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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formula for transforming the interior point of a 2d bounding box when the box is stretched by moving a single corner of the box

I would like to find out a formula (mathematical equation(s)) for adjusting the position of any point enclosed in a rectangle when its corner point is repositioned such that (1) all other corner ...
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115 views

Infinite dimensional vector space eigenvectors eigenvalues and representation

We can express linear transformations with their eigenvectors and eigenvalues in finite vector spaces if they are diagonalizable. even if they are not diagonalizable we can express them via Jordan ...
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59 views

Heat equation $\frac{\partial \theta}{\partial t}=\kappa \frac{\partial ^2\theta}{\partial x}$ using two transformations to solve

Consider the heat equation $$\frac{\partial \theta}{\partial t}=\kappa \frac{\partial ^2\theta}{\partial x}$$ for an infinite rod. We use the transformation $q_1=\frac{x^2}{kt}$ and $q_2=\frac{\theta ...
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1answer
239 views

Step in Euler's rotation theorem

I have been examining the matrix proof for Euler's rotation theorem on Wikipedia. I have deduced every step up to proving that $\det (R - I) = 0$ for any rotation matrix R. However, I'm having ...
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1answer
129 views

What does it mean by “the origin is moved by the transformation” in linear transformations?

Linear transformations have the special property that the origin is not moved by the transformation. I don't really understand what this means. The example I'm given is that the following ...
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71 views

Changing coordinate system with non standard definitions

The standard coordinate transformation to polar coordinates is $$ \begin{cases} x=r\cos(\varphi)\\ y=r\sin(\varphi) \end{cases} $$ with $r\in[0,\infty), \ \varphi\in[0,2\pi)$ The question is whether I ...
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1answer
36 views

fit of translate curve and restore translation

I have a set of data, $S$ with negative values and the function. $$ \begin{equation} y = x + s \sqrt{ m \frac{x}{s} + 1} \end{equation} $$ If I try fit the curve for get the parameters $m$ and $s$, ...
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245 views

Derive Student T distribution using transformation theorem

I am trying working on an exercise that asks me to show that If $ X_1 \in N(0,1) $ and $ X_2 \in \chi^2(n) $ are independent random variables, then $ X_1 / \sqrt{X_2/n} \in t(n) \, $ where $ ...
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2answers
124 views

Laplace Transformation Applications

In one of our Mathematics lecture our Prof told us that similar to Logarithmic Transformations we can use Laplace Transformations to solve difficult equations. What kind of equations do Laplace ...
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1answer
98 views

Linear time varying into linear time invariant.

My original problem, is to transform Linear time varying systems of the form , for example: $$\begin{bmatrix}\dot{x1} \\ \dot{x2} \end{bmatrix} = \begin{bmatrix} -3t^2 & 0 \\ 6t^5 & -6t^2 ...
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174 views

linear transformation, ker(T) and im(T) - question from final exam

Assume $T:V\to V$ is a linear transformation, $\mathrm{dim} V = n$. Let $v$ be a vector of $V$ such that for $1\leq k\leq n : v, T(v), \dots , T^{k-1}(v)$ : they are all NOT zero, but $T^k(v) = 0 $. ...
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1answer
286 views

Which matrix transforms my vector field $F(r,\theta,\phi)$ from cylindrical to spherical coordinates

I am looking for the matrix that I have to apply my vector at the position $(r,\theta, z)$ to in order to get the appropriate vector in spherical coordinates. I am totally okay, if you could give me ...
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3answers
695 views

A Möbius transformation maps circles and lines to circles and lines. What exactly does that mean?

The title pretty much says it all. I am also looking for a concrete example if possible. I have looked at the proof, but I'm not exactly sure what it means because I am kind of confused on what the ...
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0answers
77 views

Determining pose of an object in 3d space

Given a 3D model of an object centred at the origin, if I place a camera at position (x,y,z) and make it face the origin, from the image rendered the object appears ...
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2answers
1k views

Finding reflection transformation matrix

I have two 3 dimensional points. $A [x_1, y_1, z_1]$ and $B [x_2, y_2, z_2]$. I need to find a transformation matrix which when multiplied to $A$ will give me $B$ and when multiplied by $B$ give me ...
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1answer
99 views

Evaluate the following integral by transformation:

1 1-x ∫ ∫ (sqrt(x+y)(y-2x)^2)dydx 0 0 $$ \int_0^1 \int_0^{1-x} \sqrt{x+y} \, (y-2x)^2 \,dy \, dx $$ I've determined that $u = x+y$ and $v = y-2x$ and that the jacobian is $= 1/3$. and that $x ...
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83 views

coordinate transformation and scaling

I have a global coordinate system that I need to transform to a local coordinate system. The new and old coordinate systems are shown below. Using transformation rules, I came with with the ...
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1answer
61 views

Can I use the pseudoinverse of a Jacobian like I think I can?

