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5
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1answer
158 views

Jacobian of Fourier Transformation

I am trying to calculate the Jacobian determinate of the Fourier transform which I stumbled upon when studying the Path Integral in Quantum Field Theory. I know the answer should be $1$ but I don't ...
0
votes
1answer
11 views

when linear mapping keeps monotonicity of $L_2$ norm

Consider an arbitrary vector $\alpha$ from vector space $R^p$, a linear mapping $A: \alpha\rightarrow A\alpha$ transforms $\alpha$ to $A\alpha$ in space $R^q$. What condition should $A$ satisfy so ...
1
vote
1answer
16 views

Linear Transformation Problem Given 3 transformations

can anyone help me get started with this question. Right now I am guessing and checking which is not efficient. I figured out out that the transformation is (?,?,x-y-z) so far Let $T:\Bbb R^3\to ...
1
vote
1answer
31 views

Birational Transformations question

so I'm wondering, is there a birational transformation one can make to the equation $Y^2 = X^m + f_{m-1}X^{m-1} + ... + f_0$, where all $f_i \in \mathbb{Q}$ so it is of the form $Y^2 = X^m + ...
0
votes
1answer
13 views

Transform pdf in higher dimensions?

Seem to remember the following equation held: $f(u) = {dx\over du} f(x)$ if one is give the probability distribution of x and a relationship between x and u the pdf of u can be derived. Sorry can't ...
0
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2answers
32 views

What formula would you use to cast an average of several numbers into a smaller range?

Say I have four numbers ranging in value from 15-95. If I want to, for a simple example, say that if the average of the four numbers is 15 (lowest possible value), that would relate to 2 on a scale of ...
1
vote
1answer
93 views

Univariate and Matrix Representation of Affine Transformation

Let $\mathbb{F}$ be a finite field with $q$ elements and $\mathbb{E}$ an extension field of degree $n$ of $\mathbb{F}$. Let $S:\mathbb{F}^n\rightarrow \mathbb{F}^n$ be a affine transformation and ...
1
vote
1answer
42 views

For general non-symmetric square matrices is there a matrix norm that is invariant under similarity transformations?

I think that there is no similarity-invariant matrix norm for general matrices. But are there similarity invariant norms for special types of matrices (e.g. for matrices whose eigevalues are different ...
0
votes
0answers
18 views

How to find the values of transformstion and its center/axis? [duplicate]

I have system of two planes. Each plane is defined by three points, so I have their equations. One of these plane is stable, I can't perform any transformation. The second one is modifiable - rotation ...
2
votes
1answer
23 views

Linear Algebra Linear transformation Help

If $T:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is a linear transformation, then there exists a basis for $\mathbb{R}^n$ in which $T$ is diagonal. Is this true or false
1
vote
0answers
31 views

Calculating with transformation matrix

Given is the transformation of coordinates $ T_{AB} = \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix} $. 1.) What are the new coordinates for the vectors (1,0) and (0,1)? It should be: $ ...
0
votes
1answer
62 views

Help Deriving the Canonical Form of this Elliptic PDE

So I'm given the following PDE, for which I'm to derive the canonical form: $$u_{xx} + u_{xy} + u_{yy} = 0.$$ Clearly $A=1, B=1/2$ and $C=1$. Hence we have $\xi_x/\xi_y = -1/2 ...
0
votes
1answer
33 views

finding this linear transformation

i am following this guide: http://www.calpoly.edu/~brichert/teaching/oldclass/f2002217/handouts/goof.pdf my question is to find the linaer transformation that adheres to $T(1,1,1) = (1,1,1)$ ...
1
vote
1answer
23 views

Plane transformations

I need help in understanding how plane transformations work: for example, let $$A = \{(x,y) \in \mathbb{R}^2: x^2 + y^2 < 1\}$$ Now let's change coordinates like this: $$x = u^2 - v^2$$ $$y = ...
0
votes
0answers
26 views

Density of transformation of normal distribution

A data set contains real values $\left\{v_1,v_2,\text{...},v_k\right\}$, $k<\infty$. $X_n\sim \mathcal{N}(\mu ,\sigma ),\ n=1,2,...,k$ $P$ is the (not necessarily unique) permutation that ...
0
votes
1answer
185 views

Givens rotation of the following vector of 3 elements.

I have to find the givens rotation matrix that will transform the following vector $[1, 1, -1]^T$ to $[y, 1, 0]^T$ (basically to insert a $0$ on the third position without altering the second one). I ...
0
votes
1answer
55 views

Find rotation angle of given image

At first: our aim is to find the total transformation of left house to the right house. What I did it first is translating the house with the center to the origin. I already found out that the ...
0
votes
0answers
75 views

How does orthonormal basis rotating work?

