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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1answer
59 views

What does “transform among themselves” mean?

I'm reading a script on atomic physics, and there's a chapter on irreducible tensors. I can't understand the meaning of "transform among themselves" in this context: An arbitrary rotation of the ...
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1answer
35 views

Does this transformation have an inverse?

Let $f(n)$ be a complex sequence. Then for prime $p$ define $\hat{f}(p) = \sum_{n = 1}^{\infty} a_n e^{-i 2 \pi n / p}$. Then let the transformation of sequences be $T$, i.e. $Tf = \hat{f}$. Is ...
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1answer
86 views

base transformation rule significance in finding big o notation

Recall the equivalence: $$m=b^k \implies k = log_bm$$ as well as the base transformation rule: $$log_am=(log_ab)(log_bm)$$ Are the following true or false? (a) $log_2n$ is $O(log_3n)$ (b) ...
2
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1answer
117 views

Properties of Linear Transformations?

A linear transformation, $$T: \Bbb{R}^m \rightarrow \Bbb{R}^n$$ is a function that has the following properties. $$T(\text {u} + \text v) = T(\text u) + T(\text v)$$ $$T(\text{kv}) = \text ...
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2answers
39 views

Stuck on finding rank of $T$ when $n=8$ of $T(A) = A - A^T$

$T: M_{n\times n}(F) \to M_{n\times n}(F)$ is a linear transformation. I know from rank-nullity that $\text{rank}(T) + \text{nullity}(T) = \dim(M_{n\times n}(F))$. I'm trying to find $N(T)$ and then ...
0
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2answers
109 views

Transfomation of one coordinate system to a another

I have a molecule with one coordinate system ( denote as x,y,z ) where the origin is center of mass of the molecule. I have to define another coordinate system (p,q,r) for a local motion. (shown in ...
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0answers
43 views

Relation with $F$ distribution and $t$ distribution

If $X\sim F_{n,n}$ , then show that $$\frac{\sqrt n(\sqrt X-\frac{1}{\sqrt X})}{2}\sim t_n$$
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1answer
276 views

Using absolute coordinates in 2D affine transformation matrix

In my 2D animation program I have a sprite which transformation is described by a 2D affine transformation matrix (SVGMatrix): $$ \begin{bmatrix} a & c & e \\ b & ...
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1answer
113 views

Composition of linear transformations

Prove that if two linear transformations of rank 1 $f,g$ have equal kernels and images, i.e. $\mbox{Ker}f=\mbox{Ker}g$, $\mbox{Im}f=\mbox{Im}g$ then $fg=gf$. Any help would be appreciated, I don't ...
5
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3answers
207 views

non linear transformation that satisfies $T(cx) = cT(x)$

I am just curious if there is a transformation that does not satisfy $\;T(x+y) = T(x) + T(y),\;$ but satisfies $\;T(cx)=cT(x).\;$ I cannot think of any. Thanks for any help people.
0
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1answer
164 views

How to project 3D plots on a 2D coordinate system without losing the metric scale?

I've been collection data of a river bank last week and I need to plot the cross sections of the data. The issue is, that the data taken consists of 3 coordinates: easting, northing and elevation. ...
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1answer
201 views

Skew operator Squared Proof

I have a linear algebra question that is a proof and I am unsure how to approach this problem. For any vector $s$, show that $(s\times)^2 = ss^T-s^TsI$. The (sx) is the skew operator defined in ...
0
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1answer
131 views

Role of 1 in homogeneous transformation matrices

Given a 4x4 homogeneous transformation matrix that performs any useful transformation on point X to produce a transformed point X': $$ \left( \begin{array}{cccc} a & b & c & 0 \\ e & ...
0
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1answer
44 views

Matrix column addition

Suppose you have a matrix in the form of: $$\left[\begin{array}{c} a\\ b\end{array}\right]$$ How can this be represented be a two by two matrix?
2
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1answer
657 views

Exponential Function Shifts

I have some confusion about shifts concerning exponential functions. I can best describe my question with an example. Take $y = e^{-(x-3)}$. This graph has a reflection over the $y$-axis and is ...
2
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2answers
95 views

Linear Transforms, direct sum

Suppose $T\colon V \rightarrow F$ is linear. Prove that if $ v \in V $ is s.t. $v \notin \ker(T) $ then $$ V = \ker(T) \oplus \{\alpha v: a \in F\} $$ This is a question I got in an exam. I'm ...
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1answer
404 views

Linear Map extension from subspace to vector space

V is a finite dimensional vector space. How to prove that any linear map on subspace of V can be extended to linear map on V. I attempted it by taking the basis of the subspace W with Dim(m) and ...
0
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1answer
21 views

Transformation. Take n out of a root

Im kinda confused. If n is always > 0 $$ (n-a)^{\frac{x}{y}} = n*(1-\frac{a}{n})^{\frac{x}{y}} $$ is that true? Because there were some transformations in recent answers to my threads where I did ...
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0answers
51 views

Can someone explain this z transformation to me?

