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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Show that T is a linear transformation.

Let B be an element of $R^{n \times n}$ and define $T(A) = BAB$ for all $A \in R^{n \times n}$. Show that T is a linear transformation. I am completely lost and I do not know how to start this.
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Find linear transformation given kernel

Find linear transformation $F$ in canonical bases given $ F: \Bbb R^4 \to \Bbb R^3 $ $ \ker F=\operatorname{span}\left\{\begin{bmatrix}1\\2\\3\\4\end{bmatrix}, \begin{bmatrix}0\\1\\1\\1\end{bmatrix} ...
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1answer
217 views

Function inverse mapping [0, +inf) to [0, 1)

I have a measure ($x$) which the domain is $[0, +\infty)$ and measure some sort of variability. I want to create a new measure ($y$) that represents regularity and is inverse related to $x$. It is ...
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1answer
475 views

How to find the rotation matrix that will align an arbitrary vector to an axis

If I have a vector that starts at the origin, how can I find the transformation matrix that will align it with the positive y-axis. So it basically turns into a positive-y axis? EDIT: I also forgot ...
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2answers
84 views

Linear operator exists then differentiable?

Let $E_{\text{open}} \subseteq \mathbb{R}^n$, and let $\vec{x_o} \in E$. Let $\vec{f}: E \rightarrow \mathbb{R}^m$. If there exists a linear operator $A: \mathbb{R}^n \rightarrow \mathbb{R}^m$. such ...
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1answer
33 views

Linear transformation from $R^2$ to $R^2$.

Let $\vec{f}: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, where $\vec{f} (\vec{x}) = (x+y^2, x^3+5y)$ and $\vec{x} = (x,y) \in \mathbb{R}^2$. Let $\vec{h} = (h_1, h_2)$ and $\vec{a} = (1,1) \in ...
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1answer
15 views

Transformation and properties of matrices

If $T:\mathbb{R}^n \rightarrow \mathbb{R}^m$ is a matrix transformation, does $T$ depend on the dimensions of $\mathbb{R}$? i.e., is $T$ one-one if $m>n$, $m=n$, or $n>m$? Also, say if $T$ is ...
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1answer
52 views

Matrix Transformation - Using matrix multiplication

How do I use matrix multiplication to find the reflection of (-1,2) about the x axis, y axis and the line y=x?
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1answer
87 views

Do we lose everything, if the natural transformations in a monad are exactly inverse?

I'm currently explaining monads $$T:{\bf C}\to{\bf C},\hspace{1cm}\eta:1_{\bf C}\to T,\hspace{1cm}\mu:T\circ T\to T,$$ to my brain and the "only" tricky thing are really the identity relations. I ...
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1answer
2k views

Transformation of ellipsoid to sphere

So I need to find an volume-preservating mapping from an ellipsoid to a ball (solid sphere). (Specifically: x^2/9 + y^2 + z^2 <= 3, but I'd rather understand the general case than just get how to ...
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349 views

Transformation matrix from quadrilateral to rectangle

There exists a rectangle somewhere in space with some orientation. A camera from the coordinate center point is looking along the z axis and is seeing the rectangle as a quadrilateral (due to ...
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1answer
59 views

Transformation Matrix $M_B^B$ of $P_3$ for $B = (1,x,x^2,x^3)$. Is that correct?

I have the following task and just wanted to check weather this is (written) correct(ly). Let $V$ be the vector space of all polynomials of grade $\le 3$ and $f: V \rightarrow V, p \rightarrow p'$ an ...
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1answer
31 views

Image of $\phi: \mathbb{Q}^{2\times 2} \rightarrow \mathbb{Q}^{2\times 2}, \ A \rightarrow A + A^t$

This question is related to the question I previously asked: Kernel. The following function is given: $$\phi: \mathbb{Q}^{2\times 2} \rightarrow \mathbb{Q}^{2\times 2}, \ A \rightarrow A + A^t$$ ...
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3answers
51 views

Help on finding eigenvalues of transformation on matrices

T is linear transformation working on 2x2 matrices: T(A) = $\begin{bmatrix}1 & 1\\1 &1\end{bmatrix}$ A as far as I see only 0 is an eigen value but someone told me 2 is eigen value too and ...
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2answers
76 views

Nilpotent Mappings

Got completely confused with this nilpotent and JCF stuff, need some help. Matrix $A_{n\times n}$ is nilpotent of order K, $1\le k\le 4$ Need to find: a list of all possible dimensions of ...
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1answer
188 views

Take -log of a Beta distributed R.V.

