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1answer
50 views

Jacobian of a transformation in cylindrical coordinates

In an area called transformation optics, they transform Maxwell equations from one space coordinate system to another, and then somehow obtain the properties of background material $(\epsilon , \mu)$ ...
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24 views

Transformation matrices and hermitian/unitary/normal/… matrices

I need some help with the following - have I done the correct things or how can I solve the task? Let $f \in End(V)$, V a unitary space $\mathbb{C}^3$ given by: $A_{\alpha \beta} (f) = \frac{1}{7} ...
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0answers
16 views

PDE equation conversion to parabolic PDE problem

Hello guys I am new here and I accept a serious problem with my exercising. Pronunciation says that I have to transform the dependent variable of this equation $\frac{\partial V}{\partial ...
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1answer
14 views

Show there's no ordered basis $E$ with the following conditions

Let $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$ such that: $$T\left( {\matrix{ x \cr y \cr } } \right) = \left( {\matrix{ 2 & 1 \cr 3 & 4 \cr } } \right)\left( {\matrix{ ...
1
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1answer
26 views

Equation of matrices

Let $V$, a 3d vector space above $F$. Let $T:V\rightarrow V$, linear transformation and $E$, an "ordered" basis such that: $$[ T ]_E = \left( \matrix{ 0 & 0 & a \cr 1 & 0 & ...
2
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0answers
15 views

Finding the matrix ${\left[ T \right]_E}$

Let the matrix ${\left[ T \right]_{B \to E}}$, the matrix where: $${\left[ T \right]_{B \to E}}{\left[ v \right]_E} = {\left[ {T(v)} \right]_B}$$ It's given that: $${\left[ T \right]_{B \to E}} = ...
2
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2answers
32 views

Inverse Z-tranform with a complex root

The z-transform of a signal is $$ X(z)=\frac{1}{z^2+z+1}$$ I attempted to solve for the the inverse z-transform by decomposing the denominator into complex roots, $\alpha$ and $\alpha^\ast$, to get ...
2
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0answers
72 views

Finding transformation from $T : \Bbb R^5 \rightarrow \Bbb R^4 $ …

Is there a Linear Transformation from $T : \Bbb R^5 \rightarrow \Bbb R^4 $ so $$\operatorname{Ker}T = \{( x,y,z,t,w) \in \Bbb R^5 \; | \; x = 2y, \text{ and, } z = 2t = 3w\}$$ if so find an example of ...
1
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1answer
52 views

Calculating the adjustment translation to be applied after rotating and scaling so that operations pivot about a given point.

I have a matrix for transforming an image into a target frame. The matrix is a function of a scale, $s$ rotation angle, $\theta$, and a translation that is applied after rotating, $tx, ty$. The ...
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0answers
12 views

When is $c_1 \cdot f(g(x+c_2)) = f'(x)g(x)$?

We are allowed to pick and $c_1, c_2$ that helps make this question easier. So when is $$c_1 \cdot f(g(x+c_2)) = f'(x)g(x) \tag{1}$$ Also, separately, I'm wondering: $$c_1 \cdot f(g(x+c_2)) = ...
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4answers
200 views

What functions have the property that $\frac{d}{dx}f(x) = c \cdot f(x+1)$?

If we are allowed to pick any real-valued constant $c$ that helps, when does $$\frac{d}{dx}f(x) = c \cdot f(x+1)$$ In other words, when does the derivative of a function $f(x)$ equal some constant ...
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0answers
18 views

PCA - How to calculate the scores

I'm currently learning Principle component analysis and I have, so far calculated the Eigen values and vectors. Assume that I have the following: $$ E = \begin{pmatrix} 1 & 2\\ 3& 4 ...
1
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0answers
21 views

Linear Transformation - linear algebra question [duplicate]

$T:\mathbb{R}_2[x] \mapsto \mathbb{R}_2[x]$ s.t.: $$ \begin{array}{l} T(1) = 3+2x+4x^2, \\ T(x) = 2+2x^2, \\ T(x^2) = 4+2x+3x^2. \end{array} $$ Is there base $B$ of $\mathbb{R}_2[x]$ that $[T]_B = ...
1
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1answer
41 views

Impulse response and z transform question?

