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0
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1answer
349 views

Composite transformation expresing as single transformation

This is the composite transorfmation that I have this is the working that I did, but something tells me this might not be the right answer, P.S can some with more points add composite ...
0
votes
0answers
88 views

Riccati equation transformation

Is possible to transform this Riccati equation into a linear differantial one? Thank you. $$ y=y_1+\frac{1}{z} $$
1
vote
1answer
51 views

Coordinate transformation to get even function

Suppose I have the function $$f(y)=2y^4-5y^3+3y^2,$$ with zeroes $y=0$ (2x), $y=1$, $y=3/2$, which I only need on the part of the domain $0\le y\le 1$. Is there a transformation $y\rightarrow y'$, ...
5
votes
2answers
488 views

Are Legendre transforms of non-convex functions useful?

Do Legendre transforms have any applications that do not appeal to convexity? What is the intuitive interpretation of the Legendre transform of a non-convex function?
0
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2answers
2k views

Linear Transformation with 2x2 Matrix Basis

The question asks: Find the "coordinates" of $v=\begin{bmatrix} -2 & -2 \\ -2 & 4 \end{bmatrix}$ relative to the ordered basis, $F=(f_1, f_2, f_3, f_4)$ where $f_1 = \begin{bmatrix} 1 & ...
0
votes
1answer
75 views

Matrix representation of transformation in ordered bases

An example question asks me to determine $[T]_{\beta}^\gamma$ where $\beta,\ \gamma$ are standard ordered bases of $\mathbb{R}^n$ and $\mathbb{R}^m$ respectively, of $$T_1: \mathbb{R}^n \rightarrow ...
3
votes
1answer
397 views

Find Möbius transformation that send Re(z)=Im(z) to a circle and the real axis to itself

Problem 3.3.7d in Complex Variables, 2nd edition, by Stephen D. Fisher. Find a linear fractional transformation $T$ that maps the real axis onto itself and the line $y=x$ onto the circle ...
2
votes
1answer
76 views

Semigroup question

I am looking for the technical term for an element of a transformation semigroup that sends everything to one state. The best term I could think up was filter. For those that don't know a ...
0
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2answers
203 views

3D transformation between two polylines problem

Say I have 2 separate objects. One is a line defined by two points, the other is a polyline defined by three points. Line 1 consists of the set of two points: $a=(0,0,0)$ and b=$(0,0,1)$ Line 2 ...
3
votes
1answer
769 views

Transformation matrix to go from one vector to another

I've two vectors $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$. How to find transformation matrix for transform from a to b?
1
vote
2answers
922 views

Formula transformation

Is it possible to transform this equation to give R? $$y=x\left[\frac{\left(1+\frac{R}{12}\right)^{12\times{25}}}{\frac{R}{12}}-1\right]$$
0
votes
1answer
2k views

Function transformation order of operations

I am reviewing for a midterm for Pre-Calculus and I am trying to understand the concept of function transformation: Let's say I am given a function $f$ with the domain in the interval of $[1,5]$ and ...
4
votes
1answer
139 views

Is there a geometric argument that the Legendre transform of a convex function is convex?

I am trying to build intuition on Legendre transforms. Arnold's Mathematical Methods of Classical Mechanics has some nice geometric interpretations, but he does not provide a proof that the Legendre ...
2
votes
0answers
196 views

transformation of coordinate systems by rotation

I am trying to convert a set of coordinates from ECEF (Earth Center Earth Fixed) to ENU (East North Up). The operation is performed by applying a rotation matrix as shown in: ...
3
votes
1answer
201 views

Fitting Shape in Circle for Shape Classification

I need to classify arbitrary 2D shapes. The classification should be invariant to at least affine transform. To achieve this invariance, I decided to "normalize" each shape by fitting it to a unit ...
1
vote
1answer
78 views

Help in finding Jacobian

I have $$\begin{aligned}x_{1}&=r\sin(\theta_{1}),\\ x_{2}&=r\cos(\theta_{1})\sin(\theta_{2})\\ x_{3}&=r\cos(\theta_{1})\cos(\theta_{2}). \end{aligned} $$ I know how to compute the ...
1
vote
2answers
44 views

Optimally projecting a point onto a line whose orientation is known

I have a line $l$ starting at origin ending at 0,0,1 along the $z$ axis. $l$ is rotated $P$ degrees around the $x$-axis and then $Q$ degrees around the $y$-axis. So I have a new endpoint for the line. ...
1
vote
1answer
351 views

Translating Cubic with Algebra?

I'm having a little trouble figuring out how to translate $ax^3+bx^2+cx+d=y$ by vector $(1,1)$ using only algebra. If possible could someone give me a hand? An example: Translate ...
1
vote
1answer
71 views

Transforming a Continuous Function

My math is quite limited so please bear with me. I will get to the point: Is there a way to transform a continuous function into a bounded one? In essence I have a normalized Gaussian distribution ...
2
votes
2answers
170 views

A linear transformation $T$ is defined by $T(x_1, x_2)$ Find the image of the line that passes through the origin and point $(1,-1)$

I know the definition of a linear transformation, but I am not sure how to turn this word problem into a matrix to solve: $T(x_1, x_2) = (x_1-4x_2, 2x_1+x_2, x_1+2x_2)$ Find the image of the line ...
1
vote
3answers
750 views

Show that the transformation T defined by $T(x_1, x_2)\; = \;… $ is NOT linear.

I'm studying for a test, and I need help with this problem. I am not sure how to prove that this is not linear due to the notation. The comma is throwing me off. Show that the transformation $T$ ...
1
vote
1answer
51 views

Trouble with function transformation (Left and right)

I am reading this example in the book for Pre-Calculus and it is explaining how functions are shifted left or right using g(x)=f(x-1). Here is what it says in the book. Define a function g by g(x) = ...
4
votes
2answers
345 views

Transforming Differential Equation to a Kummer's Equation

I'm trying to transform an equation of the form $$ yw''(y) - [b - ay] w'(y) - [d + ey]w(y) = 0 $$ into the form of a Kummer's or confluent hypergeometric differential equation: $$ y w''(y) + [f - ...
0
votes
1answer
264 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
129 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
90 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 $\dim(\operatorname{im}(f))+\dim(\operatorname{im}(f)) - \dim(V)$? ...
3
votes
3answers
746 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
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0answers
124 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
143 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
175 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
177 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
523 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
159 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
243 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
764 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
6k 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
69 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
113 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
167 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
309 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
124 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
69 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
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2answers
5k 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
118 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
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0answers
127 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
3k 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
403 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
104 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
695 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
883 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} ...