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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Explain why: If T a linear transformation that is also onto (surjective) from V to W, then range of T = W.

I learned a corollary to theorem today. In the proof of the corollary, it states exactly what I typed in the title. Just curious how to make sense of R(T) = W (the co-domain). Thanks!
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1answer
20 views

matrix for reflection across a given line

Say I have the equation y = mx + k and I want to reflect vertices with respect to that line. How would I go about finding that matrix? Is there an example of such a matrix anywhere? I've only found an ...
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1answer
34 views

Cayley's Theorem for Semigroups

I've read and fully understand Cayley's theorem for groups, however when i get to the theorem for semigroups i come to a complete stop. I've figured that the identity and cancellative properties are ...
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2answers
18 views

Rotations/Transformations with Complex Numbers/Eulers Formula

Hello, I am not entirely sure how to do this question, as I understand a rotation in the complex plane can be described by using Euler's formula, $e^{i\theta}$. Since this is an equilateral ...
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2answers
36 views

Affine transformation that sends a conic to itself but does not preserve the focci or the axes [on hold]

So I'm trying to find an affine transformation that sends a conic to itself but does not preserve the foci or the axes. I don't know if this can be done. I'm pretty sure that if it is possible then I ...
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1answer
69 views

Profile likelihood: Box-Cox transformation

I'm trying to prove a result that shows that the maximum likelihood estimator reduces the number of parameters in a Box-Cox model. In essence, we're trying to prove that $\bar{z}$ is the nuisance ...
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1answer
29 views

Reflection matrix in $ \mathbb{R}^{3} $.

I need help in understanding how they got the transformation matrix $ Q_{L} $ from Theorem 2 and $ P_{M} $ at the bottom of the page. They skipped some steps and I find it confusing. Any help would ...
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1answer
26 views

Linear operators proof, projection and reflection matrices

I am trying to understand two parts from the picture below in my textbook, but I dont understand how they arrived at it. I am trying to understand the proof below and how they got $P_L(\vec{v}) = ...
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36 views

The Adrian Transformation of a function in $\mathbb{R}^{2}$

Recently I came upon a problem (if you would call it that, more of a thought experiment), which was phrased something like this: Rotate the area formed by $\int_{-1}^12dx$ around the curve ...
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1answer
22 views

Transformation and matrices

Two sequences $y_t$ and $z_t $ satisfy $$y_t = ay_{t-1} + bz_{t-1}$$ $$z_t = cy_{t-1} + dz_{t-1}$$ Where $a = 6$, $b = -20$, $c = -17$ and $d = -12$. From the two given equations above, ...
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84 views

If $\frac{x-1}{e^x-1} = y$ then $x=?$

I have following equation: $$\frac{x-1}{e^x-1} = y$$ I want to solve this equation such that I have the value of $x$ in the term of $y.$ i.e. inverse of the equation
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25 views

Eigenvectors and geometrical transformation

$$A= \begin{pmatrix} 2/3 & 2/3 & -1/3 \\ 2/3 & -1/3 & 2/3 \\ -1/3 & 2/3 & 2/3 \\ \end{pmatrix}$$, I need to understand that kind of ...
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10 views

Identifying a subclass of the class of monotonic transformations

Let $u$ be a continuous function from $R$ to $R$. Then $v$ is called a positive monotonic transformation of $u$ if $u(x) < u(y)$ if and only if $v(x)<v(y)$ and similarly for greater than and ...
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32 views

The curvatures of a transformed surface under a similarity transformation

Setup: Let $f:\mathbb R^3\to\mathbb R^3$ be a similarity transformation. Then $f=rA+b$ for some fixed orthogonal matrix $A$, vector $b$ and nonzero real $r$. Suppose $S$ is a surface, and $S'=f(S)$. ...
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1answer
21 views

How to prove $\sqrt{Y}$ can be variance stabilizing transformation of poisson distribution?

I am studying constant variance checking when conducting ANOVA. I know that $\sqrt{Y}$ is one of the common transformations for a Poisson distribution, but I can't prove it. I also read Anscombe ...
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33 views

Is this chain of inequalities correct?

