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3
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1answer
104 views

Why is it called the “Unscented” transform?

I have not been able to track down the reason the Unscented Transform has the name it has. Can anyone shed some light on the meaning of the term "unscented" in this context?
7
votes
0answers
173 views

Way to Tietze's Transformation Theorem

during our knot-theory lecture we have talking about the following theorem: Given two finite presentations of the same group, one can be obtained from the other by a finite sequence of Tietze ...
0
votes
1answer
177 views

What is the sense of bottom row of affine transform matrix?

Usually affine transform matrix (in 2D) is represented like where block A is responsible for linear transformation (no translation) and block ...
1
vote
0answers
49 views

Linear 2D transform in the sense of geometric figures?

Consider tranformation which turns one aligned rectangle to another: This tranformation can be written in matrix form in the following way where ...
4
votes
0answers
219 views

Geometric intuition for Jordan normal forms (invariant subspaces, shearing, scaling, etc.)

I'm trying to visualize what a linear operator does to a vector space if that operator can be put into Jordan normal form. For concrete motivation, let's take $V = \mathbb{R}^3$, with some linear ...
0
votes
1answer
63 views

Finding $u$ and $v$ in Jacobian substitutions

I've used Jacobians before in multivariable calculus to simplify integrals, but I'm lost when I need to find the substitutions myself. Today on the quiz, there was the problem $\int\int_{R} xy dxdy$ ...
1
vote
1answer
138 views

Interpret the graph of $\frac{ax+b}{cx+d}$ as a transformation of $y=\frac{1}{x}$

As part of a problem-set I'm self-studying, I'm trying to interpret the graph of $f(x)=\frac{ax+b}{cx+d}$ as a transformation of the graph of $y=\frac{1}{x}$, including determining what restrictions ...
1
vote
1answer
126 views

Local axis follows origin node rotation

I'd like to define a local axis (unit vectors l, m and n) which once defined follow the rotation of the origin node, i.e. regardless of the deformation the local axis should be basically the same as ...
1
vote
1answer
127 views

How to calculate a double integral over a triangle by transforming to polair coordinates & by using a transformation

Let T be the triangel with vetrices $( 0,0 ) , ( 1,0 )\mbox{ and } ( 0,1 ) $. Evaluate the integral : $$ \iint_D e^{\frac{y-x}{y+x}} $$ a) by transforming to polar coordinates b) by using the ...
1
vote
2answers
60 views

Invertability of a linear transformation

Given $T : \mathbb{R}^3 \to \mathbb{R}^3$ such that $T(x_1,x_2,x_3) = (3x_1,x_1-x_2,2x_1+x_2+x_3)$ Show that $(T^2-I)(T-3I) = 0$. Solution 1: I can very easily write down the matrix representing $T$, ...
2
votes
1answer
136 views

Transforming matrix-equation to overdetermined minimum problem

i have broken down my problem to plainmath and could really use some help. Basis: I have an image. In this image I have several UV-XYZ pairs. So i know the 3d position of serveral Pixels. Given the ...
1
vote
1answer
128 views

Change of coordinate matrices

Find the change of coordinate matrices: Wherein B is the standard basis for P2 $$B' = (t^2+2,t+3,t^2+t+1) \\B" = (2t^2+t+1, t^2, 2t+1) \\ B= (t^2,t,1) $$ $$P_{B'B}$$ means the transformation for the ...
0
votes
1answer
130 views

Transformation of a matrix, change of basis

Find the change of coordinate matrices: Wherein B is the standard basis for P2 $$B' = (t^2+2,t+3,t^2+t+1) \\B" = (2t^2+t+1, t^2, 2t+1) \\ B= (t^2,t,1) $$ $$P_{B'B}$$ means the transformation for the ...
2
votes
2answers
110 views

question on transformation

If a $2$d coordinate transformation function is given by $f(x,y)= 3x+1$, then what does it mean? How do I calculate the transformed coordinates for the points say $(3,4)$ in the initial space?
2
votes
1answer
82 views

Transformations and coordinate Systems

I am working on some practice exercises (not homework) on transformations and need some intuition and help. One of the questions is: $(u,v)=f(x,y)$ where $ \quad u= { e }^{ x }\cos(y), \quad v = { e ...
0
votes
0answers
60 views

transformation function using genetic programming

If I have a set of points in two spaces, say set $A$ contains 50 points and set $B$ contains 50 points. I have to find a transformation function such that if I transform the points in set $A$ using ...
0
votes
1answer
78 views

Linear Transformation Concept Question

Consider that we have a given standard matrix of T and we are asked to find the image T(X) where X is a given vector. T is 4x3 and X is 3x1. is the solution of T(x) simply T*X?
1
vote
1answer
170 views

Regarding the kernel of a linear transformation and that of the associated representing matrix

Let $V, W$ be finite dimensional vector spaces over a field $F$. Let $\mathcal{B}_{V} = \{\mathbf{v_1, \cdots, v_n} \}$ and $\mathcal{B}_{W} = \{\mathbf{w_1, \cdots, w_m} \}$ be corresponding bases. ...
1
vote
0answers
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 $r$ ...
1
vote
1answer
93 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) = ...
1
vote
2answers
421 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 ...
0
votes
4answers
107 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 ...
2
votes
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: ...
0
votes
0answers
127 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. ...
1
vote
2answers
109 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 ...
1
vote
1answer
93 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 ...
0
votes
1answer
195 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 ...
2
votes
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 ...
3
votes
2answers
432 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. ...
1
vote
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 ...
1
vote
1answer
82 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 ...
0
votes
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 ...
2
votes
0answers
115 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(?) ...
0
votes
3answers
77 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 + ...
0
votes
1answer
64 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 \\ ...
1
vote
1answer
99 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) ...
3
votes
1answer
186 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\\ ...
1
vote
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 ...
0
votes
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
0
votes
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 $$ T(ax^4+bx^3+cx^2+dx+e) = $$ $$ ...
0
votes
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 ...
2
votes
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?
0
votes
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 ...
1
vote
1answer
339 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 ...
3
votes
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 \\ ...
0
votes
2answers
192 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 ...
1
vote
1answer
137 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 ...
3
votes
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 ...
1
vote
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 ...