We have $g(k)=\{ [(1/5)^k]u(k)\text{ for $1 \le k\le3$ and $0$ for other }k\}$ The input is $x(k)=\delta(k) +3\delta(k-1)+ \delta(k-2) $ Using Z transform we have to find the output $y(k)$ and the ...
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1answer
52 views

What's the pdf of $Z=X^2 +2X$ if $X$ is a standard normal? [closed]

Le be $X$ distributed as a standard normal. What is the density function of $Z=X^2 +2X$? Thanks for your help
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2answers
38 views

Transformation of the points on a plane

How do I transform a point $(x,y,z)$ on plane $\Pi (ax + by + cz = 0)$ to a point $(x',y',z')$ on plane $\Phi(ax+by+cz+d=0)$? What matrix should I use? Here is a 2-D representation of what I'm ...
0
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1answer
21 views

Find linear map of a transformation without neither known transformed or transformation mitrix

Consider the linear map from $R^3 \rightarrow R^3$ which takes $\vec{e_1}$ to $\vec{a_1}=\begin{bmatrix} 1\\0\\-1\end{bmatrix}$, takes $\vec{e_2}$ to $\vec{a_2}=\begin{bmatrix}0\\1\\3\end{bmatrix}$, ...
1
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1answer
45 views

linear transformations with matrices $A, A^*$

Let $K$ be a field, $K\subseteq \Bbb C$. $V$ is a linear space over $K$, $\dim(V)=n(n\geq2)$. Choose ordered basis $\epsilon_1,\epsilon_2,\dotsc,\epsilon_n$ for $V$. $\bf A,B$ are two linear ...
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0answers
15 views

Generate rotations about X & Y axes between certain 3D vectors

Given a semi-arbitrary 3D vector (the z will always be positive for my purposes), how could I find rotation about the X and Y axis? Alternatively, how might I simplify an XYZ rotation to the X and Y ...
0
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1answer
35 views

Transforming a curve on an arc to a line

I have a function, actually a point cloud, (similar to a sine wave) on an arc with a known radius of curvature. I need to remove the curvature to regenerate the original function (or point cloud). ...
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1answer
20 views

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 ...
2
votes
1answer
39 views

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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0answers
11 views

New Axis calculation

I have a task whereby I use an accelerometer to calculate acceleration for a vehicle. The problem I am attempting to solve is to allow the accelerometer to be in any oriertaion. Basically I have a ...
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0answers
36 views

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 ...
1
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1answer
49 views

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 ...
0
votes
2answers
30 views

Function tranlsation $g(x) = f(x) + 15$

I can't seem to work this answer out when practicing for exams. Here's the question: You are given that $f(x) = (2x - 3)(x + 2)(x + 4) \cdots$ From this I know $f(x)$'s roots: $\frac{3}{2}$, ...
1
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1answer
40 views

matrix representation of linear transformation

For a set $N$ let $id_N:N \rightarrow N$ be the identical transformation. Be $V:=\mathbb{R}[t]_{\le d}$. Determine the matrix representation $A:=M_B^A(id_V)$ of $id_V$ regarding to the basis ...
1
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1answer
35 views

Problem with linear transformation from $\mathbb{R}^2$ to $M_{2\times 2}$

I'm trying to solve this problem, but at the end I find something's wrong with my work. Here is the problem: We're given the bases: $$ \beta = \bigg\{\begin{pmatrix}1\\1\end{pmatrix} ...
0
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2answers
21 views

Transformations on the complex plane

I'm trying to work out what the transformation $T:z \rightarrow -\frac{1}{z}$ does (eg reflection in a line, rotation around a point etc). Any help on how to do this would be greatly appreciated! I've ...
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0answers
9 views

Dual representations of affine span of a set of affine transformers.

I am not well versed in the literature of affine transformers or Farkas' Lemma. I just know the basics of the two concepts. Are there any dual representation of affine span of a set of affine ...
2
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2answers
58 views

Inverse Laplace Through Complex Roots

I have been asked to apply inverse laplace to this: $$ \frac{(4s+5)}{s^2 + 5s +18.5} $$ What I have done is; I found the roots of denominator which are : $$ (-5-7i)/2 $$ and $$ (-5+7i)/2 $$ Then I ...
0
votes
1answer
67 views

If $x$ belongs $V$ then $Tw = w$ if and only if $x = v + k$ with $k$ in $\operatorname{ker}(T)$ [closed]

Let $F$ a field, $V$ and $W$ vector spaces, $T$ a linear transformation from $V$ to $W$, if $w$ belongs $W$ and $v$ in $V$ such that $Tv = w$, if $x$ belongs $V$ then $Tx = w$ if and only if $x = v + ...
0
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1answer
40 views