Is this chain of inequalities correct? If not how to make it works? $$\frac{\ln \left( 1+x^3+y^3 \right)}{\sqrt{x^2+y^2}} \le \frac{\left( x^3+y^3 \right)}{\sqrt{x^2+y^2}} \le \frac{ \left( ...
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1answer
16 views

Let $R$ be any rotation and $P$ any reflection then $R \circ P$ and $P \circ R$ are both glide reflections

Let $R$ be any rotation and $P$ any reflection then $R \circ P$ and $P \circ R$ are both glide reflections I am having trouble showing $P \circ R$ is a glide reflection, I manage to get $R \circ P$, ...
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1answer
14 views

Identifying translations and rotations as compositions.

I am having trouble understanding the below which are the ones underline in red and blue. For the red: Why is that $R_{A,90}(A)=A$ and that $\tau_{AB}(A)=B$ As for the blue: Why is that ...
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1answer
23 views

Does the following series of transformations of inequalities holds?

I am to calculate limit of the function $f(x,y)$ i am trying to apply squeeze theorem. Is the following series of transformations of this inequality correct? If not how to do this correctly? i.e. are ...
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0answers
13 views

What is or how do you get the rotational matrix of 4-D vector onto the xyz-space?

which would make the 4-D component 0. To be honest I'm not really sure how 4-D rotations work. I know about the simple rotations but not the mechanism in how it rotates, and I'm not sure whether to ...
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17 views

Solids of Revolution around other functions.

Recently I've been thinking about solids of revlution, and thought about an interesting experiment. Can you rotate functions around, for example, the line $f(x)=x$? And consequently, could you rotate ...
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1answer
75 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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7 views

Transformation of graphs, finding the values of unknowns

I am a second grade IB student using "Mathematics Standard Level for the IB Diploma, Cambridge" book.This is the question I have a problem with: "Let f(x)=(3x-5):(x-2) a) Find the value of constants ...
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Let A,B,C be the vertices of the triangle, find the center of the following rotations

Let A,B,C be the vertices of the triangle, find the center of the following rotations: a) $R_{A,\frac{\pi}{2}} \circ R_{B,\frac{\pi}{2}}$ Two rotations that are composed together is another ...
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16 views

Find a basis for Kernel and Image of a Linear Transformation

Given: $$A = \left\{\begin{bmatrix} 0 & 1 \\ 0 & 2 \\ 0 & 1 \end{bmatrix}\right\}$$ Find a basis for $ImT_A$ and $kerT_A$ So far, I've ...
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Intersection of the composition of two glide reflections

i am taking a geometry course and we are learning about isometries. I am having a hard time with glide reflections and this problem is giving me some issue, mainly because my professor usually tells ...
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1answer
31 views

generating system of the kernel of a module-transformation

Let $G ≠ 1$ be a group and A a commutative ring. Now, the group ring $A[G]$ is naturally an A-module. Next, let's consider the transformation: $$\phi: A[G] \to A, \sum_{g \in G} a_gg \mapsto \sum_{g ...
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How can I transform coordinate systems based on quaternion data?

I have a single rigid body object, and its orientations in quaternion with respect to two coordinate systems, each is called original and prime, respectively; therefore, I have two quaternions ...
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1answer
631 views

Finding reflection of a matrix

To 2 decimal places, what is the value of the lower-right entry in the reflection matrix $Q_a $if a = 1.05? Not even sure where to begin, is there a formula? This is what I could find in my textbook ...
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1answer
38 views

How to transform function values to specific interval

I'm doing a project at university about scientific computing and I'm stuck. As in: I seem to lack quite a bit of mathematical background for this project. The program has as input an array of $x$ ...
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22 views

Why cannot the homogeneous coodinates be zero?

Given a point (x, y) on the Euclidean plane, for any non-zero real number Z, the triple (xZ, yZ, Z) is called a set of homogeneous coordinates for the point. Why can't Z be zero?
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307 views

How can I transform coordinate systems with quaternions?

I have a coordinate system 0 which I'd first like to rotate about its z-Axis which gives me system 1, and afterwards rotate system 1 about its y-axis which gives me system 2. See picture: Both ...
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9 views

transformation of variables in Melnikov method

Supposing there is a system of non-autonomous non-linear differential equations with small damping and small forcing. The unperturbed system (zero damping and forcing) is Hamiltonian but neither has a ...
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19 views

How to wrap 2D image around a circular arc?

