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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Let $X_1$ and $X_2$ be independent $n(0,1)$ random variables. Find the pdf of $(X_1-X_2)^2/2$.

I understand that $(X_1-X_2)/\sqrt2)$ ~ $n(0,1)$ since it is a linear combination of $X_1 $ and $X_2$ and hence $(X_1-X_2)^2/2$ ~ $\chi^2_1$. I'm having trouble on how to prove/show this ...
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134 views

Why does the discrete cosine transform as matrix multiplication work this way?

I have read that the DCT can be computed as a matrix multiplication. The 8x8 DCT matrix is: $D=\frac{1}{2}\left[\matrix{ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & ...
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2answers
167 views

Find invariant points, how to express using parameter

I have a matrix $$\begin{pmatrix}0&-1\\1&2\end{pmatrix}$$ where I have to find the invariant points for a transformation using this matrix. I have no problem working through to two ...
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2answers
65 views

inverse of similarity transformation

If $S$ is a similiarity transformation, i.e. there exists $c>0$, such that $$ \lvert S(x)-S(y)\rvert = c\lvert x-y\rvert. $$ Then, apparently, we have that $$ \big\lvert ...
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1answer
68 views

Odd polynominal

Let's define an odd polynominal be a polynominal which has odd degree, and ALL of its terms have odd exponential (except the constant), for example: $x^5+x^3+1$, or $x^7+2x^5+3x^3+4x+5$. We all know ...
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2answers
162 views

double integral over an arbitrary triangle

Assume we have an arbitrary triangle ABC in x-y plane and we want to integrate a function $f(x,y)$ over surface of this triangle as shown in fig. 1: We can define another coordination system [x' ...
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2answers
91 views

Why isn't $f(x)=\sqrt{2-x}$ reflected across the y-axis?

If I try to graph this function, it does not appear to reflect across the y-axis when it comes time to do the reflection. Rather, it is reflected around the point where the function begins on the ...
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1answer
80 views

How to transfrom my equation to $Y=KX^2$

In general , $$\vec{C}(u)=\vec{a_0}+\vec{a_1} u+\vec{a_2} u^2$$ is a parabolic arc between the points $\vec{a_0}$ and $\vec{a_0} + \vec{a_1} + \vec{a_2}$. So I'd like to prove it by myself: My trial ...
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0answers
28 views

General Tensor Assistance

Sorry if this is a stupid question, but it might help me grok things if I can connect from something that's intuitive to me. Consider a transformation from Cartesian coordinates to spherical ones: ...
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1answer
32 views

What is the new point reflected with a respect to fixed line through origin?

If I have $(x,y)$, what would be the reflected point with respect to fixed line through origin? does it depend on what line I have, If I have $x$ or $2x$ $3x$, they all pass through origin.
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90 views

Surface area of transformed sphere

So if I have a sphere with center C and radius R and then apply one or more affine transformations (so any combination of rotating, scaling and translating), how would I go about finding the surface ...
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1answer
35 views

Find the Laplace transform of integral(from 0 to x) sin(2t) dt

Find the Laplace transform of $\int_0^x\,\sin\,(2t)\,dt$ So basically, $$\int_0^x\,\sin\,(2t)\,dt = -\frac{1}{2}(\cos\,(2x) - 1)$$ So $$\mathcal{L}\{\cos\,(2x)\} = \dfrac{s}{s^2 + 4}$$ ...
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2answers
66 views

Prove that an element of the basis is an element of the Kernel after linear transformation

Let $T:R^4\rightarrow R^4$ and basis $B=(v_1,v_2,v_3,v_4)$. $$T(v_1)+T(v_2)=T(v_3)\; \text{ and } \; T(v_1)+T(v_3)=T(v_2)$$ Prove that $v_1\in Ker(T)$ What I wrote is: $$T(v_1)=T(v_3)-T(v_2)\; ...
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2answers
63 views

Transformation Matrix project

My task is to find the Transformation Matrix, that projects, any point of the xy-plane, on the line $$ y = 4x$$ The solution should be: $$T=\pmatrix{0.06&0.235\\0.235&0.94}$$ But somehow i ...
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0answers
24 views

Performing a shift on a piecewise function

Following is a convolution of $y(t) = h(t)*x(t)$ $$y(t) = \cases{\frac{1}{2}e^{2} &\text{ if } t\ge 1 \cr\cr \frac{1}{2}e^{2t} &\text{ if }0 \lt t\lt 1 \cr\cr 0 &\text{ if } t\le 0}$$ ...
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1answer
58 views

How to find an unitary transformation of $A$ that minimize $(A'_{i,i}-1)^2$?

Is there a way to find an unitary transformation $$ A'=U^+AU $$ that minimize: $$(A'_{i,i}-1)^2$$ In other words, the diagonal elements must be similar to one: $A'_{i,i} \approx 1$ Any hint? ...
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1answer
60 views

Linear transformation

Let there be a linear transformation $T:R^3\rightarrow R^2$ Is there a linear transformation so that: $Ker(T)=Span((1,2,1),(0,3,-1))$ and $Im(T)=Span((5,-7))$ Answer: $Dim(V)=Rank(T)+Null(T)=2+1=3$ ...
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0answers
15 views

Does logged data have to be transformed back to its original form before testing the accuracy?

I have run my data through a model in r, i ran ARIMA to forecast. The model forces a log transformation to be applied to the data. To test the accuracy of the fitted model formed by ARIMA would i need ...
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1answer
81 views

Why does the discrete cosine transform compact the information at the “low frequencies”?