When you insert an orthonormal set into the column vectors of a matrix, you create a rotation matrix. I can't understand how this works, by simply placing the the vectors in there you have a rotation ...
1
vote
2answers
177 views

Books on geometric transformations and/or analytic geometry?

I've been looking to expand my knowledge in geometry as it's not covered in my undergraduate curriculum. For some reason I'm repelled by the classical approach (hopefully it will pass) as I feel it's ...
1
vote
4answers
77 views

Quadratics, transformations, and formulas

Two-part question. Feel free to answer just one part, or both (write which letter part you are answering) a) If the quadratic function $g(x)=a(x-h)^2+ k$ does not touch the $x$-axis, what can be ...
1
vote
2answers
31 views

Find the transformation.

I have to define (find?) the linear transformation $ f:\mathbb{R}^{3}\rightarrow \mathbb{R}^{2} \ \ \ where:$ $f(1,1,0)=(1,1)$ $f(0,2,-1)=(-1,0)$ $f(1,2,-1)=(0,2)$ How to achieve this? It is hard ...
1
vote
1answer
98 views

Find kernel and image of linear transformation.

I am given transformation : $f:R^3 \rightarrow R^2$ $ f(x,y,z)=(-x+y+z,x-y+z)$ I am requested to find kernel and image of this transformation. I am finding kernel: $ (-x+y+z,x-y+z)=(0,0 )$ ...
0
votes
1answer
62 views

Transforming partial differential equations

$13.$ Consider the change of variables $$x = e^{−s} \sin t,\space y = e^{−s} \cos t, \space \text{such that} \space u(x,y) = v(s,t)$$ (i) Use the chain rule to express $∂v/∂s$ and $∂v/∂t$ in terms ...
5
votes
1answer
73 views

How to compute (and check) this transform matrix?

Background: This is a homework exercise which asks to compute a transform matrix. The answer has been published by our teacher. However, my approach goes a different way and gets a different solution. ...
2
votes
3answers
126 views

How to Evaluate $\int^\infty_0\int^\infty_0e^{-(x+y)^2} dx\ dy$

How do you get from$$\int^\infty_0\int^\infty_0e^{-(x+y)^2} dx\ dy$$to $$\frac{1}{2}\int^\infty_0\int^u_{-u}e^{-u^2} dv\ du?$$ I have tried using a change of variables formula but to no avail. Edit: ...
0
votes
5answers
140 views

Is it true that every orthogonal transformation , even over $\mathbb R$, is diagonalizable?

Is it true that every orthogonal transformation , even over $\mathbb R$, is diagonalizable? I didn't succeed to get any information about it. Could anyone explain please?
5
votes
0answers
135 views

How to use polynomial or conformal transformation

In my research, I came to a transformation problem. The simple version is an initial circle (or sphere) region is advected by some deformational flow. After some time the circle will be deformed into ...
0
votes
1answer
61 views

Jacobian of an inverse

Suppose that we have an invertible map $T(u,v)=(x,y)$. The Jacobian of $T$ is given by $ \text{Jac}(T)= \left| {\begin{array}{cc} x_u & x_v \\ y_u & y_v \\ \end{array} } ...
-1
votes
1answer
55 views

Find conjugate transpose of linear transform

A difficult question I've been trying to tackle but I seem to hit a dead end. let $V$ be an inner product space over $\mathbb R$. We are required to find $T^{*}$ such that $<T(u),v> = ...
1
vote
1answer
48 views

I need hints on showing a matrix with certain properties defines a special transformation

Given the matrix $$A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$$ with integer coefficients, rational eigenvalue, and determinant $1$, show $A$ acts as a shearing along its eigenvector. Here is ...
1
vote
1answer
22 views

Obtain hamiltonian from a lagrange functional.

Assume that we have a Lagrange functional $L = L(\psi, \partial_t\psi,\partial_x\psi)$ with $\psi:(x,t) \rightarrow \psi(x,t)$. From the this I want to calculate the Hamiltonian. I was wondering how ...
0
votes
1answer
399 views

How do I find transformation matrix with respect to standard basis?

I know that in order to find transformation matrix with respect to a basis, I need to apply the transformation to said basis and the result is the column of the transformation matrix. But what ...
3
votes
1answer
74 views

Show that T is a linear transformation and find a, b, c

I'm having trouble understanding this question and the proper way to solve it. I don't understand the solution given and why this was the right way to answer it. Problem: For the vector space ...
0
votes
1answer
57 views

Z-transform a transfer function

Could someone help me invers Z-transform of this transfer function. $H_k(z) = \frac{Y_k(z)}{X(z)} = \frac{1}{1-cos(\frac{2·\pi ·k}{N})·z^{-1}+z{^-2}}$
0
votes
3answers
112 views

If two sides of a triangle are equal, and the angle between them is $60^\circ$, prove the third side is equal to the first two sides.

