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0
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1answer
215 views

Rotation matrix for a set of points

I've got a set of $N$ points $p_1,\dots,p_N$ that all belong to a real object. Consequently, there are $N-1$ vectors $\vec{v}_i$ when $\vec{v}_i$ points from $p_1$ to $p_i$. Now, the object is ...
4
votes
1answer
116 views

Can a transformation matrix be expressed in terms of the vector to be transformed?

I'm currently learning linear algebra with my friend via an online course, and we have a disagreement that we would like settled. Upon learning that vectors can be projected onto lines by a simple ...
0
votes
1answer
89 views

Can the dimension of the image of a linear map “increase”?

Suppose we have a linear transformation $f: V \to V$. How is it possible that $\dim(\operatorname{im}(f \circ f))$ is larger than $\dim(\operatorname{im}(f))+\dim(\operatorname{im}(f)) - \dim(V)$? ...
2
votes
3answers
650 views

Geometric interpretation of linear transformation

I have a linear transformation, given by the following matrix $$ \begin{pmatrix} x_1\\ x_2 \end{pmatrix} \mapsto \begin{pmatrix} 2 & 2\\ -1 & -1\\ \end{pmatrix} \begin{pmatrix} x_1\\ x_2 ...
1
vote
0answers
117 views

Determine transformation from set of points

I have unknown perspective transformation matrix and unknown coordinates of the points in xy-space, but have coordinates of the points in uv-space and know that some points have the same distance ...
0
votes
1answer
130 views

Graph transformations.

This is the exact problem from the worksheet. Now I understand that it is giving the parent function for $f(x)$. The only formula I know of for transformations is $y=f(x)\rightarrow y=af(bx+c)+d$ ...
0
votes
2answers
166 views

Long-term behaviour of a linear transformation (Is the domain eventually mapped onto the dominant eigenspace?)

As far as its coordinate representation is concerned, the domain of a linear transformation will eventually (i.e. after infinitely many iterations of the transformation) be mapped onto the dominant ...
0
votes
1answer
169 views

Legendre Transform - Convexity question

I know that the Legendre transform $F(p)$ of a given function $f(q)$ is well defined only if $f(q)$ has a definite convexity. Furthermore I know that I can take the Legendre transform twice to recover ...
0
votes
1answer
404 views

Justification for transforming explanatory variables

I am using linear and generalised linear models, and have transformed my explanatory variables using $log10(\bullet)$ and $sqrt(\bullet)$ transformations, and my response variable using an arcsine ...
1
vote
1answer
124 views

Calibration of an eye tracking device: transformation from known gaze points

I am creating a calibration system for an eye tracking device. This calibration involves having the user look at five points on a screen. The eye tracker then reports where it believes the user was ...
0
votes
1answer
217 views

Use a Jacobian matrix to differentiate between linear and non-linear transormations

When determining whether or not a map/transformation is linear or non-linear, how can the Jacobian matrix be used? A linear equation in two variables is one that may be written in the form y = ax + b, ...
1
vote
2answers
640 views

Calculate an encoding matrix from inputs and outputs

I have a list of inputs and outputs of what I believe is encoded with a matrix (similar to this method). I was wondering if its possible to reproduce the matrix used to transform the inputs into the ...
0
votes
1answer
4k views

Arcsine squareroot transformation for data ranging from -$1$ to $1$

According to the Handbook of Biological Statistics, the arcsine squareroot transformation is used for proportional data, constrained at $-1$ and $1$. However, when I use ...
2
votes
1answer
65 views

Transform data distributed around zero

I have data that ranges continuously from $-1$ to $+1$, with lots of zeros in the middle. I want to transform the data to a normal distribution. How would I do this? My normal approach with data ...
1
vote
0answers
107 views

transform base of bilinear form

If $B$ and $B'$ are the matrix representations of a bilinear form in two bases, then these matrices are related by the equation $T^t B T = B'$ for an invertible matrix $T$. Is it the case that ...
2
votes
1answer
159 views

Why should coordinate transformations be reversible?

Intuitively I understand why coordinate transformation should be reversible. New coordinates should cover the same area covered by the initial coordinates, i.e. there should be one-to-one mapping. ...
5
votes
3answers
288 views

Fraction of two binomial coefficients

In an exercise I was asked to simplify a term containing the following fraction: $${\binom{m}{k}\over\binom{n}{k}}$$ The solution does assume the following is true in the first step, without ...
0
votes
2answers
119 views

What is the generalization of Lie group of transformation?

What is the generalization of Lie group of transformation? I found $a_1x+a_2$ and $(a_1x+a_2)/(a_3x+a_4)$ are also called Lie group of transformation!! It contradicts with what we learn about the ...
0
votes
1answer
68 views

Is there any sensible way to simplify this pde?

