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56 views

Help in implementing of peak function in Fourier transform

I have a function Peak function I know how to implement it in time range just need to caclulate $r$. first I initial $x$, y with a range and meshgrid them, after it calculate $r$ ...
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1answer
92 views

Fourier Transform - Time Shift

Could someone help me understand how a simultaneous time shift on two separate functions is possible? I am having trouble linking a property to this solution. Given the function: $$x(t) = ...
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2answers
413 views

Matrix representation of the dual space

Let $V$ be an $n$-dimensional vector space over $F$, with basis $\mathcal{B} = \{\mathbf{v_1, \cdots, v_n}\}$. Let $\mathcal{B}^{*} = \{\phi_1, \cdots, \phi_n\}$ be the dual basis for $V^{*}$. Let ...
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4answers
104 views

Linear map between duals induced by linear maps between vector spaces

Let $V, W$ be vector spaces over a field $F$ and let $\psi: V \to W$. Show that $\psi$ induces a linear map $\psi^{*}: W^{*} \to V^{*}$ naturally. Although the question asks for a naturally induced ...
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3answers
53 views

X has pdf $f(x) = \frac{x^{2}}{18}$ for -3<x<3, what is the pdf of $X^{2}$

So this was my solution: Say, $Z = X^{2}$, then $X=\pm \sqrt{Z}$ and, $$P(Z=z)=P(X = \sqrt{z}) + P(X = -\sqrt{z}) = \frac{z}{18} + \frac{z}{18} = \frac{z}{9}$$ for $$0<z<9$$ However: ...
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0answers
126 views

Proof of identity of magnitude of rational function in z-domain

In the set of rational polynomial functions $H(z)$ of a complex number $z$, there exist functions whose magnitude $|H(z)|^2$ is a constant $C$, but whose denominator and numerator are not constants. ...
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2answers
107 views

Basic questions regarding matrix algebra.

I had two true/false questions on my exam of which I missed. $1)$ The map $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by $T(x)=x+e_1$ is a linear transformation. I know this to be false, because ...
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1answer
92 views

Set with plane having two axes of symmetry with angle of intersection

Proof that a closed and compact subset of plane having two axes of symmetry with angle of intersection not a rational multiple of $\pi$ is either a disc or a whole plane. Proof The composite of two ...
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1answer
193 views

transformation between square and a polygon?

I have a square and a polygon. I want to transform all the points inside this square such that they are mapped inside the polygon. I was trying using scale and rotate matrices but I am not able to ...
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0answers
49 views

Is there an intuitive understanding of what a walsh coefficient is?

I am working with Walsh coefficients. I know the intuitive understanding is almost that that they are the degree of connectivity, but it is there a better way of thinking about it? What is the ...
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2answers
421 views

How to compute a matrix for rotating and centering rectangle in viewport?

I have a rectangle given by 4 points. I'm trying to compute a transformation matrix such that the rectangle will appear straight and centered within my viewport. I'm not even sure where to begin. ...
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1answer
2k views

How to find a transformation matrix having several original points and their respective transformed results?

I have three original points $pt_1, pt_2, pt_3$ which if transformed by an unknown matrix $M$ turn into points $gd_1, gd_2, gd_3$ respectively. How can I find the matrix $M$ (all points are in ...
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1answer
81 views

The Legendre Transform of Bernoulli r.v.s

This question is most likely related to calculus/algebra and tricks regarding supremums rather than actual understanding of large deviations and the Legendre transform, but anyway. For a random ...
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1answer
53 views

Transpose of 2 matrices together

So if I have an $m\times n$ matrix $A$ and I represent that matrix as $\displaystyle A = QR$, how do I write $A^{T}$ (transpose) in terms of the original $\displaystyle QR$? Does it become ...
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0answers
114 views

Proof for a summation-procedure using the matrix of Eulerian numbers?

I've discussed a procedure for divergent summation using the matrix of Eulerian numbers occasionally in the last years (initially here, and here in MSE and MO but not in that generality and thus(?) ...
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3answers
76 views

A simple question on linear algebra and linear transformations

Let $f = (f_1, \cdots, f_m)$ be a function from $\mathbb{R}^n \to \mathbb{R}^m$. Prove that $f$ is linear if and only if for each $i$, $f_i$ is of the form $$f_i (x_1, \cdots, x_n) = a_1x_1 + ...
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1answer
62 views

How to transform this matrix & swap its columns?

