# Tagged Questions

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), (rigid-...

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### Arbitrary transformation of a 1D function into another

I have a 1D function (a spectrum in fact, n points representing amplitude vs. wavelength, call it sp1). This spectrum changes shape into another spectrum (sp2) as a function of experimental ...
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### Variance stabilization for Poisson data

Intro Let $Z > 0$ be a random variable with the mean and variance defined as $\mathbb{E}\{ Z \}$ and $\operatorname{Var}\{ Z \}$, respectively. The variance stabilization transform (VST) $f(z)$ ...
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### Matrix for orthogonal projection with respect to ordered and canonical bases

Orthogonal projection onto the line $y = 2x$ gives a linear transformation $T: R2 → R2$ such that $$T(1,2) = (1,2)$$ and $$T(−2,1) = (0,0)$$ Then the matrix of T with respect to the ordered basis ...
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### matrix of orthogonal projection with respect to the ordered basis.

Orthogonal projection onto the line $y = 2x$ gives a linear transformation $T: R2 → R2$ such that $$T(1,2) = (1,2)$$ and $$T(−2,1) = (0,0)$$ Then the matrix of T with respect to the ordered basis ...
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### Transformation between 2D coordinate systems

Let there be two coordinate systems: unit coordinates at $(0,0)$, rotated by 45 degrees the same, but at $(5,5)$ How would I go about to create a transformation to convert #2 coordinates into #1 ...
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### Variations of transformation of inversion?

Is there a transformation analogous to inversion, that is based on something other than circle (or sphere in higher dimensions), and has some interesting properties or applications? The motivation ...
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### Question on transformations in the complex plane

In the image (part (b)), Since $z < |3|$ before the transformation, does that simply imply that the region to be shaded after the transformation is definitely the inside of the circle and not it's ...
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### Defining a Linear Transformation Given a Basis for the Domain

I'm having difficulty understanding the proof for the following theorem: Suppose $B$ = $\{$$v_1$$, ... , $$v_n$$\}$ is a basis for a vector space V. Then for any elements $w_1$$, ... ,$$w_n$ of a ...
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### Matrix representations of linear transformation between bases

Let V and W be vector spaces, and let L: V -> W be a linear transformation between them. A basis for V is E = {$v_1$,...,$v_5$}. A basis for W is F = {$w_1$,...,$w_4$}. On the basis vectors the linear ...
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### abs(x)cos(x) in Fourier space

I am working on some problems concerning Fourier Transform and I am facing something I don't understand. I am trying to understand what is the representation of the function f(x)=abs(x)cos(x) in the ...
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### Basic Matrix Transformation

The information I have is for a matrix transformation from R^3 to R^3 (denoted by L()), L(a_1) = 3(a_1) and L(2(a_1))= (5,-3,6). Find L(3a_1-22a_1), L(-4a_1), L(0), L(4a_1). What I tried to do was ...
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### Looking for a formula to calculate DCT/FFT frequencies when cropping a matrix/image.

Given: A is a matrix of dimensions W1 x H1 . Cropping: Few rows and/or few columns were deleted from matrix A. We got matrix B of dimensions W2 x H2. Not more than 5% of matrix A rows/columns ...
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### How to prove this property of a projective transformation?

The copy below is from this book: Sophus Lie, Vorlesungen über Differentialgleichungen mit bekannten Infinitesimalen Transformationen, bearbeitet und herausgegeben von Dr. Georg Wilhelm Scheffers,...
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### How do digital filters work in time domain?

I am trying to understand how do digital filters work and how to actually calculate the output numerically. I have read that they are characterised by a transfer function $H(z)$ which results in a ...
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### How does the transformation on a point affect the normal at that point?

Say I have a point in 3D with coordinates $\begin{bmatrix} p_1 \\ p_2 \\p_3 \end{bmatrix}$ and the normal on the point with coordinates $\begin{bmatrix} n_1 \\ n_2 \\n_3 \end{bmatrix}$. Now I apply ...
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### Non constant function of two points invariant under Affine transformation proof

Here is the question; Prove that there does not exist any nonconstant function of pairs of distinct points $P,Q\in\mathbb{R}^2$ or of triples of distinct non collinear points $P,Q,R\in\mathbb{R}^2$ ...
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### Matrix representations of Transformation with change of basis (Fraleigh Beauregard)

I'm having problems understanding section 7.2 of FB's Linear Algebra, 3rd edition, and I can't find the solution online since no specific name is given to the matrices. Sorry for the long explanation,...
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### Impact of the transformation matrix distribution on linear transformation

Let $X$ be a $m\times n$ ($m$: number of records, and $n$: number of attributes) normalized dataset (between $0$ and $1$). Denote $Y=XR$, where $R$ is an $n\times p$ matrix, and $p<n$. I understand ...
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### Find the basis of a transformation matrix for an endomorphism

I have a 3x3 transformation matrix $D_{BB} (f)$ with $B$ as a basis of vector space $V$ and $f$ as a diagonalizeable endomorphism $f : V \to V$ given. Basis $B$ is not explicitly given. The entries of ...
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### Fourier Transform of rational function

So I have this function: $$f(t)=\frac{1}{(1-it)^{n+1}}$$ And I have the Fourier Transform defined as $$\hat{f}(\lambda)=\frac{1}{\sqrt{2\pi}}\int_\mathbb{R}f(t)e^{-\lambda.it}dt$$ Now my ...
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### How does a cropping of a 2D matrix/image affect its DCT transform?

I apologize in advance: since I am not a mathematician, maybe my question is not well defined, but I hope that some of you will still understand my meaning. Given a 2D matrix, or an image of ...
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### Rotation operator for a point in a coordinate system linearly derived from Cartesian coordinates

For some experimental and practical reason, I have created a new coordinate system in the form $$x^\prime_i=T_{ij}x_j$$ where $T_{ij}$ isn't a square matrix. $x_i$ is standard Cartesian coordinates, ...
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### Rigid Deformation

I'm trying to parse through this paper on using the method of moving least squares for rigid transformations - http://www.cs.rice.edu/~jwarren/research/mls.pdf Under section 2.3, the author mentions ...
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### Transformation Matrix of a function

I have the following: (Note: $V^{*}$ is defined as: $V^{*} = \{ L: V \rightarrow \mathbb{R} | \text{L is linear} \}$) Let $V$ be an $\mathbb{R}$-Vectorspace. Let $\phi \in V^{*} \text{ \ } \{0 \}$ ...
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### scale transformation is invariant for H_1

Consider the subspace $H_1$ of $C_0(0,\infty)$, where $\phi=\int_0^t\dot{\phi}(s)ds$ and $\int_0^{\infty}{\dot{\phi}}^2ds<\infty$. The transformation is $(T\phi)(t)=t\phi(\frac{1}{t})$. How to ...
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### Which, if any, of the following polynomials are in Range(t)?

Let T: P^2 ----> P^2 be a linear transformation defined by T(p(x)) = xp'(x) (i) 2 (ii) x^2 (iii)1-x I was hoping someone would show me how to find the range of one of them so I know how to do the ...
Let B be an element of $R^{n \times n}$ and define $T(A) = BAB$ for all $A \in R^{n \times n}$. Show that T is a linear transformation. I am completely lost and I do not know how to start this.