Problem: Try to simplify $$x^2\frac{\partial^2w}{\partial x^2}+y^2\frac{\partial^2w}{\partial y^2}+z^2\frac{\partial^2w}{\partial z^2}+yz\frac{\partial^2w}{\partial y\partial ...
7
votes
2answers
4k views

How to transform a set of 3D vectors into a 2D plane, from a view point of another 3D vector?

I googled around a bit, but usually I found overly-technical explanations, or other, more specific Stackoverflow questions on how 3D computer graphics work. I'm sure I can find enough resources for ...
1
vote
1answer
106 views

determinant of matrix of transformation from Cartesian to orthogonal curvilinear

Let $(x_1, x_2)$ and $(y_1, y_1)$ be two orthogonal coordinate system with unit vectos $(\hat i_1, \hat i_2)$ and $(\hat e_1, \hat e_2)$ respectively defined by the $x_1 = x_1(y_1,y_2)$ and $x_2 = ...
2
votes
0answers
119 views

Transform 3D vectors between planes using a matrix

I've got 6 points in 3D space: $A,B,C,D,E,F$, that represent 4 vectors. $AB$ is perpendicular to $AC$ and $DE$ is perpendicular to $DF$. I need to find a transformation matrix M, that transforms $AB$ ...
2
votes
1answer
2k views

building transformation matrix from spherical to cartesian coordinate system

How to arrive at the following from given $ x = r\sin \theta \cos \phi, y = r\sin \theta \sin \phi, z=r\cos\theta $ $$ \begin{bmatrix} A_x\\ A_y\\ A_z \end{bmatrix} = \begin{bmatrix} \sin ...
0
votes
2answers
385 views

convert values from one coordinate system (x,y) to another coordinate system (x', y')

Following is a graph that contains both coordinate systems (x,y) and (x',y'). x, y, x', and y' are all axes ...
0
votes
0answers
96 views

What happens to Fourier Transform of function when the function's time scale is changed?

When a function $f(t)=exp(-|t|)$ for example undergoes Fourier Transformation, it gives $F(w)=\frac{-2}{1+w^2}$ But what happens to the result if the time scale is scaled and shifted, so that $t ...
1
vote
1answer
608 views

Finding Fourier series with function not centered at the origin

I am trying to find both Fourier cosine and sine series which represent the function F(t) in the interval $(0, \pi)$ where $F(t)=\begin{cases} \frac{\pi}{2} & \ \ 0<t< \frac{\pi}{2}\\ 0 ...
0
votes
1answer
776 views

Linear transformation for projection of a point on a line

This is what my textbook wants me to do: The matrix of the linear transformation $P_L$ that projects $\mathbb{R}^2$ on de straight line $l \leftrightarrow y = mx$ is: \begin{pmatrix} ...
0
votes
1answer
5k views

Transformation of unit vectors from cartesian coordinate to cylindrical coordinate

Let $ (\hat i, \hat j, \hat k) $ be unit vectors in Cartesian coordinate and $ (\hat e_\rho, \hat e_\theta, \hat e_z)$ be on spherical coordinate. Using the relation, $$ \hat e_\rho = ...
2
votes
1answer
1k views

transformation of unit vectors between coordinate systems

Let the transformation rule between two coordinate systems $ (x_1, x_2, x_3) $, and $ (u_1, u_2, u_3) $ be $$ x_1 = a_{11} u_1 + a_{12} u_2 + a_{13} u_3 \\ x_2 = a_{21} u_1 + a_{22} u_2 + a_{23} u_3 ...
0
votes
1answer
404 views

Finding the transformation when given transformation matrix

Lets say, there is a transformation: $T:\Re ^{n}\rightarrow \Re ^{m}$ transforming a vector in $V$ to $W$. Now the transformation matrix, $A=\begin{bmatrix} a_{11} & a_{12} &...&a_{1n} \\ ...
4
votes
2answers
568 views

3d transformation two triangles

I have two triangles in 3d. I need to calculate transformation matrix(3X3) between two triangles in 3D. 1)How can I calculate the transformation matrix(rigid) while fixing one of the points to the ...
0
votes
2answers
129 views

Z-Transform Identity

I've come across an identity and would like to know if it has some sort of formal name or derivation or explanation or something! Also, I'm curious as to whether others are aware of such an identity. ...
3
votes
1answer
371 views

Möbius Transformation

Hey I am doing a basic undergraduate course in complex analysis and need some help on Möbius transformations. When determining the Möbius transformation does it really matter what 3 points I'm ...
3
votes
2answers
108 views