How to transform rectangular image to image that is bent at certain radius? On the plot the center line follows the radius which is 2. Conformal transformation would be this, but there should be ...
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1answer
22 views

Is the transformation $T: (r, \theta) \to (r, \theta + \phi)$ linear? Here $\phi$ is a given angle

Let $T$ rotate every point through the same angle $\phi$ about the origin, $i.e.$ $T: (r, \theta) \to (r, \theta + \phi)$ where $\phi$ is given. If in addition that $T(O) = O,$ namely, if $T$ maps the ...
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1answer
2k views

Negative & Positive Shear Factor

My question relates to constructional geometry & matrices aren't to be involved in the solution because stated Math level is up to O Levels... The figure below shows shear with y=3 as invariant ...
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1answer
20 views

Terminology with linear transformation

I am working on a problem that asks me to "Write C for the matrix whose ij entry is $(1/2)^{ij}$" given that $M$ is the vector space of all $n x n$ matrices and $l$ is a linear transformation on $M$. ...
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11 views

Composite Transformation expressed as single transformation

Is this right following the composition rule. Also, i know that we can add or subtract 2 pie so as to make the number nicer, by adding 2 pie in the case i shouldn't be wrong right ?
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How to solve an inverse relationship (cooking temp/time)

How to figure out exactly the "add a little more time" to the question: cook at 425 deg for 18 minutes ... if I have several things in the same oven and need to set the oven at 375. I can't use a ...
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0answers
30 views

mobius transformation form $M= B(0,1)\setminus\overline{B(1,\sqrt2)}$ to a sector.

I know that the intersections of the two circles need to be sent to $0$ and $\infty$ in order to get a sector $S = \{x+iy: x>0,0<y<x\}$ The intersections of the two balls were i and -i. So i ...
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The Composition of Two rotations

So far I rewrote the halfturns of d,c,b,a to halfturn (p,n)(m,l) where n=m because lines c and d are parallel so I can make ambiguous lines n and p parallel too. I also know that lines c,d can be ...
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1answer
15 views

How principal component analysis ensures component orthogonality when using zero co-variance as the restriction to maximize variance?

I am currently learning the mathematics behind PCA and I found when PCA maximizes variance to find out the 2nd, 3rd, ... components, it uses zero co-variance as the restriction, as shown below, ...
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389 views
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Transformation that f(A+B)=f(A)+f(B) and f(AB) = f(A)f(B)

if $f:M_{n*n}(F) \rightarrow M_{m*m}(F) $ and f transformation identity to identity matrix and $f(AB) = f(A)f(B) , f(A+B)=f(A)+f(B)$. now we want to prove there is an integer like $k$ that $m = k*n$
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Transforming a function by a sequence geometric operations on its graph.

I am solving the following problem: Let $f(x) =\sqrt{x}$. Find a formula for a function $g$ whose graph is obtained from $f$ from the given sequence of transformations: shift right $3$ ...
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2answers
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Matrix for rotation around a vector

I'm trying to figure out the general form for the matrix (let's say in $\mathbb R^3$ for simplicity) of a rotation of $\theta$ around an arbitrary vector $v$ passing through the origin (look towards ...
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Fourier Tansform of derivative on Wolfram Alpha

If I'm not mistaken, the Fourier Transform of the nth order partial derivative of a function with respect to x, using the transform variable k is: (i*k)^n * [F(k)] so for the 1st order derivative ...
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3answers
56 views

Prove that the map $f(z)=\frac{1}{z}$ sends any line onto either a line or a circle.

Show the cases in which the image is a line and the case in which the image is a circle. I understand that representing the equation of line, ($ax+by+c=0$ $a,b,c\in\Bbb R$ $a,b\neq0$ at the same ...
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1answer
29 views

Find the image of the unit vector at the point $z_0=i$ under the function $f(z)=z^2+2z$

From this image find the rotation angle and the expansion factor. $z\in\Bbb C$ I am unsure how to find a way to accurately plot the image. I understand that the image should expand because the ...
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23 views

Prove that the image of the center of a given circle is never the same as that of the circle image under a linear fractional map

Given some linear fractional map$f(z)=\frac{az+b}{cz+d}$, that is $ad-bc\neq0, c\neq0$ and a circle on $\Bbb C$ not passing through $z_0=\frac{-d}{c}$, so that the image is another circle, prove that ...