0
votes
0answers
5 views

Transform gradient to reference element

Minimal example of the problem My attempt I think this is not a linear solution like \begin{equation} \nabla u = \nabla A_K x + \nabla b_K \end{equation} which must be wrong because $A_K$ is a ...
0
votes
0answers
19 views

Transformation matrices and hermitian/unitary/normal/… matrices

I need some help with the following - have I done the correct things or how can I solve the task? Let $f \in End(V)$, V a unitary space $\mathbb{C}^3$ given by: $A_{\alpha \beta} (f) = \frac{1}{7} ...
0
votes
2answers
69 views

Question on transformations in the complex plane

In the image (part (b)), Since $z < |3|$ before the transformation, does that simply imply that the region to be shaded after the transformation is definitely the inside of the circle and not it's ...
2
votes
1answer
40 views

Meaning of entries of a transformation matrix in practical terms [Homework related]

I'm having a bit of trouble understanding what the matrix entries mean practically in this problem: 100 kg of a highly toxic substance is spilled into three lakes. The state, t weeks after the ...
0
votes
0answers
17 views

Basic Matrix Transformation

The information I have is for a matrix transformation from R^3 to R^3 (denoted by L()), L(a_1) = 3(a_1) and L(2(a_1))= (5,-3,6). Find L(3a_1-22a_1), L(-4a_1), L(0), L(4a_1). What I tried to do was ...
0
votes
0answers
26 views

is it all right my pf?

PB: Give a proof that the image of a circle under a linear transformation is a circle. (Let $z$ be a $z=z_{0}+Re^{it}$, $t$ is a angle.) I tried it. Can you check my pf? (is it all right?) My Pf) ...
1
vote
1answer
111 views

Joint density of two functions of random variable

This is online homework, and I'm not always clear on which chapter questions are from, so I might be completely off base. I have two random variables, $X_1$~UNI(5,10) and $X_2$~UNI(4,10), and then ...
0
votes
3answers
54 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 \\ ...
0
votes
1answer
24 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?
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
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 ...
0
votes
1answer
20 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 $V$ ...
0
votes
0answers
24 views

Fitting data with an AR-model

An experiment involves a discrete time dynamical system with inputs u and outputs y. $ U = \left( \begin{array}{c} -2\\ 1\\ 0\\ 0\\ -1 \end{array} \right)$ and Y = $ \left( \begin{array}{c} -1\\ ...
1
vote
1answer
48 views

transformations of $\mathbb R^2$

Consider the transformation $(u,v)=f(x,y)=(x-y,xy)$. Demonstrate the effect of this transformation on the lines $x-y=\text{constant}$, $x+y=0$, and the curves $xy=\text{constant}$. In particular ...
1
vote
1answer
141 views

Finding equation of the image under a linear transformation

The equation of C is $x^2 + y^2 =1 $ How do I find the equation of the curve $C'=f(C)$ This is the image of $C$ under the linear transformation $f$ represented by the matrix $A=\begin{bmatrix}2 & ...
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

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
74 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. ...
-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> = ...
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 ...
1
vote
1answer
101 views

Rotate a triangle specified by vectors around its center

Well, I know that, in order to rotate a triangle specified by three vectors in $R^2$ we just rotate each vector in the same angle, and to do this we apply the rotation matrix in ...
0
votes
0answers
29 views

Affine transformation to find the limit of a function

In an exercise, one has to show that $\lim_{\rho\to 1} \frac{x^{1-\rho}}{1-\rho} = \ln (x)$ with $x>0$, which, supposedly, is to be done by applying an affine transformation, writing ...
0
votes
0answers
269 views

Matrix Transformation composite without homogeneous matrix

I have to give the transformation matrix, then composite some of them. Here are my matrices. a) scale $(1/2) x, (1/3) y$ => $\vec T_1 = \left[ \begin{array}{cc} 1/2 & 0 \\ 0 & 1/3 ...
0
votes
1answer
57 views