Put a transformation under the form of a rotation in the complex plane

On the complex plane, I have a transformation "T" such that : $z' = (m+i)z + m - 1 - i$ ($z'$ is the image and $z$ the preimage, $z$ and $z'$ are both complex number) and $m$ is a real number. ...
4
votes
1answer
99 views

Beta integral transformation

It's a homework task and I can't get past the last step. Task is to prove that $$ B(x,y)=\int\limits_0^1 \frac{\tau^{x-1}+\tau^{y-1}}{(1+\tau)^{x+y}} \mathrm{d}\tau $$ By substituting ...
3
votes
1answer
366 views

Coordinate transformation

I have some problems with a geometrical calculation. I want to know the coordinates of the point $P_2$ in my coordinate system $A \ (x,y,z)$ as shown in the following figure. Point $P_1$ (in $A \ ...
0
votes
1answer
456 views

Does null space always exist for a transformation?

This is inspired from this post as I was mentally playing with the concepts. The statement is the same just the transformation different, though for the benefit of everybody, I am repeating it, with a ...
5
votes
1answer
1k views

Finding the Dual Basis

Define the four vectors in $\mathbb{R}^4$ by $$v_1=\left( \begin{array}{ccc} 1 \\ 0 \\ 0 \\ 0 \end{array} \right), v_2=\left( \begin{array}{ccc} 1 \\ 1 \\ 0 \\ 0 \end{array} \right), v_3=\left( ...
1
vote
1answer
422 views

What is the Z-transform of a process shifted by a constant?

If $X(z)$ is the Z-transform of a discrete timeserie $x(t)$, what is the Z-transform of $x(t)+k$ where $k$ is a constant? From the linearity property of the Z-transform I would expect it to be $X(z) ...
1
vote
1answer
360 views

Given a matrix, find a linear transformation that uses it

The matrix is: $$\begin{pmatrix} 3+l & 8 & 3 & 3+l \\ 8 & 9 & 3 & 7 \\ 3 & 3 & 7 & 8 \\ 3+l & 7 & 8 & 13 \end{pmatrix}$$ I'm given the above ...
1
vote
0answers
178 views

2D Cartesian Matrix / coordinate transformation.

I has initially asked this question in the programming site but did not get an answer that worked. This is my first question on this site so please bear with me. Consider a page with three distinct ...
0
votes
1answer
176 views

Transformation Matrices [duplicate]

Possible Duplicate: Why can any affine transformaton be constructed from a sequence of rotations, translations, and scalings? Assuming that I have a set of points in a co-ordinate system (I ...
3
votes
1answer
2k views

Matrix for rotation around a vector

I'm trying to figure out the general form for the matrix (let's say in $\mathbb R^3$ for simplicity) of a rotation of $\theta$ around an arbitrary vector $v$ passing through the origin (look towards ...
0
votes
1answer
186 views

Prove equivalency of orthogonal transformation, $h(f(x),f(y))=h(x,y)$ and $f$ maps an orthonormal basis to another?

Please help! How do I go about proving this please? Let $f: M \longrightarrow M$ be a linear transformation. Then the following are equivalent: a) $f$ is an orthogonal transformation. b) ...
5
votes
1answer
797 views

How to figure of the Laplace transform for $\log x$?

I was looking at a table of common Laplace transforms of functions when I came across the transform for $\log x$. Apparently, the transform is as follows: $$\mathcal{L} \left\{ \log ...
2
votes
1answer
77 views

Change of coordinate codomain from $[-1,1]$ to $[0,1]$

Greetings to everyone! I've been searching around and thinking about how one translates coordinates from $[-1,1]$ to $[0,1]$ with little success. I am hoping someone here could help me with my little ...
0
votes
2answers
68 views

Normalize $X$ to $0$ to $10$ scale with asymptotes at either end

I am trying to find a scaling function that mimics the gas gauge in a car. I would like to map a value to a $0$ to $10$ scale, on which I have two known points. For example: $X_1 = 2$, $Y_1 = 2.5$, ...
1
vote
1answer
187 views

what is the “skewX” and “skewY” transform specified by flash's motion XML?

Flash has the ability to export animations into a format they call motion XML. Its specification is here I am trying to write a python renderer for these animations using pyglet. I understand ...
0
votes
1answer
317 views

Affine transformation matrixes

I could use some advise with the following problem: Lets say there is a cuboid that has two distinguished points - that is one of its vertexes ($A$) and the other one is somewhere on the surface ...
1
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0answers
514 views

2D Projective transform

Let's say we're transforming a square to an arbitrary 4 points via projective transform. Is there a way to ensure that the resulting points have homogeneous coordinates that are >0 ? i.e. sending ...