X1.....Xn~Beta(a,1) Y = -log(X) Use the transformation formula to calculate the pdf of Y. What named distribution does it have? I am confused what method to use here. A beta does not converge to a ...
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1answer
10 views

Regarding some elementary transformations

I was trying to follow a math forum thread when suddenly I stumbled upon a transformation that I just couldn't understand. It goes as follows: $v = \sqrt{ 2 U / r - 2 U / r_0}$ $v = \frac{dr}{dt} = ...
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95 views

I'm looking for the name of a transform that does the following (example images included)

I'm in the usual situation that if I would know what the name of the thing was, then I could find the answer. Since I dont know the name, here is what I'm looking for: Suppose I have the following ...
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2answers
393 views

Isomorphism between symmetric and upper triangular matrices

Question: Determine if the vector spaces $V=S_{3}$, the 3x3 symmetric matrices, and $W=U_{3}$, the 3x3 upper triangular matrices, are isomorphic. If they are, give an explicit isomorphism $T: V ...
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165 views

What does the Yoneda lemma say for the identity functor and finite sets?

So I try to plug in the simplest arguments into the Yoneda lemma and see how to interpret it. I'll try it for the identity functor and the category of finite sets, in particular, I use an three ...
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1answer
171 views

Linear Transformation from $\alpha$ to $\beta$

T: $R^3$ $\to$ $R^2$ $$[T]_{\beta\alpha} = \begin{matrix} 2 & 3 & 1 \\ 1 & 2 & 1 \\ \end{matrix} $$ $\alpha$ = {(1, -1, 1), (0, 1, 0), (1, 0, 0)} ...
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1answer
69 views

Complex Transformation

$z_1 = 1 + i$ and $z_2 = -1 + i$ I am told: $w = \dfrac{az + b}{z + d}$ where $z \not= -d$ Where a, b and d are complex numbers, maps the complex number $z$ onto the complex number $w$. Given that ...
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0answers
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Manipulating this probability distribution function

I have a probability distribution function as follows: $$ P(y|x,w, \phi) = \frac{\phi}{2\pi} \exp ^{-0.5 (y-t(x, w)'\phi (y-t(x,w)) } $$ Here $y$ and $x$ are two observed values. $\phi$ is also some ...
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1answer
42 views

$3D$ projection onto a plane

I have an engineering problem involving math so I figured I ask it here. I have two sets of data: Acceleration in $3$ dimension is given by $\langle X,Y,Z \rangle $. Change of orientation along ...
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1answer
1k views

How can I transform coordinate systems with quaternions?

I have a coordinate system $0$ which I'd first like to rotate about its $z$-axis which gives me system $1$, and afterwards rotate system $1$ about its $y$-axis which gives me system $2$. See picture: ...
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2answers
144 views

How can I calculate the origin of a scale transformation, given the starting and ending coords and dimensions?

Background: I have two sets of coordinates/dimensions. One for the red rectangle and one for the blue rectangle, as shown below. The blue rectangle is quite simply the red rectangle transformed by ...
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3answers
143 views

Composite linear map Rank and Image

I have been pondering on this question, I did part $(a)$ wherein you had to prove that $\operatorname{Im}(T)= \operatorname{Im}(T^{2})$ , but I am struggling to get the concept of part $(b)$, any help ...
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70 views

Characteristic polynomial of a mapping from matrices space to matrices space

Let $T$ be the linear map from $M_n \to M_n$ given by TX=AX, while A is as well a matrix $n \times n$ (a) Write out the characteristic polynomials for $T$ (b) Show that if A is ...
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1answer
147 views

Transfer Transformation from Physics to Vector Graphic

Upfront, I am not a professional in Maths and hope that the formulation of my question describes the problem well enough. I am creating a jump'n'run game, which uses a physics engine (Box2D) and SVG ...
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51 views

$T:V \rightarrow V$ And $U \cap Ker(T)={0}$ prove that if$ (u_1,..u_n)$ linear Independent so does $T(u_1)…T(u_n)$

There will be $T:V \rightarrow V $ Linear Transformation U is sub-space of V so that $U \cap Ker(T)={0}$ Prove that if $(u_1,u_2,...,u_n)$ are linear independent so does $T(u_1),T(u_2),...,T(u_n)$. ...
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2answers
55 views

upper bound for equation

Let $0 < p < 1$ be some constant. I am looking for an $M$ such that $$f(n) = \left(1-p^{\log{n}}\right)^{n} < M(n)$$ I am looking for a tight bound, something of the form: $2^{-n/\log{ ...
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133 views

Existence of a linear transformation in an infinite dimension vector space.

If $V$ and $W$ are vector spaces, $\beta=\{v_1, \ldots , v_n\}$ is a finite a basis for $V$ and $\{w_1, \ldots , w_n\}\subset W$, we know there is an unique linear transformation $T:V\rightarrow W$ ...
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1answer
448 views

Expectation of (1/x)-1 possible transformation involved??

