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36 views

One double integral elated problem

The bit I am stuck is the limits in the double integral. I tried X from 0 to uy and Y from 0 to infinity, this is obviously incorrect. I just want to know the complete double integral in the order ...
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2answers
20 views

Showing a set is a basis

Let $V, W$ vector spaces and $f, g:V\rightarrow W$, linear transformations. $\ker f \subset \ker g$. Now, let $\{v_1,...,v_n\}$ a basis for $\ker f$ and we'll complete it with $\{u_1,...,u_m\}$ to a ...
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4answers
87 views

$\{ v_1,v_2,…,v_n\}$ is basis of $V$ if and only if $\{ v_1,v_1 + v_2,…,v_1 + v_2+…+v_n,\}$ is a basis of $V$

Let $V$ a vector space over a field $K$. Is it true $\{ v_1,v_2,...,v_n\}$ is basis of $V$ if and only if $\{ v_1,v_1 + v_2,...,v_1 + v_2+...+v_n,\}$ is a basis of $V$ ? I made some examples and ...
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1answer
39 views

$V$ and $W$ finite vector spaces with dimension $n$ and $r$ with $\{ v_1,v_2,…,v_n \} \subset Ker T$ and $\{ u_1,u_2,…,u_s \} \subset V$

Let $V$ and $W$ finite vector spaces with dimension $n$ and $r$ respectively and $T: V \rightarrow W$ linear transformation, $\{ v_1,v_2,...,v_n \} \subset Ker T$ and $\{ u_1,u_2,...,u_s \} \subset ...
1
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3answers
52 views

Diagonalizable operator of a finite vector space

Let $V$ a vector space of finite dimension, $dim (V) = r$, and $T: V \rightarrow V$ a diagonalizable operator with $ \lambda _1,\lambda_ 2,...,\lambda _r$ distincts eigenvalues of $T$ then $ (T- ...
1
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1answer
46 views

Linear transformation matrix representation with differentiation answer confirmation

I hope you liked the title. I have a question that is as follows: Consider the linear transformation $T: P_3(\mathbb{R}) \to P_3(\mathbb{R})$ given by $$T(f(x))=f(0)+f'(x)+f''(x)$$ Where the ...
4
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1answer
89 views

Finding a basis for $\ker(T)$

I have this question: Let $Z\in M_{2\times2}(\mathbb{R})$ be defined as $$Z = \left( \begin{align} 1 &&1\\1 &&1 \end{align} \right)$$ and consider $T: ...
0
votes
1answer
25 views

Lorentz transformation and Minkowski metric

For the exam I'm trying to solve some problems. Today I found this exercise and need some help: For the group S0(1,1) of the Lorentz transformation I have $\phi \in \mathbb{R}$ and $A_{\phi}: ...
0
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4answers
68 views

Showing some transformation is linear

Let $T: P_3(\mathbb{R}) \to P_3(\mathbb{R})$ be an operation defined by $$T(a+bx+cx^2+dx^3) = a + dx + (a+d)x^2 +(b-c)x^3$$ Show that $T$ is linear What I have done so far is look at it like ...
1
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2answers
48 views

$\dim(V) = \dim T(V) + \dim T^{-1}(0)$

Let $T\colon V \rightarrow W$ a linear transformation between the real vector spaces $V$ and $W$ both with finite dimension. How can i prove that $\dim(V) = \dim T(V) + \dim T^{-1}(0)$. I can't ...
1
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1answer
28 views

Sequence of Mobius Transformation

Let $ T(z) = \frac {z+2}{2z+1} $. Now it follows that: $ T_1(z) = T(z), T_2(z) = T(T_1(z)), T_3(z)=T(T_2(z)) .... T_{n+1}(z)=T(T_n(z)) $ I'm trying to prove this sequence at the nth terms, but I ...
2
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1answer
70 views

Solving a cubic polynomial equation.

Overview I have tried finding a solution to this problem myself and I have flailed. Its just a challenge for me. could you please tell me how far am I in solving this question? My approach for ...
0
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1answer
36 views

Can someone help ? I have this answer of linear transformation.

Onto? What we have in class that if $n=2$ and $m=3$ that clear $2<3$ ,$T$ will be not onto. He said make three point u have with three variable $=(y,z)$ .then u will have one free variable that ...
1
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1answer
24 views

Transformation shifts parallelogram to trapezoid - fairly simple

We are given the region $D= {\{(x,y) | 1 \leq x-y \leq 2, x \leq 0, y\leq 0\} \subseteq \mathbb R^2}$ I drew this region on a piece of paper, it resembles an infinite parallelogram on the third ...
0
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1answer
49 views

Do T and T* have the same eigenvalues with the same algebraic multiplicity?