I need to compute the Jacobian for a transformation that maps parameters $p_1,...,p_n \to q_1,...,q_m$, $n\neq m$. For this, I need to compute the derivatives $\frac{\partial p_i}{\partial q_j}$. ...
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2answers
346 views

How to enlarge a circle?

if you are given a circle with equation $(x-a)^2 + (y-b^2) = r^2$ and it is enlarged by a factor of $3$ what would the new equation be? Would you put $2x$ an $2y$ in the place of $y$?
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1answer
2k views

How to find an all-in-one 2D to 3D Transformation Matrix for perspective projection, rotation, and translation?

I have read Finding a 3D transformation matrix based on the 2D coordinates but I think my situation is different because I think I need a 4x3 matrix, not a 3x3 matrix. I'm not sure but this might be ...
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2answers
359 views

Find a vector $\mathbf x$ whose image under $T$ is $b$.

I am having trouble with this question and how to get the answer. With $T$ defined by $T(\mathbf x)=A\mathbf x$, find a vector $x$ whose image under $T$ is $b$. $$ A = \begin{pmatrix} 1 & -3 ...
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1answer
100 views

Stereographic projection when the “North/South Pole” is not given by $(0,…,\pm 1)$?

Straight forward enough... what if My point is arbitrary, how can I get a new stereographic projection?
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40 views

Looking for a “Neat” Transform to Yield a Convex Set

Optimizing on a unit sphere $\mathbb{S}^n$ is almost a convex problem (if the function is convex in the new set) if we make our "new" set $\mathbb{R}^n$, via the stereographic projection. Clearly ...
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2answers
138 views

Fourier, Laplace, … and other Integral-transformations

I know Laplace, Fourier and Mellin-Transformation. Is there a general theory of transformations? My main interest is about classification of transformations satisfying specified properties like ...
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1answer
226 views

Mirror anamorphosis for Escher's Circle Limit engravings?

You are probably familiar with "mirror anamorphosis," the rendering in a painting of a distorted figure that can be undistorted by viewing in an appropriately tilted or curved mirror. The skull in the ...
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1answer
71 views

Determine all the linear transformation of $S^3$

I encounter the following problem: Let $S^3$ denote all the $3\times 3$-real symmetric matrices. If $F:S^3\to S^3$ is a linear mapping, with $F(OAO^T)=F(A)$, for any ...
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1answer
114 views

Transformation of Cubic Polynomial

I'm stuck on transforming this equation and am not sure where to begin. I know I need to define $x$ as some multiple of $u$ and somehow cancel the coefficient of the $x^2$ term but am not sure how to ...
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1answer
245 views

Transform between cartesian coordinate system and abstract coordinate system

I am trying to find a transformation that takes me between Cartesian coordinates and a pseudo-coordinate-system I have developed which is described as follows: Please first see the diagram below. ...
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1answer
128 views

condition for upper triangular matrix

Consider the following condition from this other post Define $S_k = {\rm span} (e_1, \ldots, e_k)$, where $e_i$ the standard basis vectors. Clearly, the linear map $T$ is upper triangular if and ...
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97 views

Update rotation matrix

Imagine you have a two noded beam in space, defined by extreme nodes 1 and 2. Image is owned by Jean-Marc Battini. To ...
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3answers
149 views

Prove an equality between dimensions of kernels

Let $V$ be a inner product space over field $\mathbb{R}$ with $\dim(V)<\infty$, and $T\in \text{Hom}(V,V)$. I'm trying to prove:$$\dim(\ker T)=\dim(\ker T^*)=\dim(\ker TT^*)$$ Also, as a conclusion ...
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1answer
234 views

Homothetic transformation

suppose that,we have Homothetic,in rectangular coordinate system,with center origin $(0,0)$ and $k$ ,and this homothetic send point $A(2,3)$ to point $B(2*x-1,x)$.aim is to find $k$,first i have ...
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1answer
80 views

Range of transformation $T: C[0, 1] \to \mathbb R$ defined by $T(f) = \int_0^1 f(x)\,dx$

What is the range of the transformation $T: C[0, 1] \to \mathbb R$ defined by $T(f) = \int_0^1 f(x)\,dx$ $C[0, 1]$ is the vector space of all continuous functions $f: [0,1] \to \mathbb R$. So the ...
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1answer
165 views