Compute the transformation in the given Basis

I forgot how to compute the transformation in a given basis. :'( For example, say I have the transformation \begin{equation}(a, b) \mapsto \begin{bmatrix}10a - 6b \\ 17b - 10b ...
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0answers
46 views

Z - transform of a transfer function

I have to apply a z-transformation to my transfer function which looks like this: $$\frac{K}{s} - \frac{K\cdot T}{T\cdot s}+1$$ I have tried it and this is my result: $$K \cdot \frac{z}{z-1} - K ...
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0answers
22 views

3D cartesian cordinates to 3D isometric cordinates

I am working on Isomer opensource project https://github.com/jdan/isomer and I need to convert 3D cartesian cordinates to 3D isometric cordinates. We already have 3D iso to 2D cartesian ...
0
votes
1answer
12 views

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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0answers
24 views

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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2answers
69 views

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 ...
1
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0answers
38 views

Using the Modulation property of the Fourier Transform

I'm working on a problem: Let $X(w)$ be the Fourier transform of $x(t)$. Find the transform of $y(t)=x(5t+3)\sin(2t)$ in terms of X(w). I am table to take the Fourier transform of $x(5t+3)$ and ...
0
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1answer
34 views

How can I convert an n-dimensional vector to a 2d point?

I have a n-dimensional vector / sequence of values, how can I convert it to a 2D representation of such vector? Follow-up: if I had a time-sequence in which every frame is n-dimensional, how can I ...
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1answer
29 views

The linear map $ T: \mathbb R^3{\rightarrow} \mathbb R^3$ with given matrix is a rotation about some line. Find the line.

Finals studying continued. $ T: \mathbb R^3{\rightarrow} \mathbb R^3$ with matrix $$A= \begin{pmatrix} -2/7 & 6/7 & 3/7 \\ 3/7 & -2/7 & 6/7 \\ 6/7 & 3/7 & -2/7 \\ ...
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0answers
19 views

Differential probabilities of transformed random variables

Let $X$ be a random variable with probability density function (pdf) $f(x)$ and let $Y = h(X)$ be a transformation on rv $X$. While calculating the pdf of $Y$, it is assumed that $Prob(x \le X \le x + ...
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0answers
26 views

Complex Variables Conformal Mapping in Complex Plane of harmonic Functions

Consider the harmonic function $u(x,y) = 1 - y + x/(x^2+y^2)$ on the upper half plane $y > 0$. What is the corresponding harmonic function on the first quadrant $x>0$, $y>0$, under the ...
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0answers
55 views

From inverse Weierstrass function to Jacobi elliptic/inverse elliptic functions?

As a conclusion to a previous question on integrals, I get an answer in terms of inverse Weierstrass elliptic function : $$ f\left(x\right)=\wp^{-1}\left( \beta + \frac{9\beta^2-1}{3(x-\beta)} \right) ...
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1answer
25 views

Finding an image (Linear Transformations) $T(2,-1,1)$

Let $T:R^3 \rightarrow R^3$ be a linear transformation such that $T(1,1,1) = (2,0,-1), T(0,-1,2) = (-3,2,-1)$ and $T(1,0,1)=(1,1,0)$. Find the indicated image $T(2,-1,1)$ I used the rule that: ...
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0answers
51 views

Help determining whether a function is a linear transformation $T:M_{2,2}\rightarrow R, T(A)=|A|$

Again, here is the function: $T:M_{2,2}\rightarrow R, T(A)=|A|$ I was able to prove that its not a linear transformation because $T(A+B) \neq T(A)+T(B)$ in fact, $T(A+B) = C$ where $C$ is a new ...
0
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1answer
14 views

Find linear transformation of a given matrix of linear transformation

I have two basis. The first is a $R2$ basis ${(1,0) , (0,2)}$. Lets call it basis of $U$. The second is a $R3$ basis ${(1,0,-1), (0,1,2), (1,2,0)}$ Lets call it basis of $V$. Is given a matrix of a ...
2
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1answer
65 views

Meaning of entries of a transformation matrix in practical terms [Homework related]

I'm having a bit of trouble understanding what the matrix entries mean practically in this problem: 100 kg of a highly toxic substance is spilled into three lakes. The state, t weeks after the ...
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0answers
32 views

Making a change of variable to transform an equation

$\displaystyle m\frac {dv} {dt} = mg - kv^2$ $\displaystyle\frac {dV}{dT} = 1 - V^2$ Make a change of variable $v=aV$ and $t=bT$, show that for suitable choices of the parameters $a>0$ and ...
0
votes
1answer
29 views

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 ...