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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Complex analysis question, maximum principle application

Let $\Omega=\{z, \text{Re}z>0\}$ Suppose that $f$ is continuous in the closure of $\Omega$ and $f$ is holomoprhic on $\Omega$ and there are constants $A<\infty $ and $\alpha<1$ such that ...
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Equation of smooth spline curve

This is a homework question a)Assume an equilateral triangle ABC of a side AB = a = 10.The coordinate of A is (5, 3).The slope of the segment AB is 2.This triangle controls a curve.This smooth ...
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98 views

transformation of conic section

Given is the conic section $x^2 +xy + y^2 +2x +3y -3 = 0$. The following tasks: 1.) What is the coordinate matrix $A_1 = M_{\beta} (\sigma) $ of the bilinearform? 2.) do the transformation and ...
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How to write this term compactly?

How can I write this term in a compact form where $a$ only appears once on the RHS (in particular without cases)? $T(a) = \begin{cases} a^2 &,\text{ if $a \leq 0$}\\ 2a^2 &,\text{ ...
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31 views

Probability function and random variables

Given a Bernoulli r.v., $W$, which is derived from r.v. $T$ (Poisson) (a) if $T=0$ then $W=1$ and (b) if $T>0$ then $W=0$. One has to show that the sample mean (the proportion of $0$s in the ...
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38 views

Probability function (p.f) of a random variable

If we have a Bernoulli random variable $W$ that is derived from a Variable $T$ (Poisson $\lambda$), by the following rules $W =$ (if $T=0$ then $W=1$ and if $T>0$ then $W=0$), I am having trouble ...
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55 views

probability: transformation of a random variable $Y = X^4 + 1$

Find the PDF of $Y = X^4 + 1$ if $X\sim\exp(\lambda)$. When a transformation is not one-to-one, we have multiple solutions for $X$. Take for example $Y = X^2$. Then \begin{align*} x_1 &= ...
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294 views

Box Muller Transform - Proving that Z is Normal Distribution

I'm studying the Box Muller transform and I cannot see how Z0 and Z1 represent standard normal distributions. I've looked at the wikipedia page for the box-muller function but they don't seem to have ...
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122 views

how to obtain transformation matrix A in y = Ax + b notation?

I'm trying to obtain original transform matrix A and its translation vector b From y=Ax+b equation. I have original values of vectors before transform and translation (x) and vectors after transform ...
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52 views

Finding the image of a linear transformation given other images

Suppose there is a linear translation $T: \mathbb{R^3} \rightarrow \mathbb{R^3}$ such that $$T(\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}) = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}, \ ...
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54 views

Matrix representation, linear transformation , general question on linear algebra

Let $X$ be a finite dimensional vector space over $K$ and define $T:X\rightarrow X$ to be a linear transformation on $X$. If $\alpha, \beta$ are two different basis for $X$ then we know that the ...
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36 views

Integration matrix

I want to do integration(summation) of a signal(x) using matrix multiplication. I am looking for a transformation matrix, I corresponding to integration such that F = I * x , where x is the signal ...
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One question on Matrix Equation

Assume $\hat{M}_1, \hat{M}_2, \hat{T}_{11}, \hat{T}_{12}, \hat{T}_{21}, \hat{T}_{22}$ are $2\times 2$ matrix. And $a, b, A, B, C, D$ are all numbers, satisfying the following relation: \begin{align} ...
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1answer
29 views

Homeomorphism from $[0,1]\times[0,1]$ to $\overline{D}(0,1)$?

I'm trying to construct a homeomorphism from $[0,1]\times[0,1]$ to $\overline{D}(0,1)$. I'm pretty sure there is one. I've been trying to work geometrically : mapping $[0,1]\times[0,1]$ to ...
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172 views

Comparing function to parent function without graphing

How can I compare this function to the parent function without graphing? Where did the 5/4 come from and what steps do I need to take to solve this?
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34 views

Prove the linear transformation that takes all linear maps T: V → W to their respective matrix representations is an isomorphism.

Let V, W be finite dimensional vector spaces. Prove the linear transformation that takes all linear maps T: V → W to their respective matrix representations is an isomorphism. Thanks in advance! I ...
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3answers
118 views

Rewrite formula

I have the formula: $$ R(T)=R(T_0)e^{-B(\frac{1}{T_0} - \frac{1}{T})} $$ How can I write this to T=...? I came this far: $$ \ln(\frac{R(T)}{R(T_0)})= -B(\frac{1}{T_0} - \frac{1}{T})$$ $$ ...
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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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1answer
110 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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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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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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1answer
204 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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49 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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1answer
39 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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“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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171 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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113 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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1answer
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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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299 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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324 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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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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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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380 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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129 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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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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176 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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46 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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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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81 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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118 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 ...