Mapping behavior of imaginary axis via $v=\frac{z-a}{z+a}$

I would like to know what the bilinear transform $v=\frac{z-a}{z+a}$ does to the imaginary axis, where $a$ is a real number. I substituted $z=yi$ and calculated $|v|$ giving me $|v| =1$. Is this ...
0
votes
1answer
19 views

Transformation of Functions

Given that $ g= 2f(-2(x+1))-2$ and the point (4, -1) is on the graph of f(x) What point must exist for g(x)? Any hints that can help me start to solve this question?
2
votes
0answers
56 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
votes
1answer
81 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 ...
1
vote
1answer
48 views

Image of $A \subset \mathbb{R}$ under transformation $(x,y) \rightarrow (u,v)$

What is the image of the set $$A=\{ (x,y) : 0\le x \le a\ , \ 1\le y\}$$ under the transformation $(x,y) \rightarrow (u,v)$ where $$u=x/y$$ $$v=x$$ The parameter $a$ is positive. I got a 'triangle' ...
1
vote
1answer
72 views

Möbius transformation question

Möbius transformation copies the annulus $\{z:r<|z|<1\}$ to the domain between $\{z:|z-1/4|=1/4\}$ and $\{z:|z|=1\}$ Please help me to find what is $r$.
0
votes
1answer
73 views

What is the image of $D=\{z:0<\operatorname{Re}z<\pi\}\setminus\{\pi/2\}$ under $f(z)=\tan z$?

What would be the image of the domain $D = \{z:0<\operatorname{Re}z<\pi\} \setminus \{\pi/2\}$ under $f(z) = \tan z$? I havn't met with tan(z) transformation so I don't really know how to ...
2
votes
1answer
212 views

Simple Graph Transformation Question $\rightarrow$ $1/f(x)$

for the graph: such that the function is : $ y = \frac{a+x}{b+cx} $ where a = -2, b = 1 and c = 1/2 how do you sketch the graph of $ y = |\frac{b+cx}{a+x}| $ ?? i got that the VA of the new ...
2
votes
1answer
77 views

Basis of kernel and image of a linear transformation - verification

The transformation matrix I found is: $$\begin{pmatrix} 1 & -1 \\ 1 & 1 \\ 0 & 0\end{pmatrix}$$ Is this how a basis for $\ker$ and $\mathrm{im}$ is calculated? $$\begin{pmatrix} 1 & ...
2
votes
0answers
61 views

Perhaps an easy algebra problem, but it still evades me

I need help spotting a corresponding transformation Let $x,y$ be some variables and $$z=z(x,y)$$. We have a transformation $X(\lambda):(x,y,z)\to (x',y',z')$, such that $$x'= x\exp(a\lambda)\\ ...
2
votes
1answer
83 views

Finding a hyperbolic isometry that fixes the point $x = 2$ and $x = 17$

I know that a Möbius transformation is hyperbolic if the trace is $> 2$ which is $a + d$. But I'm not sure of the next steps involved to arrive at the answer.
1
vote
1answer
61 views

Prove a transformation is a variational symmetry for J

The following problem is from The Calculus of Variations by B.von Brunt (page 215, Exercise 9.2.1) Let $$ J(y)=\int_a^b xy'^2\mathrm{d}x. $$ Show that the transformation $$ X=x+\epsilon2x\ ...
1
vote
2answers
43 views

Please, I need a more detailed explanation of the particular solution of the problem with vectors

Here is the problem and its solution (link to the source if you are interested): Two different points $A$ and $B$ are given. Find a set of such points $M$, that ...
7
votes
0answers
168 views

Way to Tietze's Transformation Theorem

during our knot-theory lecture we have talking about the following theorem: Given two finite presentations of the same group, one can be obtained from the other by a finite sequence of Tietze ...
0
votes
1answer
60 views

