Transformation has many meanings in mathematics. If using this tag, add another tag related to the object being transformed. If there is a tag for your specific kind of transformations, use that one instead: e.g., (laplace-transform), (fourier-analysis), (z-transform), (integral-transforms), ...

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4
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1answer
152 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
206 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
243 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
79 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 ...
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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. ...
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vote
1answer
419 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
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1answer
72 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
183 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 ...
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3answers
922 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
55 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
429 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
309 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 ...
5
votes
1answer
142 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
93 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 $\dim(\operatorname{im}(f))+\dim(\operatorname{im}(f)) - \dim(V)$? ...
4
votes
3answers
874 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 ...
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0answers
127 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
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1answer
154 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
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2answers
183 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
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1answer
191 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 ...
1
vote
1answer
613 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
179 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
270 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, ...
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vote
2answers
939 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
7k 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 ...
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0answers
117 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
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1answer
178 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
321 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
127 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
70 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 ...
8
votes
2answers
7k 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 ...
2
votes
1answer
130 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
131 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 $AB$ ...
3
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
421 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
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0answers
112 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
781 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
1k 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} ...
1
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1answer
7k 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
563 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
727 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
141 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
407 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
118 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
108 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
428 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
765 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
2answers
2k 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
633 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
382 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 ...