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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Proof that $V^*$ is isomorphic to $V$.

In my notes for a linear algebra course there is proof that $V^*$ is isomorphic to $V$. However I am unclear on a few of the steps. We begin by choosing a basis $B = \{v_1,...,v_n\}$ for $V$. We now ...
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0answers
77 views

Fast Hankel Transform

Can someone please explain what would be the expression for weights(Ho) in a Fast Hankel Transform.I found this in a paper and could not find any satisfactory answers .
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1answer
28 views

Confusion with the notation $L_A$

My linear algebra class went from 0-100 real quick. I've attended every single lecture (so I know I haven't missed out on anything); however, very recently he has been using the notation $L_A$ for a ...
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2answers
34 views

transformation of single random variables with absolute value ??

integral I got the final answer to be fy(y)= 1 0< y < 1 I am not sure could anyone correct me if its wrong !
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0answers
23 views

Remove Multiplicative Constant from Hypergeometric Function

I have a function of the form $$f(x;\lambda) = {}_2F_1\left(a,b;c;-\frac{e^{2x}}{\lambda}\right)$$ I need to invert this function to solve for the constant $\lambda = f\left(x\right)$. I could do ...
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2answers
58 views

Transformation to polar coordinates

I know this is very simple and I'm missing something trivial here... I'm having trouble converting this set of equations to polar form: $$ \dot{x_1}=x_2-x_1 (x_1^2+x_2^2-1)\\ \dot{x_2}=-x_1-x_2 ...
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1answer
57 views

3D rotation of an object with respect to another object's rotation

I am writing a python code to translate and rotate an object with respect to another object. Please take a look at the picture bellow: The smiley face and the arrow have initial poses (position ...
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1answer
34 views

Integral transformation

I'm trying to do a transformation of an integration, I have that $$\int_{0.5}^1\int_0^{0.5}e^{xy}xydxdy$$ And I want to get that integrate $$\int_0^1\int_0^1 f(x,y)dxdy$$ Where the value of the ...
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18 views

Find a matrix to represent the mapping of a factor module

I have a problem from my past paper I can't figure the logic to, even after seeing the answers. The question goes 【Q】Let $V=\mathbb{R}[X]_{<4}$ be the vector space of real polynomials of degree ...
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2answers
116 views

Features of phase and magnitude spectrum?

I have read in many books that whether the signal is 1D or multidimensional , The magnitude spectrum tells you how strong are the harmonics in the signal and The phase spectrum tells where this ...
2
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1answer
35 views

Proving surjectivity and injectivity of two transformations, knowing the rank of their composition.

I have got another question concerning linear algebra. The excercise is: Let ...
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0answers
43 views

$K=\frac{1}{2}mV^{2}$, random variable transformation.

An object has random velocity $V$ and kinetic energy $K = \frac12mV^2$, where $m$ is the mass of the object. Suppose that the velocity has the Laplacian distribution with probability density ...
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1answer
24 views

How to go about finding a transformation $T$ in order to solve an integral.

I have the integral $$\int\int_R\left(2x+y\right)dA$$ Where $R$ is the region bounded by $$x+y=-1, x+y = 3, 2x=y,2x-4=y$$ So my first though was drawing the region, which gave me this odd region, so ...
2
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1answer
31 views

Projective transformation a parabola to a circle

Take the parabola $x^2 - y = 0$ in the cartesian plane. I'm not entirely sure about this, but we can express this using homogenous coordinates as $X^2 - Y = 0$ (the $W$ coefficient is $0$?) With the ...
3
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2answers
126 views

What do $a_0$ ,$a_m$ and $b_m$ terms mean in the Fourier series formula?

We know that a Fourier series for signal $x(t)$ is given as $$\frac {a_0} 2 + \sum \limits _{m=1} ^\infty (a_m \cos \frac {2 \pi m t} T + b_m \sin \frac {2 \pi m t} T)$$ So my question is what ...
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0answers
32 views

transform orthonormal coordinate system to another

I have one orthonormal coordinate system ABC that it's origin is the point p0. I would like to transform it to another orthonormal coordinate system A'B'C', that it's origin is p1. I know how to ...
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3answers
40 views

Show whether this linear transformation is one-to-one and onto.

$T:P_2$ $\rightarrow$ $R^3$ is a linear transformation defined by $$T(a+bx+cx^2) = \left[ \begin{array}{ccc} 2a-b \\ a+b-3c \\ c-a \end{array} \right]$$ This linear transformation is neither ...
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1answer
30 views

Eliminating 3rd term in $\frac{d^2u}{dx^2} + \frac{d^2u}{dy^2} + 3u = 0$ through change of variables?