Finding $u$ and $v$ in Jacobian substitutions

I've used Jacobians before in multivariable calculus to simplify integrals, but I'm lost when I need to find the substitutions myself. Today on the quiz, there was the problem $\int\int_{R} xy dxdy$ ...
1
vote
1answer
169 views

Regarding the kernel of a linear transformation and that of the associated representing matrix

Let $V, W$ be finite dimensional vector spaces over a field $F$. Let $\mathcal{B}_{V} = \{\mathbf{v_1, \cdots, v_n} \}$ and $\mathcal{B}_{W} = \{\mathbf{w_1, \cdots, w_m} \}$ be corresponding bases. ...
1
vote
2answers
413 views

Matrix representation of the dual space

Let $V$ be an $n$-dimensional vector space over $F$, with basis $\mathcal{B} = \{\mathbf{v_1, \cdots, v_n}\}$. Let $\mathcal{B}^{*} = \{\phi_1, \cdots, \phi_n\}$ be the dual basis for $V^{*}$. Let ...
0
votes
4answers
104 views

Linear map between duals induced by linear maps between vector spaces

Let $V, W$ be vector spaces over a field $F$ and let $\psi: V \to W$. Show that $\psi$ induces a linear map $\psi^{*}: W^{*} \to V^{*}$ naturally. Although the question asks for a naturally induced ...
0
votes
3answers
76 views

A simple question on linear algebra and linear transformations

Let $f = (f_1, \cdots, f_m)$ be a function from $\mathbb{R}^n \to \mathbb{R}^m$. Prove that $f$ is linear if and only if for each $i$, $f_i$ is of the form $$f_i (x_1, \cdots, x_n) = a_1x_1 + ...
0
votes
2answers
45 views

An explanation about terminology in vector spaces

Call a linear transformation $\rho: V \to V$ ($V$ is a vector space) idempotent if $\rho^2 = \rho$. Prove that if $\rho$ is idempotent, then it acts as the identity on $\rho(V)$. If I understand the ...
2
votes
2answers
82 views

Meaning of $p(\phi)$ where $\phi (x,y) = (x+y, x- 2y)$ and $p(x) = x^2 -2x + 1$

Consider the linear transformation $\phi : \mathbb{R}^2 \to \mathbb{R}^2$ defined by $\phi (x,y) = (x+y, x- 2y)$. Let $p(x) = x^2 -2x + 1$. Does $p(\phi)$ make sense and if yes what is it?
3
votes
1answer
104 views

Question about special orthogonal Lie group construction

Working through homework and I run into this problem: Suppose the Lie group $SO^{+}(2,2)$ is presented as the group of all transformations in its associated space. How do you determine whether a ...
1
vote
1answer
92 views

Region of convergence of Z-Transform connected area?

Shouldn't the Region of Convergence of the Z transform be a connected area ? In Oppenheim solution manual, I've found this answer of a question that asks to determine the different forms of the ...
1
vote
1answer
217 views

Subspaces, transformation matrices exercise

I have trouble understanding the following exercise so I would really appreciate any help you could give me: Let $k$ be a non zero vector in $\mathbb R^n$, written in standard basis. Let $H$ be ...
17
votes
1answer
271 views

Show that $\phi: \mathbb{R}_3[x]\rightarrow\mathbb{R}^3, \phi(p):=[p(-1), p(0), p(1)] $ is a linear transformation

Let $\mathbb{R}_3[x]$ be a vector space of polynomials p with degree $\leq3$ and show that $\phi: \mathbb{R}_3[x]\rightarrow\mathbb{R}^3, \phi(p):=[p(-1), p(0), p(1)] $ is a linear transformation. ...
0
votes
3answers
6k views

Image and Kernel of a Matrix Transformation

So I had a couple of questions about a matrix problem. What I'm given is... Consider a linear transformation $T: \mathbb R^5 \to \mathbb R^4$ defined by $T( \overrightarrow{x} )=A\overrightarrow{x}$, ...