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3answers
55 views

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation. Find $T(x)$

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation with $T \left(\begin{bmatrix} 1 \\ -2 \\ -1 \\ \end{bmatrix}\right) = \begin{bmatrix} 1 \\ -1 \\ 2 \\ ...
1
vote
1answer
31 views

Linear Transformations, kernels, and Basis for Range

I have the following question and am completely lost on how to start: Let $T: P_{2} \to P_{3}$ be the linear transformation $[T(p)](x) = p^{\prime}(x) + xp(x)$ Find $\ker(T)$ and find a basis for ...
1
vote
0answers
20 views

Rigid Deformation

I'm trying to parse through this paper on using the method of moving least squares for rigid transformations - http://www.cs.rice.edu/~jwarren/research/mls.pdf Under section 2.3, the author mentions ...
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0answers
13 views

Transformation Matrix of a function

I have the following: (Note: $V^{*}$ is defined as: $V^{*} = \{ L: V \rightarrow \mathbb{R} | \text{L is linear} \}$) Let $V$ be an $\mathbb{R}$-Vectorspace. Let $\phi \in V^{*} \text{ \ } \{0 \}$ ...
1
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0answers
12 views

scale transformation is invariant for H_1

Consider the subspace $H_1$ of $C_0(0,\infty)$, where $\phi=\int_0^t\dot{\phi}(s)ds$ and $\int_0^{\infty}{\dot{\phi}}^2ds<\infty$. The transformation is $(T\phi)(t)=t\phi(\frac{1}{t})$. How to ...
0
votes
2answers
21 views

Which, if any, of the following polynomials are in Range(t)?

Let T: P^2 ----> P^2 be a linear transformation defined by T(p(x)) = xp'(x) (i) 2 (ii) x^2 (iii)1-x I was hoping someone would show me how to find the range of one of them so I know how to do the ...
1
vote
2answers
38 views

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.
2
votes
1answer
88 views

Find linear transformation given kernel

$ F: R^4 -> R^3 $ $ kerF=span\{\begin{bmatrix}1\\2\\3\\4\end{bmatrix}, \begin{bmatrix}0\\1\\1\\1\end{bmatrix} \} $ Find linear transformation in canonical bases given above information. I tried ...
1
vote
1answer
32 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 ...
0
votes
1answer
78 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 ...
1
vote
2answers
39 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 ...
0
votes
1answer
30 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 ...
1
vote
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 ...
0
votes
1answer
25 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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0answers
13 views

Inverse coordinates on a matrix

I am trying to "inverse" some coordinates on a matrix. For example, take this grid: ...
2
votes
1answer
79 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 ...
0
votes
1answer
138 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 ...
1
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2answers
113 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 ...
1
vote
1answer
40 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 ...
0
votes
1answer
25 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: $$\phi: \mathbb{Q}^{2\times 2} \rightarrow \mathbb{Q}^{2\times 2}, \ A \rightarrow A + A^t$$ ...
1
vote
3answers
45 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 ...
1
vote
2answers
59 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 ...
0
votes
1answer
47 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 ...
0
votes
1answer
9 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} = ...
5
votes
3answers
68 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 ...
2
votes
2answers
94 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 ...
5
votes
2answers
112 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 ...
1
vote
1answer
49 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)} ...
1
vote
1answer
30 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 ...
1
vote
0answers
31 views

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 ...
0
votes
1answer
26 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 ...
0
votes
1answer
34 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: Both ...
0
votes
1answer
29 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 ...
0
votes
3answers
29 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 ...
4
votes
2answers
51 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 ...
1
vote
1answer
60 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 ...
1
vote
3answers
28 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 $T(u_1),T(u_2),...,T(u_n)$. ...
2
votes
2answers
43 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{ ...
1
vote
2answers
55 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 $T:V\rightarrow W$ ...
1
vote
1answer
75 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 ...
0
votes
0answers
42 views

State space transformation matrix that involves the 'input' vector.

I have a simple state space system $x(k+1)=Ax(k)+Bu(k)$, where: \begin{align*} A &= \left[ \begin{array}{ccc} 0 & 0 & 0\\ 1 & 0 & 0\\ 0 & 1 & 0 \end{array} \right]\\ ...
4
votes
0answers
51 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 ...
0
votes
1answer
20 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 ...
2
votes
1answer
131 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 ...
0
votes
0answers
26 views

Find angle-preserving transformation matrix given 2 points

I asked a similar question yesterday about finding an affine transform matrix given the same 2 points in both coordinate systems. I was told that there was only a unique solution, if the scaling was ...
0
votes
0answers
92 views

Conversion of covariance matrix from Cartesian to Spherical coordinates for integration

I have to perform a convolution of a function in polar coordinates $\rho(\textbf{x}) = \rho(r,\theta,\phi)$ with a function $P(\textbf{x}) = P(x,y,z)$ in cartesian coordinates. $\int ...
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0answers
12 views

stabilizing variance

Let $X_n$ be a Markov sequence such that $X_n=(1+X_{n-1})Y_n$ with $X_0=0$ and $Y_n$ - iid and independent of $X_n$'s. Suppose $\mathbb{E}[Y_1]=1$ and $\mathbb{E}[Y_1^2]=\varkappa$ with ...
0
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0answers
26 views

Derivation of F distribution

Prove that the PDF of Snecdor's F distribution, given by: $$F=\frac{U/n_1}{V/n_2}$$ Where $U=\chi^2(n_1)$ and $V=\chi^2(n_2)$, is given by: ...
0
votes
1answer
111 views

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? ...
0
votes
0answers
11 views

Bijective mappings in [0, 2^n-1] that are not too expensive in computation costs

I like to have a good collection of bijective mappings in [0, 2^n-1] that are not too expensive in computation costs. The following are certainly trivial (all values are n bits): (1) x --> a*x + b ...