I've been investigating about the discrete cosine transform. I think I understand the practical applications it has and how it is used in image/audio compression. I also know it is related with the ...
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1answer
60 views

How do you prove a hilbert transform?

I am stuck with this question below, I need help;
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2answers
849 views

Writing a composite transformation as a matrix multiplication

I am confused about a question on matrix multiplication of a transformation. I have two matrices, P and Q as follows: $$P = \begin{pmatrix}\frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} ...
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32 views

What is the proof/show that the post of linear transformation generated by LDA is at most k-1

What is the proof/show that the matrix $Sw$ generated by LDA is at most rank $p-k$, where $p$ is the dimension of the data and $k$ is the number of classes. LDA: ...
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2answers
458 views

How to find the equation of the graph reflected about a line?

Consider the graph of $y = e^x$ (a) Find the equation of the graph that results from reflecting about the line $y = 4$. (b) Find the equation of the graph that results from reflecting ...
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0answers
35 views

Show that a transform is involutive

Let $\mathcal C$ be the class of continuous, nonnegative, not identically equal to zero, concave, positive homogeneous of first order functions from $\mathbb R^n_+ = \{ x \in \mathbb R^n \colon x \geq ...
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1answer
93 views

Transformation of a surface normal

I'm taking a university level course in discrete geometrics and graphical programming, and I'm having trouble understanding this exercise. Let p be a point in R^3, n a surface normal, and M a ...
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1answer
65 views

using Boltzman transformation to change PDE to ODE

using Boltzman transformation $\phi=z/\sqrt{t}$ to transform a patrial Problem: $$\frac{\partial \theta}{\partial t} =\frac{\partial }{\partial z} \left(D{\frac{\partial \theta}{\partial ...
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0answers
39 views

Finding transformation function for a distribution that looks like exponential

Suppose that we have two data sets, R and P. R is larger than or equal to ...
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1answer
52 views

Laplace transform, Inverse Laplace transform

Let $(\mathcal{L}f)(s)$ be the Laplace transform of a piecewise continuous function $f(t)$ defined for $t\geq 0$. If $(\mathcal{L}f)(s)\geq 0$ for all $s\in\mathbb{R^+}$ does this imply that $f(t)\geq ...
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1answer
26 views

Linear transform $T$ such that $T(b^x)=b(b-1)^x$

The title pretty much says it all. I'm trying to find a linear transform, maybe a vague analog of a derivative, that has the property that if $f(x)=ab^x$, then $T(f)=ab(b-1)^x$, analogous to the ...
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1answer
55 views

Transforming a nonlinear system to a linear system

Suppose I have two points in $\mathbb{R}^2$ and I wish to find values of parameters $a$ and $b$ such that I obtain the power law $y=ax^b$ which goes through the two given points. I can solve the ...
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2answers
47 views

What does it mean when dim(V)=rankT

I have a question relating to a linear transformation and have ended up with the result that $dim(V)=rank(T)$. I got to this because I'm told that $V$ and $W$ are finite dimensional vector spaces, ...
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0answers
51 views

Affine Transformation and Continuous Deformation

How do these two concepts relate? Thus far I have a (what I think is a) good intuitive idea of a continuous deformation- the visual basically looks like the boundary being stretched so that it never ...
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1answer
29 views

Transformation Matrix of a linear function

Consider the function $f: \mathbb{R}^3 \rightarrow \mathbb{R}^2$. Let $A = \{ (1,2,3)^t, (1,0,4)^t,(0,0,2)^t \}$ a base of $\mathbb{R}^3$ and $B = \{ (1,1)^t , (2,1)^t) \}$ a base of $\mathbb{R}^2$. ...
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1answer
49 views

Is a linear transformation just a mathematical description of a straight line?

On Physics Stack Exchange, the question was asked: Are lorentz transformations linear? The up-votes given to an answer seemed to be in proportion to how mathematically sophisticated it was, with mine ...
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1answer
40 views

how to find triangular point from a side

i have two triangles. Say a , b , c and p, q, r and the projection of the abc to pqr a - > p b - > q c - > r here known point values are a b c p q and r unknown. $\overline{PR}=\overline{AC}$ ...
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337 views

Shift numbers into a different range

I was wondering how can I shift my data that fall between a range lets say [0, 125] to another range like [-128, 128]. Thanks for any help
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0answers
126 views

Trick for Jordan-Matrix and transformation of basis

some time ago I found a 'trick' for getting a basis-transformation-matrix for jordan. I'd like to understand it, but at a certain point I stuck. Maybe you can help me? Given is a matrix A: ...
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0answers
23 views

how to extract frequency from a set of numbers

Given the numbers 4,1,0,4,0,0,4,1,0,4 it is obvious there's a dominating frequency of 4 appearing every four numbers. Given 5,1,1,3,0,0,6,1,0,4 again it looks that there's a spike of about 4 (4.5 to ...
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3answers
93 views

Transformation(?) of Random Variables

There are two independent Gaussian R.Vs: $U:N(-1,1)$ and $V:N(1,1)$ How do I go about finding the PDF of the following transformations? X = U+V T = (U+2V, U-2V) W = U (with 50% chance), V (with ...
3
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1answer
120 views

Computing the derivative of a transformation matrix

I am trying to find a geometric transformation between two images, where the transformation is a simple scaling matrix. So, if I denote the two image functions as $r$ and $f$ and the scaling matrix as ...
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1answer
62 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 ...
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2answers
47 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 ...
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1answer
60 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 ...
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1answer
61 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 ...
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1answer
109 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
63 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
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0answers
98 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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1answer
39 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 ...
2
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2answers
58 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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2answers
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 ...