I want to eliminate the third term in this pde $$\frac{d^2u}{dx^2} + \frac{d^2u}{dy^2} + 3u = 0$$ Here's what I thought of, is this ok? Let $a = \lim \limits_{k \to \infty} kx$ Let $b = \lim ...
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1answer
19 views

Transforming pde to nicer form?

I have a second order differential equation for $u$ $$\frac{d^2u}{dx^2} + \frac{d^2u}{dy^2} + 5u = 0$$ I am looking for a transformation $u(x,y) \rightarrow v(x,y)$ that gives $$\frac{d^2v}{dx^2} + ...
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1answer
20 views

Transformed pde but my answer doesn't match solution?

$$\frac{d^2u}{dx^2} + \frac{d^2u}{dy^2} + \frac{du}{dx} + 2\frac{du}{dy} + 3u = 0$$ Let $u = ve^{ax + by}$ and find $a, b$ such that we can transform to the following equation $$\frac{d^2v}{dx^2} + ...
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1answer
89 views

Have some queries about Fourier Transform

I have some queries about the Fourier transform In most of the cases, the Fourier transform of a signal is symmetric with respect to positive and negative frequency. I think the computational ...
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0answers
121 views

Converting fractional coordinates to cartesian coordinates

I have a set of fractional coordinates with the following vectors: 2.950 -1.475 0.0000 0.000 2.5547 0.0000 0.000 0.0000 77.5379 with the ...
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61 views

Proof that Determinant is Scale Factor

I've seen a lot of supposed properties of linear transformations that're never proven -- just often repeated. These include: The determinant is the scale factor between the volume of region in your ...
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2answers
102 views

Whether the job of Fourier Transform is just to convert signals from time domain to frequency domain only or more than it?

I am a beginner . We convert a signal in time domain to frequency domain by applying Fourier transform on the signal to obtain frequency and phase spectrum. So,whether the job of Fourier transform ...
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21 views

Education tool for learning 3D angles

I hope it is not an off-topic. I have started working on 3D frame transformation and transforming a vector such as acceleration or angular velocity from one coordination to earth coordination. My ...
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3answers
40 views

transforming $(A,B,C)$ to $(0, 0, 1)$ by rotations

I'm trying to reflect the "world" through a specified plane $p:Ax+By+Cz=0$. I know how to reflect the "world" through the $xy$-plane, so I want to rotate $p$ in the $3$ axes ($x,y,z$-axes) so it will ...
4
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1answer
41 views

Lens Conformal Map

Please help me find a conformal map of the set $ A = \left \{\; z: \; |z-1| < \sqrt{2} \; and \; |z+1| < \sqrt{2} \; \right \}$ one-to-one onto the open first quadrant. First, I noticed that ...
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0answers
34 views

Transformation of a Partial Differential Equation

How can we convert $$\frac{\partial c}{\partial t} = M\left[\frac{\partial}{\partial x}\left(c\frac{\partial c}{\partial x}\right)+\frac{\partial }{\partial y}\left(c\frac{\partial c}{\partial ...
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1answer
24 views

Distribution of Logistic of Normal

If $X \sim N(\mu_X, \sigma^2_X)$ and $Y= \frac{\exp(X)}{1+\exp(X)} $, what is the distribution of $Y$? I thought logit-normal would fit the bill, however the logit of $Y$ is ...
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681 views

What are the limitations /shortcomings of Fourier Transform and Fourier Series?

I am fond of Fourier series & Fourier transform. But every approach has some outcomes and some shortcomings. It's limitations lead to innovation of new approach. So, can anybody explain about ...
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0answers
20 views

Proving a theorem on a rotation about a line followed by the inversion to show that it is a reflection

A theorem in my textbook is : A rotation about a line followed by the inversion about a point on that line is a reflection or a rotary reflection. I can picture this theorem in my head on a 3D space ...
0
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1answer
38 views

Rotation in high dimension in the direction of given vectors

Given two vectors $A$ and $B$ (with high dimension), and an angle $\alpha$. How can one find the vector $C$ which is $A$ rotated over $\alpha$ in the direction of $B$? If it changes anything: the ...
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0answers
13 views

Transforming coordinates according to Point of View.

Imagine I have two cameras (A and B) in the same axis and looking at the same white wall with a black dot. Both cameras see the black dot. I'd like to know how to proceed in order to derive the black ...
3
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2answers
46 views

Dynamical System transformation

How can the system $$\frac{dx}{dt}=-y+\epsilon x(x^2+y^2)$$$$\frac{dy}{ dt}=x+\epsilon y(x^2+y^2)$$ be transformed into $$\frac{dr}{dt}=\epsilon r^3$$ $$\frac{d\theta}{dt}=1$$ via polar coordinates? ...
4
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1answer
106 views

How to determine when this two variable transformation is invertible?

