(T)$

I have this question: Let $Z\in M_{2\times2}(\mathbb{R})$ be defined as $$Z = \left( \begin{align} 1 &&1\\1 &&1 \end{align} \right)$$ and consider $T: ...
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1answer
36 views

Lorentz transformation and Minkowski metric

For the exam I'm trying to solve some problems. Today I found this exercise and need some help: For the group S0(1,1) of the Lorentz transformation I have $\phi \in \mathbb{R}$ and $A_{\phi}: ...
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4answers
72 views

Showing some transformation is linear

Let $T: P_3(\mathbb{R}) \to P_3(\mathbb{R})$ be an operation defined by $$T(a+bx+cx^2+dx^3) = a + dx + (a+d)x^2 +(b-c)x^3$$ Show that $T$ is linear What I have done so far is look at it like ...
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2answers
56 views

$\dim(V) = \dim T(V) + \dim T^{-1}(0)$

Let $T\colon V \rightarrow W$ a linear transformation between the real vector spaces $V$ and $W$ both with finite dimension. How can i prove that $\dim(V) = \dim T(V) + \dim T^{-1}(0)$. I can't ...
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1answer
36 views

Sequence of Mobius Transformation

Let $ T(z) = \frac {z+2}{2z+1} $. Now it follows that: $ T_1(z) = T(z), T_2(z) = T(T_1(z)), T_3(z)=T(T_2(z)) .... T_{n+1}(z)=T(T_n(z)) $ I'm trying to prove this sequence at the nth terms, but I ...
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1answer
243 views

Solving a cubic polynomial equation.

Overview I have tried finding a solution to this problem myself and I have flailed. Its just a challenge for me. could you please tell me how far am I in solving this question? My approach for ...