In other words, given points $A$ and $X$. Rotate $X$ $\,-60^\circ$ around $A$ to get point $X'$. How would you prove $XX' = AX = AX'$? I know this is true.
1
vote
1answer
60 views

Small Lorentz Transformation

This is very simple and I can 50% understand it but would like to properly understand why it is. If we have an infinitesimal Lorentz transformation $\Lambda^\mu _\nu = \delta^\mu _\nu + \omega^\mu ...
1
vote
0answers
65 views

Calculating convolutions of probability density functions

I have a PDE: $$\frac{\partial N (x,u)}{\partial x}=\int _0^uN(x,u)f(u-u')du'$$ $$N(0,u) = \delta (u)$$ Here $f(u)$ is a probability density function for $0 \le u \le u_{max}$, $\int _0 ^ {u_{max}} ...
0
votes
0answers
95 views

Proving Kummer's Transformation

I'm working with a Kummer equation of the form: $z\frac{\partial^2 w}{\partial z^2}+(b-z)\frac{\partial w}{\partial z} -aw=0,$ which is solved by Kummer's Confluent Hypergeometric function: ...
-1
votes
1answer
120 views

The image under mapping $w=(z+i)/(z-i)$, of the third quadrant?

The title says it all. I am not sure how to approach this problem. The only related problems i have done is mapping a (unbounded)line /circle to a line/circle. Regards Exatic
0
votes
0answers
70 views

Transforming FCC, BCC and HCP lattice types to cubes.

I was wondering if it is possible to transform the FCC, BCC and HCP into SC, or simple cubic lattices while preserving the lengths between the nodes? I would like to transform each into this: ...
0
votes
0answers
37 views

$K$-linear transformation $f:Mat (3 \times 2, K) \rightarrow K^3$

Could somebody please check my solution? Let $K$ be a field. Given is the $K$-linear transformation $f:Mat (3 \times 2, K) \rightarrow K^3$ $\begin {pmatrix} a_{1 1} & a_{2 1}\\ a_{2 1} & ...
0
votes
1answer
63 views

Prove or disprove that there exists a linear map given a set of vectors and their mapping

I'm stuck on this seemingly simple homework question, but I just don't know how to approach it at all :( Here is the question: " Prove or disprove that there is a Linear map ...
0
votes
2answers
41 views

How can I show that the kernel of $f-id_V$ is equal to the image of $f$?

How can I show that $\ker (f-id_V)=\Im f$ given that $f:V\longrightarrow V$ is a linear transformation such that $f\circ f=f$? Thank you.
4
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1answer
2k views

“Well defined” function - What does it mean?

What does it mean for a function to be well-defined? I encountered with this term in an excersice asking to check if a linear transformation is well-defined.
0
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0answers
86 views

Proof of the affine property of normal distribution for a landscape matrix

The widely used/mentioned/assumed affine property of multivariate normal distributions says that: Given a random vector $x \in R^N$ with a multivariate normal distribution -- $x \sim N_x(\mu_x, ...
0
votes
1answer
136 views

Proving a subspace under a linear transformation by the closure of standard addition and scalar multiplication

$T(x,y,z)= (3x-2y, -2x+3y, 5z)$ be a linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^3$ Show that $A= \{(u,v,z) \in \mathbb{R}^3~|~(u,v,w)=T(x,y,z)\}$ for some $(x,y,z)$ in $\mathbb{R}^3$ is ...
1
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0answers
83 views

some corollaries of the rank - nullity theorem

Here is a problem which I encountered in linear algebra. I realized that it might be a corollary of the "rank - nullity theorem" but I don't know how to work with it. Hope you can help! Thank you! ...
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2answers
62 views

Using transformations and basis to find standard matrices

Let $A =\{(1,3), (2,5)\}$ be a basis of $\mathbb{R}^2$. Let $M =\left[\begin{array}{rr} 1 & -2\\ 3 & 0\end{array}\right]$ be the standard matrix for the linear transformation from ...
0
votes
1answer
61 views

Representation of Linear Transformation with respect to basis please helppp

Let $A = (1,3) (2,5)$ be a basis of $\mathbb{R}^2$. Let $M =\left[\begin{array}{rr} 1 & -2\\ 3 & 0\end{array}\right]$ be the standard matrix for the linear transformation from $\mathbb{R}^2$ ...
0
votes
1answer
50 views

Linear Algebra Transformation Question

Suppose 3x3 matrix A = [122,011,012](commas separating rows) And suppose T: R^3 -> R^3 be defined by T(x) = A(x) for every x in R^3 (a) Find A^-1 (Easy) (b) Suppose T^-1 be the inverse transformation ...