I have a signal $h[n]=\frac{1}{z+3}$ and the solution is $H(z) = (-3)^{n-1}\delta[n-1]$. Looking the solution up in a transformation table, I come to the conclusion that I need to transform $h[n]$ ...
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0answers
169 views

Infinite dimensional vector space eigenvectors eigenvalues and representation

We can express linear transformations with their eigenvectors and eigenvalues in finite vector spaces if they are diagonalizable. even if they are not diagonalizable we can express them via Jordan ...
2
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0answers
64 views

Heat equation $\frac{\partial \theta}{\partial t}=\kappa \frac{\partial ^2\theta}{\partial x}$ using two transformations to solve

Consider the heat equation $$\frac{\partial \theta}{\partial t}=\kappa \frac{\partial ^2\theta}{\partial x}$$ for an infinite rod. We use the transformation $q_1=\frac{x^2}{kt}$ and $q_2=\frac{\theta ...
0
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1answer
294 views

Step in Euler's rotation theorem

I have been examining the matrix proof for Euler's rotation theorem on Wikipedia. I have deduced every step up to proving that $\det (R - I) = 0$ for any rotation matrix R. However, I'm having ...
0
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1answer
186 views

What does it mean by “the origin is moved by the transformation” in linear transformations?

Linear transformations have the special property that the origin is not moved by the transformation. I don't really understand what this means. The example I'm given is that the following ...
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0answers
83 views

Changing coordinate system with non standard definitions

The standard coordinate transformation to polar coordinates is $$ \begin{cases} x=r\cos(\varphi)\\ y=r\sin(\varphi) \end{cases} $$ with $r\in[0,\infty), \ \varphi\in[0,2\pi)$ The question is whether I ...
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1answer
41 views

fit of translate curve and restore translation

I have a set of data, $S$ with negative values and the function. $$ \begin{equation} y = x + s \sqrt{ m \frac{x}{s} + 1} \end{equation} $$ If I try fit the curve for get the parameters $m$ and $s$, ...
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0answers
343 views

Derive Student T distribution using transformation theorem

I am trying working on an exercise that asks me to show that If $ X_1 \in N(0,1) $ and $ X_2 \in \chi^2(n) $ are independent random variables, then $ X_1 / \sqrt{X_2/n} \in t(n) \, $ where $ ...
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2answers
242 views

Laplace Transformation Applications

In one of our Mathematics lecture our Prof told us that similar to Logarithmic Transformations we can use Laplace Transformations to solve difficult equations. What kind of equations do Laplace ...
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1answer
147 views

Linear time varying into linear time invariant.

My original problem, is to transform Linear time varying systems of the form , for example: $$\begin{bmatrix}\dot{x1} \\ \dot{x2} \end{bmatrix} = \begin{bmatrix} -3t^2 & 0 \\ 6t^5 & -6t^2 ...
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2answers
213 views

linear transformation, ker(T) and im(T) - question from final exam

Assume $T:V\to V$ is a linear transformation, $\mathrm{dim} V = n$. Let $v$ be a vector of $V$ such that for $1\leq k\leq n : v, T(v), \dots , T^{k-1}(v)$ : they are all NOT zero, but $T^k(v) = 0 $. ...
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1answer
359 views

Which matrix transforms my vector field $F(r,\theta,\phi)$ from cylindrical to spherical coordinates

I am looking for the matrix that I have to apply my vector at the position $(r,\theta, z)$ to in order to get the appropriate vector in spherical coordinates. I am totally okay, if you could give me ...
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3answers
1k views

A Möbius transformation maps circles and lines to circles and lines. What exactly does that mean?