I'm a bit confused with the first steps in this problem: $F(x)=x^4$ for $0<x<1$ a) Find $E[(1/X)-1]$ b) Let $Y=(1/X)-1$. Find the support of $Y$, its pdf and CDF. Name its ...
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63 views

Transform recurrence relation

Is it possible to transform following recurrence relation $a_n=4a_{n-2}-a_{n-4}$, $a_0=1$, $a_1=0$, $a_2=3$, $a_3=0$ so that it will have nonnegative coefficients? Number of terms, of course, can be ...
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1answer
26 views

general rotations

Let $R$ be the rotation about the point $(1,0)$ by an angle of $45$ degrees. By using matrix methods: Find the image of the line $2x-3y+1=0$ under $R$ I would really appreciate it if someone ...
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1answer
2k views

How can I combine affine transformations into one matrix?

So from what I understand from this picture, the box is stretched to twice its width. And it is then flipped from the x-axis. And then it is rotated 30 degrees anticlockwise. So these three ...
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1answer
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Find 2D affine transform matrix given a pair of points

I have the coordinates of two points in an initial 2d coordinate system and the corresponding coordinates in a target system. Is is possible to determine the affine transform matrix from these values? ...
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1answer
23 views

Function that transforms a Matrix to different dimensions

What is the name of a function that transforms a matrix into different dimensions? Say I have a matrix M of dimensions $(x,y)$ and I want to transform it to dimensions $(w,v)$. I can accomplish this ...
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1answer
123 views

Constructing a similarity matrix between points

I have two images with two sets of corresponding points. In order to align the images I'm trying to compute the similarity matrix that describes the relationship between the corresponding points. I ...
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0answers
146 views

Invariant functions under integral transforms

We all know Fourier transform has invariants such as $e^{-x^2}$, and another MSE post has shown the non-existence of invariant function under Hilbert transform using Fourier transform. I am wondering ...
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1answer
52 views

Show that $T$ is a linear transformation given Orthonormal basis

Suppose that $T:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and suppose that $\{v_1,v_2,\cdots,v_n\}$ and $\{Tv_1,Tv_2,\cdots,Tv_n\}$ are orthonormal basis of $\mathbb{R}^n$. Prove that $T$ is a linear ...
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2answers
41 views

Diagonalization of a strange transformation

Let be $V$ a vector space on $\mathbb C$ and $\dim V=4$ and let be $f \in \operatorname{End}(V)$ such that $\operatorname{Im}(f^2+a \cdot \operatorname{id}) \subset \ker(f+id)$ where $a=\det f$, ...
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1answer
166 views

Linear Programming Transformations

What is the process of performing a transformation from a given problem to another linear programming problem such that the transformed problem has an optimal solution iff the initial problem has a ...
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2answers
71 views

Does congruence guarantee length conversion?

Suppose that a linear transformation $M:R^2 \rightarrow R^2$ maps a triangle $ABC$ to a congruent triangle $A'B'C'$ ($\{A, B, O\}, \{B, C, O\},\{C, A, O\}$ are not colinear, and $A,B,C\neq O$) Is it ...
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1answer
28 views

About the matrix of two linear transformations

I have an exercise to answer, and I don't know if I've done it the right way. This is only a little part of the exercise, but I have to know if what I've done so far is correct. Here we go: Let $V$ ...
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3answers
74 views

Linear Algebra Vector Space matrix help

Let $M_{2\times2}$ be a vector space of all $2\times2$ matrices. If the transformation from $M_{2\times2}$ to $M_{2\times2}$ is $t(A)=A+A^T$ and $A$ is a $2\times2$ matrix with the top row $a,b$ and ...
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1answer
46 views

Increasing length of closed spline (scaling)

I have a 2D closed spline and I need to increase its total length by a factor k, without changing its curvature, basically scaling. If this spline was a circle, I ...
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48 views

Is it possible to prove a non linear transformation to linear transformation?

Is it possible to prove a non linear transformation to linear transformation.For example,F(x,y)=(3x,2xy).F(0,0) results linear transformation,but F(2,2) gives non linear transformation.Which result I ...
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0answers
82 views

Diagonalization of a linear transformation in the polynomial vector space

Let $V = R_3[X]$ be the vector space of polynomials with real coefficients of degree at most 3 and consider the linear transformation $V \rightarrow V$ defined by $f_a(p(x))=p(1-ax)$ for each $p(x) ...
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1answer
50 views

If $T,S \in L(V)$ are positive operators, how can I show that $TS$ is self-adjoint?

If we let $V$ be a finite dim. real/ complex inner product space, and $T \in L(V)$ and $S \in L(V)$ we let be positive operators, how can I prove that $TS$ is self-adjoint? I tried to decompose $TS ...