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1answer
50 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 ...
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0answers
38 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 ...
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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
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} ...
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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 ...
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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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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 + ...
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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 ...
0
votes
1answer
80 views

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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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 ...
1
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2answers
152 views

Combine a rotation matrix with transformation matrix in 3D (column-major style)

I'm trying to concatenate a rotation matrix with a 4x4 homogeneous transformation matrix with column-major convention. I want this rotation matrix to perform a rotation about the X axis (or YZ plane) ...
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0answers
39 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
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 \\ ...
0
votes
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 ...
0
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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 + ...
1
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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 ...
1
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2answers
79 views

Integral transform with Dirac delta

Let $f,g: \mathbb{R}^n \to \mathbb{R}$. Let $\delta$ denote the Dirac delta function. How can I write the integral over $\mathbb{R}^n$ (on the left hand side) as an integral over $g^{-1}(0)$ $$ ...
2
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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) ...
0
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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: ...
0
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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
votes
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
votes
1answer
66 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
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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 ...
0
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1answer
17 views

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 ...
0
votes
1answer
90 views

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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0answers
17 views

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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2answers
47 views

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 ...
3
votes
1answer
69 views

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 ...
0
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2answers
35 views

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 ...
1
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2answers
127 views

Transformation matrix from quadrilateral to rectangle

There exists a rectangle somewhere in space with some orientation. A camera from the coordinate center point is looking along the z axis and is seeing the rectangle as a quadrilateral (due to ...
0
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0answers
33 views

How to calculate the combined frequencies of a DCT matrix?

Given a 2D matrix of dimensions w1,h1. I preform a DCT 2D transform on the matrix (DCT = DCT type 2). I get a 2D result matrix. This matrix has two frequency axes - x,y (which are simply the ...
0
votes
1answer
19 views

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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0answers
14 views

Transformations on a $4\times4$ matrix

Let's say a company produces chips containing $16$ elements, ordered in a $4\times 4$-matrix: $$\begin{bmatrix} \hline ...
0
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0answers
29 views

Determine if the following linear transformation is surjective or injective

Let $S \left(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}\right) = $ $\begin{pmatrix} x_1 & -2x_2 & x_3 & x_4\\ 2x_1 & - 4x_2 & -3x_3 & ...
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1answer
46 views

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 ...
0
votes
1answer
11 views

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 ...
0
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1answer
73 views

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 ...
0
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0answers
39 views

Using basic transformations to derive matrix for the reflection in a line?

Using basic transformations (translation, scaling and rotations), show all the steps to derive the transformation matrix for the reflection of points n the line : y = 3 - x I know that a directional ...
2
votes
1answer
80 views

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 ...
0
votes
1answer
18 views

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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0answers
15 views

About representations and transformations under an $SU(n)$ Lie Group

I think my problem is that I misunderstand what "transforms under" really means. Let's take $SU(3)$, for the $\mathbf{3}$ with Dynkin indices $(1,0)$, a state transforms like : $ψ→gψ$. For the ...
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2answers
5k views

Value range of normalization methods? min-max, z-score, decimal scaling

I am working my way through Normalization (data transformation) of data and was curious about four methods: min-max normalization, 2. z-score, 3. z-score mean absolute deviation, and 4. decimal ...
1
vote
1answer
112 views

Joint density of two functions of random variable

This is online homework, and I'm not always clear on which chapter questions are from, so I might be completely off base. I have two random variables, $X_1$~UNI(5,10) and $X_2$~UNI(4,10), and then ...
0
votes
3answers
55 views

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation. Find $T(x)$

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation with $T \left(\begin{bmatrix} 1 \\ -2 \\ -1 \\ \end{bmatrix}\right) = \begin{bmatrix} 1 \\ -1 \\ 2 \\ ...
1
vote
1answer
34 views

Linear Transformations, kernels, and Basis for Range

I have the following question and am completely lost on how to start: Let $T: P_{2} \to P_{3}$ be the linear transformation $[T(p)](x) = p^{\prime}(x) + xp(x)$ Find $\ker(T)$ and find a basis for ...