The title pretty much says it all. I am also looking for a concrete example if possible. I have looked at the proof, but I'm not exactly sure what it means because I am kind of confused on what the ...
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0answers
99 views

Determining pose of an object in 3d space

Given a 3D model of an object centred at the origin, if I place a camera at position (x,y,z) and make it face the origin, from the image rendered the object appears ...
0
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2answers
1k views

Finding reflection transformation matrix

I have two 3 dimensional points. $A [x_1, y_1, z_1]$ and $B [x_2, y_2, z_2]$. I need to find a transformation matrix which when multiplied to $A$ will give me $B$ and when multiplied by $B$ give me ...
2
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1answer
114 views

Evaluate the following integral by transformation:

1 1-x ∫ ∫ (sqrt(x+y)(y-2x)^2)dydx 0 0 $$ \int_0^1 \int_0^{1-x} \sqrt{x+y} \, (y-2x)^2 \,dy \, dx $$ I've determined that $u = x+y$ and $v = y-2x$ and that the jacobian is $= 1/3$. and that $x ...
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0answers
102 views

coordinate transformation and scaling

I have a global coordinate system that I need to transform to a local coordinate system. The new and old coordinate systems are shown below. Using transformation rules, I came with with the ...
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1answer
69 views

Can I use the pseudoinverse of a Jacobian like I think I can?

I need to compute the Jacobian for a transformation that maps parameters $p_1,...,p_n \to q_1,...,q_m$, $n\neq m$. For this, I need to compute the derivatives $\frac{\partial p_i}{\partial q_j}$. ...
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2answers
581 views

How to enlarge a circle?

if you are given a circle with equation $(x-a)^2 + (y-b^2) = r^2$ and it is enlarged by a factor of $3$ what would the new equation be? Would you put $2x$ an $2y$ in the place of $y$?
3
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1answer
3k views

How to find an all-in-one 2D to 3D Transformation Matrix for perspective projection, rotation, and translation?

I have read Finding a 3D transformation matrix based on the 2D coordinates but I think my situation is different because I think I need a 4x3 matrix, not a 3x3 matrix. I'm not sure but this might be ...
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2answers
498 views

Find a vector $\mathbf x$ whose image under $T$ is $b$.

I am having trouble with this question and how to get the answer. With $T$ defined by $T(\mathbf x)=A\mathbf x$, find a vector $x$ whose image under $T$ is $b$. $$ A = \begin{pmatrix} 1 & -3 ...
0
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1answer
139 views

Stereographic projection when the “North/South Pole” is not given by $(0,…,\pm 1)$?

Straight forward enough... what if My point is arbitrary, how can I get a new stereographic projection?
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0answers
42 views

Looking for a “Neat” Transform to Yield a Convex Set

Optimizing on a unit sphere $\mathbb{S}^n$ is almost a convex problem (if the function is convex in the new set) if we make our "new" set $\mathbb{R}^n$, via the stereographic projection. Clearly ...
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2answers
181 views

Fourier, Laplace, … and other Integral-transformations

I know Laplace, Fourier and Mellin-Transformation. Is there a general theory of transformations? My main interest is about classification of transformations satisfying specified properties like ...
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1answer
332 views

Mirror anamorphosis for Escher's Circle Limit engravings?

You are probably familiar with "mirror anamorphosis," the rendering in a painting of a distorted figure that can be undistorted by viewing in an appropriately tilted or curved mirror. The skull in the ...
0
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1answer
71 views

Determine all the linear transformation of $S^3$

I encounter the following problem: Let $S^3$ denote all the $3\times 3$-real symmetric matrices. If $F:S^3\to S^3$ is a linear mapping, with $F(OAO^T)=F(A)$, for any ...
0
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1answer
156 views

Transformation of Cubic Polynomial

I'm stuck on transforming this equation and am not sure where to begin. I know I need to define $x$ as some multiple of $u$ and somehow cancel the coefficient of the $x^2$ term but am not sure how to ...
0
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1answer
341 views

Transform between cartesian coordinate system and abstract coordinate system

I am trying to find a transformation that takes me between Cartesian coordinates and a pseudo-coordinate-system I have developed which is described as follows: Please first see the diagram below. ...
0
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1answer
187 views

condition for upper triangular matrix

Consider the following condition from this other post Define $S_k = \operatorname{span} (e_1, \ldots, e_k)$, where $e_i$ the standard basis vectors. Clearly, the linear map $T$ is upper triangular ...
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0answers
133 views

Update rotation matrix

Imagine you have a two noded beam in space, defined by extreme nodes 1 and 2. Image is owned by Jean-Marc Battini. To ...
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3answers
164 views

Prove an equality between dimensions of kernels

Let $V$ be a inner product space over field $\mathbb{R}$ with $\dim(V)<\infty$, and $T\in \text{Hom}(V,V)$. I'm trying to prove:$$\dim(\ker T)=\dim(\ker T^*)=\dim(\ker TT^*)$$ Also, as a conclusion ...
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1answer
323 views

Homothetic transformation

Suppose that, we have a homothetic transformation in a rectangular coordinate system, with center origin $(0,0)$ and $k$. This homothetic sends point $A(2,3)$ to point $B(2x-1,x)$. My aim is to find ...