I'm looking for a transformation matrix (or set of transformation matrices) that transforms matrix $\mathbf A = \begin{pmatrix} a&b&i&j\\ c&d&k&l \\ e&f&m&n \\ ...
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1answer
98 views

A simple question on linear transformations between vector spaces

Let $\phi: V \to W$ a linear transformation between vector spaces and $v_1, \cdots, v_k \in V$. Suppose that the following condition is fulfilled: $$\phi \left (\sum_{n=1}^{k}c_n\mathbf{v_n} \right) ...
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1answer
181 views

How to “flip” and change the sign of one particular row of this matrix?

I would like to transform the following matrix : $\mathbf A$ =$\ \begin{bmatrix} a&b\\ c&d\\ e&f\\ g&h \end{bmatrix}\ $ into this one : $\mathbf B$ = $\ \begin{bmatrix} g&-h\\ ...
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2answers
2k views

RYB and RGB color space conversion

I am working on a project where I need to convert colors defined in RGB (Red, Green, Blue) color space to RYB (Red Yellow Blue). I managed to solve converting a color from RYB to RGB space based on ...
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1answer
71 views

Define the linear transformation TA(v) = A(v) where (v) is the co-ordinate vector

I'm having some problems with this question. Could someone point me in the right direction? Thanks
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0answers
141 views

Linear transformation from P4 to M2x2

Can someone verify the formula in the last line. It seems that the LHS is not equal to the RHS The linear transformation of P4 ---> M2x2 is given by $$ T(ax^4+bx^3+cx^2+dx+e) = $$ $$ ...
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2answers
45 views

An explanation about terminology in vector spaces

Call a linear transformation $\rho: V \to V$ ($V$ is a vector space) idempotent if $\rho^2 = \rho$. Prove that if $\rho$ is idempotent, then it acts as the identity on $\rho(V)$. If I understand the ...
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2answers
82 views

Meaning of $p(\phi)$ where $\phi (x,y) = (x+y, x- 2y)$ and $p(x) = x^2 -2x + 1$

Consider the linear transformation $\phi : \mathbb{R}^2 \to \mathbb{R}^2$ defined by $\phi (x,y) = (x+y, x- 2y)$. Let $p(x) = x^2 -2x + 1$. Does $p(\phi)$ make sense and if yes what is it?
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1answer
22 views

Rescale a function

Lets assume that $A_1$ a function with maximal value $M$ and minimal value $m$ with $M \neq m $ How can i find the transformation that maps this function to $A_2$ with $N$ as new maximal value and ...
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1answer
333 views

Are the following transformations linear?

I'm preparing for my exam and I am stuck at these two exercises in which I must prove that the given transformatios are linear. I know that a transformation is linear, if it's closed under adition and ...
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3answers
1k views

Find the spanning set of the range of the linear transformation $T(x)=Ax$.

Let $$ A= \begin{bmatrix} -4 & -4 & 12 & 0 \\ -4 & -4 & 12 & 0 \\ 4 & -2 & 0 &-6 \\ 1 &-4 &7 &-5 \\ ...
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2answers
191 views

Standard Basis of the Finite Field of Prime Numbers

A little info regarding this field: Addition and multiplication in $Z^n_p$ behave as usual but with the remainder taken upon division by $p$. Ex: $Z_3$ will only consist of the three integers ...
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1answer
133 views

How to convert a sequence of numbers in the formula?

I'm trying to understand different sorting algorithms and their BigO notation. Suppose, I'm using insertion sort and I have the worst case: 6 | 5 | 4 | 3 | 2 | 1 ...
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1answer
104 views

Question about special orthogonal Lie group construction

Working through homework and I run into this problem: Suppose the Lie group $SO^{+}(2,2)$ is presented as the group of all transformations in its associated space. How do you determine whether a ...
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2answers
1k views

Diff eq. transformation polar coordinates

I have $(x',y')=(x-y-x(x^2+y^2)+\frac{xy}{\sqrt{x^2+y^2}},x+y-y(x^2+y^2)-\frac{x^2}{\sqrt{x^2+y^2}} )$ Now I want to use polar coordinates $(x,y)=(r\cos(t),r\sin(t))$ to get ...
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0answers
304 views

Graph Transformations Order of Operations

I'm a little confused regarding order of operations for shifted graph functions. For part b, which one is correct, and why? Shifting two to the left, and flipped over the x-axis, or flipped over ...
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1answer
113 views

