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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Transformation as function of time, Solve for time

I'm trying to create a flawless a priori collision solver. I have two local coordinate systems which map to global coordinates using $[translate][rotate][scale]$, and map to eachother using ...
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109 views

What's the inverse of the Weierstrass-Mittag-Leffler-Transform $\exp\left[g(z) + \int_\mathbb C f(y)\ln(z-y)\,dy\right]$?

As mentioned in another post, as a consequence of Mittag-Leffler's theorem combined with the Weierstrass factorization theorem, after reducing to the common denominator, any meromorphic function can ...
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Applying PCA on covariance matrix in order to generate a new random variable.

Let $\mathbf{x}$ be a random $n\times1$ real vector, $\mathbf{x}\in\Bbb{R}^n$, which is distributed normally with mean $\bar{\mathbf{x}}$ and covariance matrix $\Sigma_x\in\Bbb{R}^{n\times n}$, i.e. ...
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37 views

Calculate projection of a line in a square

Said that we have two points (P1, P2) that form a line, and 3 points (S1,S2,S3) that form a square, how would we calculate the position X and Y of the point resulting from the intersection of the line ...
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36 views

What is special about a transformation if the matrix of that transformation is symmetric?

If the matrix of a linear transformation T$\colon \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$ with respect to some basis is symmetric, what does it say about the transformation? Is there a way to ...
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19 views

Questions about a special tensor transformation

Suppose tensor $U_{i\alpha\beta}$ with dimension $M*N*N$ satisfy following condition: $$U_{i\beta\alpha}=W^1_{\alpha\alpha'}W^2_{\beta\beta'}U_{i\alpha'\beta'}$$ where $W^1$ and $W^2$ are $N*N$ ...
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46 views

Finding the transformation matrix R

Please help me in solving this problem, I am not sure what a transformation matrix R is and how to proceed.. Any help is appreciated. Find the transformation matrix R that relates the (orthonormal ) ...
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88 views

finding if a linear transformation exists, and proving it.

We just started the topic of linear transformations and I have this hw question that I just don't understand. Does there exist a non-trivial linear transformation, represented by some 2x2 matrix, ...
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187 views

Jacobian determinant of unitary transformation

Is the Jacobian determinant of a unitary transformation equal to one? I ask because I get that impression from the appendix of this paper. They have spherical coordinates for two particles, ...
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48 views

A conformal mapping from a sector to a strip

What is the simplest function that maps the sector $r < 1$, $0 < \theta < \pi$ conformally onto the strip $0 < u < \pi/2$, $v > 0$? Here, $r$, $\theta$, $u$, $v$ have their usual ...
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38 views

Inverse Laplace transformation of (s^2-4s-2)/((s^2+2)^2)

I approached this problem as follow: $1.$ rewrote $(s^2-4s-2)$ into $(s-2)^2-6$ $2.$ Now break the function into 2 parts: $\frac{(s-2)^2}{(s^2+2)^2} + \frac{6}{(s^2+2)^2}$ the Laplace inverse ...
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28 views

“compression” transform

Is there a mathematical transform that cuts off a signal at two extreme values? Here is code to do what I want: ...
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167 views

Legendre transform concave function

Let $f$ be a concave function and define $f^*(y) := \inf_{x}(yx-f(x))$. Is this in any sense related to the Legendre transformation? -If yes, is $f^*$ also concave? Is this transformation invertible ...
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30 views

If S and T are transformattion mappings, what is [ST]?

S and T are transformation mappings, what does [ST] and [TS] mean? Does it mean transform via S and then apply T to the result and vice versa?
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45 views

Conformal mapping of nonsimply connected domains

The question asks: Map the complement of the arc $|z|=1$, $y\geq 0$ on the outside of the unit circle so that the points at $\infty$ correspond to each other. How would you construct such conformal ...
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1answer
48 views

Linear Algebra Dimension

Let $L(U,V)$ = $\{T:U\rightarrow V\ :\ T\ \text{linear}\},$ and dim $(U)=n$, dim $(V)=m$. Then show that $$ \dim L(U,V) = mn. $$ I don't know how to begin and I already searched the internet to find ...
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111 views

Linear Algebra - - Linear transformation

The matrix $$ A=\left[\begin{array}{ccc} 1 & -a & a \\ -1 & a & a+2 \\ 1 & 2a+3 & -3a-4 \end{array}\right], $$ where $a \in \mathbb{R}$, represents a linear transformation $T: ...
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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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285 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
298 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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104 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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83 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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353 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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126 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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85 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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1answer
43 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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1answer
168 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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45 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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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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73 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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64 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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76 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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114 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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69 views

How do you prove a hilbert transform?

I am stuck with this question below, I need help;
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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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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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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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141 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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87 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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40 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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65 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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28 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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61 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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57 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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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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37 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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52 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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42 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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579 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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154 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: ...