I am given: $$ X= U \cos(V) \tag{1}\\ $$ $$ Y = U \sin V \tag{2}$$ Now, I need to: a) Give the respective ranges for $U$ and $V$ in order that the transformation defined is one to one. and ...
0
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1answer
58 views

How do i scale 2D vector using matrix

I know that scale matrix is 2x2 { x, 0, 0, y } basis. My vector { 100, 2 } and i want to scale it using custom 2x2 matrix. I've read that if left operand is 2D row vector, then multiplying it on a ...
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0answers
11 views

Elementary Transformation of Mean density

Sir Im confused about dealing with some transformed mean density. I cant follow the logic how the author derived a specific result out from the following given: $$p(t) = \frac{1}{nπ} ...
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0answers
26 views

transformation of mean density

Sir Im confused about dealing with some transformed mean density. I cant follow the logic how the author derived a specific result out from the following given: $$p(t) = \frac{1}{nπ} ...
0
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0answers
21 views

What are The applications of Fast Walsh–Hadamard Transform.

There is a problem requiring the expect value of the intersection of two random subsets selected from a universal set, with the values and the probabilities of subsets given. My friend said it could ...
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1answer
20 views

Effects of Isomorphic Transformations on Vector Spaces.

Let $V$, $W$ be finite-dimensional vector spaces and let $T: V\rightarrow W$ be an isomorphism. Let $X$ be a subspace of $V$. Show that $T(X)$ is a subspace of $V$. My attempt: I know two vector ...
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3answers
48 views

Transforming differential equation to polar coordinates, example.

... For example, if we write $\dfrac{\operatorname dy}{\operatorname dx} = \dfrac{-x}{y}$ in polar coordinates, we obtain the equation $\dfrac{\operatorname dr}{\operatorname d\theta} = 0$ whose ...
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0answers
27 views

Fourier transform from Laplace transform

So what I did was Laplace transform $f(t)$ to $F(s)$ and then plug in $s=jw$. However, when I tried this for $cos(3t)$, $$F(jw)={jw\over9-w^2}$$ was the answer. I don't know if this is correct, and ...
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61 views

Strange results on training ANN by BP to “do” a kind of simple Discrete-Fourjer-Synthese

Here's the result of an experiment I'm trying to do atm. I've checked it, It really does a FourjerSythese! Can someone please explain me how these strange matrixs' can do that ? For those of you ...
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28 views

Compute the solutions of the following equation in Fourier space:

$$\frac{d^3u}{dx^3} − αxu = 0, x ∈ R, $$ where $ α > 0$ is some constant and $u(x)$ is assumed to satisfy $\int_R u(x) dx = π.$ I know this is a ODE so this is what I came up with so far: ...
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2answers
49 views

Transform of the Cartesian plane that maps hyperbolic arcs $xy = C$ to line segments

I have the finite set of curves: $$y = \frac{C}{x}, \qquad C = 2, 3, \ldots, C_{\max},$$ with $C$ and $x$ positive integers, $2 \le x \le C$ ($x$ varies on a finite domain). Is it possible to apply ...
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1answer
20 views

Homomorphisms inbetween factor modules

Consider the Ring $\mathbb{Z}$ and the two ideals $(n), (m)$, where $n, m \in \mathbb{N}$, and consider $GCF(m, n)$ (the greatest common factor of $m, n$). Let p: $\mathbb{Z}/(m) \to \mathbb{Z}/GCF(m, ...
2
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2answers
114 views

Fourier Decompositon problem

have a look at this video of Fourier Decomposition of an image (otherwise you can also refer the image, which shows few plots of different extracted waves from an image). We also know that a Fourier ...
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1answer
16 views

Determine the Fourier transform of $f(x) &

f(x)=1 if |x| < a or f(x) = 0 if |x| > a We use the formula $$ {1\over 2\pi} \int_{\infty}^\infty f(\bar x)e^{i\omega \bar x} $$ So is $f(\bar x)$ the same as $f(x)$ ?? In an answer they ...
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22 views

Inverse z transform - help

How can I find inverse z transform of $$X(z)=\frac{5}{(z-2)^{2}}$$ ? It is known that discrete signal x[n] is causal. Here is what I have done. Since signal x(n) is causal, convergence of z ...
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1answer
19 views

Construct a triangulated adjunction from a right triangulated adjoint functor?

In the Lemma 40. of the note on triangulated categories by Daniel Murfet, one finds the construction of a triangulated adjunction from a left (triangulated) adjoint triangulated functor, whose proof ...