Householder reflections

Let $x=\begin{bmatrix} 1\\ 2\\ 3 \end{bmatrix}$ I want to use a Householder reflector U to keep only first element in vector x, and make everything else zero but I'm doing something wrong... ...
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0answers
50 views

distance to sheared right circular frustum

How do I calculate the distance to a sheared right-circular frustum? In particular, I'm shearing in a direction perpendicular to the axis, so the cross sections remain parallel circles. I know I can ...
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1answer
535 views

Causal Inverse Z-Transform of Fibonacci

Say the Fibonacci sequence is defined by: $y(n) = y(n-1) + y(n-2)$ initial conditions: $y(0)=0, y(1)=1$ I incorporate those initial conditions as: $y(n) = y(n-1) + y(n-2) + \delta(n-1)$ ...
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0answers
129 views

question about coordinate transformation

In isoparameteric finite element of second order tetrahedron element, the original coordinate $(x,y,z)$ would be transformed to standard coordinate $(\xi, \eta, \zeta)$ by a shape function. As a ...
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43 views

Further Properties of Adjoints

I recently posted a question here about adjoint maps. I now want to take this further and show that - with the assumptions as in the original - the minimum polynomials of $T$ and $T^{*}$ are the same. ...
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1answer
96 views

Properties of Adjoints

Suppose we have a linear transformation $ T $ on some real inner product space $ V $, with adjoint $ T^{*} $. How can we go about showing that $$ (T^{n})^{*} = (T^{*})^{n} $$ for a positive integer ...
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3answers
515 views

Rank of matrix AB when A and B have full rank

Define $A$ as $m\times n$ matrix with rank $n$, and $B$ as $n\times p$ matrix with rank $p$. Calculate the rank of matrix $C=AB$. --edit-- Rank of a matrix is the number of linear independent rows.
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1answer
130 views

If $AB=0$, then $A+A^T$ or $B+B^T$ is singular

Define $A$ and $B$ as being square matrices of dimension $2011$. Prove that if $AB=0$, then at least one of matrices $A+A^{T}$ or $B+B^{T}$ have rank below $2011$. -- edit -- Rank of a matrix is ...
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1answer
92 views

Region of convergence of Z-Transform connected area?

Shouldn't the Region of Convergence of the Z transform be a connected area ? In Oppenheim solution manual, I've found this answer of a question that asks to determine the different forms of the ...
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2answers
150 views

Finding if the equation is even or odd

I am learning fourier transform and I came across this question in which author right away says the given equation is "even". How does this equation become "even"? $$x[n]=\begin{cases}A & -M\le ...
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1answer
167 views

Fourier Transform of an Operator

I need to calculate the fourier transform of an Operator. meaning I need to calculate the transform of the Operator's corresponding convolution kernel. so the question is: 1.given a 2d fourier ...
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1answer
206 views

How to deal with non random data in statistical analysis?

I have a set of monthly water quality data, and I want to use them in a few statistical analysis (such as finding distribution or using in copula models) which require random variables as input. I ...
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3answers
321 views

Image of function definition notation

In my Linear Algebra and Geometry textbook, it defines the image of a linear transformation $T$ as: $$\operatorname{Im}\, (T) := \{\; w \in W : \; w=Tv \;\;\text{ for some } v \in V \} $$ As far as ...
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99 views

what's the difference between Cohen-Daubechies-Fauraue 9/7 transformation and Discrete cosine transform?

can someone explain, what's the difference between this 2 tranformations? DCT and CDF Greetings
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0answers
212 views

Multiple integral variable substitution using Jacobian matrix and matrix rotations

Question: By an appropriate choice of new variables evaluate the integral ${\int\int}_R(x^2+y^2)dxdy$ over the interior of the square bounded by $y=\pm x$ and $y=\pm (x-2)$. I sketched the square ...
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3answers
165 views

Find the necessary and sufficient conditions on $A$ such that $\|T(\vec{x})\|=|\det A|\cdot\|\vec{x}\|$ for all $\vec{x}$.

Consider the mapping $T:\mathbb{R}^n\mapsto\mathbb{R}^n$ defined by $T(\vec{x})=A\vec{x}$ where $A$ is a $n\times n$ matrix. Find the necessary and sufficient conditions on $A$ such that ...
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2answers
561 views

linear transformation and angles?

Does linear transformation, prevent preserve angle between two vectors? I guess that it is true, so if we translate normal vector of a plan, it will be orthogonal to translated plan.
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2answers
451 views

Kernel density estimation for heavy-tailed distributions using the champernowne transformation

I am trying to follow this paper to estimate the density for a heavy-tailed distributions using the champernowne transformation. Alternative link to the paper Another alternative link to the paper ...