Formula for Adding Time expressed as decimal

If I record the duration of an event that was 97 minutes as 1.37 and then record the duration of another event that was 162 minutes as 2.42 and then add the two together to get 3.79, is there a ...
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1answer
92 views

Distribution of the Inverse of a Random Variable

I am trying to figure out how to find the distribution of the inverse of a random variable. Say, $Y=X^{-1}$ where X can take negative values. The two ways I know to find the distribution of a random ...
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1answer
96 views

log transformation for dummies

I have a question which is probaly very simple to answer for most people here: We have a formula: y = -log(x) Then this happens to x: ...
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144 views

Rewriting a formula (hopefully just basic algebra..)

I have a question which actually involves statistics, but I think it comes down to some basic algebra, so hopefully someone here can help me out (as I am obviously not a mathematician). Let’s say I ...
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1answer
37 views

Calculating transformation from origin to point

I have an icosahedron of radius $x$ with 12 vertices at known coordinates. If I have a point at $(0,0,x)$ where $x > 0$ and a vertex of this icosahedron at $(a,b,c)$ how can I find the rotation ...
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1answer
24 views

Algebraic Transformation query…

I'm boning up on Algebra, and I'm looking into Algebraic Transformation. I understand the basic concept - but I'm confused by two self assessment questions. The two questions, from what I can see, ...
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2answers
145 views

Transformation between the same rotation expressed in different coordinate systems

EDIT Lets assume my transformation does the following mapping: x = -y y = -x z = -z Which produces this transformation matrix $R_t$: $ \begin{array}{lll} 0 & -1 & 0 \\ -1 & 0 & 0 \\ ...
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1answer
248 views

my plane is not vertical, How to update 3D coordinate of point cloud to lie on a 3D vertical plane

I have a bunch of points lying on a vertical plane. In reality this plane should be exactly vertical. But, when I visualize the point cloud, there is a slight inclination (nearly 2 degrees) ...
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1answer
122 views

Dependence of vectors : before and after linear transformation

I have a pretty simple question that confused me: V is a vector space of a finite dimension. $T: V \to V$ is a linear transformation. The information that's been given in question: ...
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3answers
74 views

geometry: linear transformation

I know I do it wrong but where is the mistake??? In E3* are given the points $A(1,0,0,0)$, $B(0,1,0,0)$, $D(0,0,1,0)$, $O(0,0,0,1)$ and $E(1,1,1,2)$. The linear transformation $\Phi$ operates ...
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1answer
66 views

Interpretation of Linear Algebra and superposition

I have been told and have seen that knowing Linear Algebra is foundational in pursuing advanced mathematics. After dealing with it enough I have gotten "used to it" (I still need a lot more practice!) ...
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2answers
64 views

Explanation of an example of linear transformation

This is an example from a text (Linear Algebra, Freidberg). I am trying to follow along, and I feel like I should know this from vector calc but I am missing something silly. The example is: ...
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1answer
49 views

Image of $A \subset \mathbb{R}$ under transformation $(x,y) \rightarrow (u,v)$

What is the image of the set $$A=\{ (x,y) : 0\le x \le a\ , \ 1\le y\}$$ under the transformation $(x,y) \rightarrow (u,v)$ where $$u=x/y$$ $$v=x$$ The parameter $a$ is positive. I got a 'triangle' ...
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4answers
100 views

Showing that a transformation $T:\mathbb R^3 \to \mathbb R^2$ is linear

OK, I am trying to prove the following transformation is linear, and find the basis for $\ker(T)$ and Im$(T)$ (also denoted in our textbook by $N(T)$ and $R(T)$ ). Then we're suposed to find the ...
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4answers
134 views

Why do we define a linear transformation to have the property that $f(cW)=c f(W)$?

Why we define a lin tranfs to have the property that $f(cW)=c f(W)$ ? let $V,T$ be any two vector spaces and let $f:V\rightarrow T$ be a linear transformation between $V $and $T $ why do we ...
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1answer
146 views

Fourier transform for a 2D curve

I am stuck on the following problem about a Fourier transform of a 2D curve: I have to calculate the Fourier transform (using 1D complex FT) (and the opposite of it) for a 2D curve z(t). The curve is ...
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1answer
57 views

Function to map a range of $[-1,1]$ to a range of $[0,1]$ [duplicate]

I'm not capable of 'rigorously' defining the problem I have but this is the best I can do. If I have a set of points that range from $-1$ to $1$ inclusive and I have to transform the data so that it ...