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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Prove polynomial transformation is linear

Suppose a polynomial transformation: How do I prove the "closed under addition" property of linearity? I am trying this: I try to expand the equation on the left hand side, but I don't get ...
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Fourier transform of real part part of a complex signal

Let $$g(t) = g_r(t) + j g_i(t)$$ be a complex signal with $g_r(t)$ and $g_i(t)$ real signals. We can write the Fourier transform $$G(f) = G_r(f) + jG_i(f),$$ where $G_r(f)$, $G_i(f)$ and $G(f)$ are ...
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sum of singular vector dyadics derived from the matrix itself

I have an m*n (m>n) n-rank matrix (let's denote it by A), with nonnegative elements. SVD decomposition says, that A=UDV', where U and V are orthogonal matrixes, and their columns are the singular ...
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Transform and domain such that phase shift of the original function results in shift in transformed domain variable

It is known that the Fourier transform of a phase-shifted function results in a constant shift of the dependent variable of the phase spectrum: If $F(x(t)) = X(w) = |A(w)| \cdot e^{-i \cdot \phi(w)}$...
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Relative Motion between two rotating frames

I am looking for mathematical relations between equations of motion between two rotating frames. Relating the said motion by first going to a globally fixed frame is not an issue but how to approach ...
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Scale a rectangle about a point considering reflection

Given a rectangle of width $(w_0)$, height $(h_0)$, left $(x_0)$, top $(y_0)$. How do I scale it from an origin $(x_1,y_1)$ with a scale factor of $(w_1, h_1)$ taking into account reflection? This ...
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Find equation of log function given graph

How would you find the equation of this log graph with respect to transformation of the function $$a\log[k(x-d)] + c$$ I had trouble solving this accurately.
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Certain symmetrized product of cosines - can it be transformed into more manageable form

I am interested in the following expression: $$F_{k_1,\ldots,k_n}(t):=\sum_{\sigma\in S_n}\cos(\sigma(1)k_1t)\cos(\sigma(2)k_2t)\cdots\cos(\sigma(n)k_nt)$$ where $k_1, \ldots, k_n$ are natural ...
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Fast procedure to know which transformations are linear

I have the following transformations: How can I tell if the transformation is linear? Is there a fast way of knowing (by looking at equations)? or do I need to check through the theorems: ...
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Find and sketch the image of the straight line $z = (1+ia)t+ib$ under the map $w=e^{z}$

I need to find and sketch the image of the straight line $z = (1+ia)t +aib$, where $-\infty < t < + \infty$, $a,b\in \mathbb{R}$, and $a \neq 0$, under the map $w = e^{z}$. In order to ...
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Going from a point in a sphere from a point in an ellipsoid

Given a certain vector $v$ on the surface of a sphere centered at $0$, I'm trying to find another vector $w$ such that $w$ and $v$ are colinear and $w$ is on the surface of an ellipsoid also centered ...
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How do projections in 3D with homogeneous coordinates work?

Affine 3D transformations can be expressed in homogeneous coordinates by a matrix $M \in \mathbb{R}^{4 \times 4}$. This means we have 16 parameters to calculate. The first thing I asked myself is how ...
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Problem finding Jacobian when computing density

I'm having trouble finding the Jacobian when trying to compute a distribution. If $(X,Y)$ is a point on a unit disk with radius $1$, I'd like to find the density of the distance between the point and ...
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Explain Similarity Transformation

For two n-by-n matrices $X$ and $Y$, we know that they are similar if the following is true for some invertible n-by-n matrix $M$: $X = M^{-1}YM$. Can anyone explain what the $M$ and $M^{-1}$ are ...
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How do I graph these curves?

My teacher taught me this in class but I still don't understand it. Could someone please explain how to graph $y= x +\frac1x$ and $y = x - \frac1x$? Thank you :)