I know that the eigenvalues of T* are the conjugates of T's eigenvalues , but how can I see each eigenvalue of T and it's conjugate , the eigenvalue of T*, have the same algebraic multiplicity?
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2answers
30 views

Prove that this transformation is a reflection

Let $ v \in V$ be a unit vector in a Euclidean vector space. Prove that the endomorphism $$\phi: V \to V, \qquad \phi(x)=x - 2<x,v> v$$ is a reflection. I know that a reflection is an ...
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0answers
16 views

Existence of a particular transformation

I've a set of data points $S = \{ x | x\in [0,1]\}$ (i.e. real values from the unit interval). In some cases I've big clusters in the data and I want to spread the values in between the unit interval ...
0
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1answer
30 views

Randomly generating special affine transformations

I want to generate many random special affine transformations, that is, affine transformations that preserve volume (determinant equal to 1). I need quite a few of them. Is there a better way than ...
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0answers
27 views

change of a basis

lets say i have following bases: $C=\left\{\begin{bmatrix}1\\ 1\end{bmatrix},\begin{bmatrix}1\\ -1\end{bmatrix}\right\}$ $S=\left\{\begin{bmatrix}1\\ 0\end{bmatrix},\begin{bmatrix}0\\ ...
1
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1answer
115 views

Analytic geometry - rotation + translation

In $K=O\vec{e_1}\vec{e_2}\vec{e_3}$ I have to find the analytical representation of the screw motion( rotation + translation) $\psi$ with a rotation axis $g$ given by the points $A(5,-4,3)$ and ...
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2answers
30 views

trace function ($2\times2$) with ordered bases as linear transformation

We got trace function as following: $$\operatorname{tr}\begin{pmatrix} a & b\\ c & d\\ \end{pmatrix}=a+d$$ So now have to write down $[\operatorname{tr}]_{S_1,S_2}$, ...
1
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3answers
38 views

Find $T(x,y,z)$ for $(x,y,z)$ in $\mathbb R^3$

I have a question with the following facts: $T: \mathbb R^3\to\mathbb R^3$ is a linear transformation represented by the basis $B=((1,0,0),(1,1,0),(1,1,1))$, by the matrix $[T]_{B}$, where $[T]_{B}$ ...
1
vote
1answer
30 views

Möbius transformation image

Let $f(z)=\frac{az+b}{z+d}$, when $d\in\mathbb{R}$, $d\not=0$ $a,b\in\mathbb{C}$ and $f$ is not constant. I want to find the image of the real and imaginary axes under $f$. I've found that the image ...
0
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0answers
19 views

Find the reference point required to transform scale two elements uniformly

This is actually a programming issue I am having but the answer is rooted in matrix mathematics so this seems like the best place to ask it. I am no mathematician so I apologise if some of my concepts ...
0
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1answer
26 views

Finding if a linear transformation exist [duplicate]

I have to following question which I find hard to solve. Is there a linear transformation from $\mathbb{R}^5$ to $\mathbb{R}^4$ such that: $T:\mathbb{R}^5→\mathbb{R}^4$ Such that: ...
0
votes
3answers
54 views

find the vector $(x,y,z) \in \mathbb{R}^3$ and the constants $\lambda \in \mathbb{R} $ such that $T(x,y,z) = (\lambda x, \lambda y, \lambda z )$

Let $T : \mathbb{R}^3 \rightarrow \mathbb{R}^3$ defined by : $$T(x,y,z) = (x-y+4z,3x+2y-z,2x+y-z)$$ How can i find the vector $(x,y,z) \in \mathbb{R}^3$ and the constants $\lambda \in \mathbb{R}$ ...
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1answer
26 views

Endomorphisms and Invariant Subspaces

I have a question or two regarding the following exercise: Let $\alpha$ be the endomorphism of $\Bbb{Q}^4$ defined by: $$\alpha : \left[\begin{matrix}a \\ b \\ c \\ d \end{matrix}\right] \mapsto ...
0
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1answer
20 views

dimension of kernel and image of isomorphism

T:V ->V is isomorphism, dim V = n. The kernel of isomorphism has only vector 0 in it, so by rank nullity theorem does it mean that dim of kernel is 1 and dim of image is n-1? the question seems a ...
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3answers
32 views

Query regarding Linear Transformation…

As we always read in Complex Analysis, Linear Transformation (L.T.) is a combination of Translation, Rotation and Magnification i.e. $T(z)=az+b$ is a L.T. in complex. However, It doesn't satisfy the ...
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0answers
47 views

One to one Bivariate Transformation

Why does the below show the transformation is one to one? These are lecture notes ( the text and the blue writing)
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1answer
80 views

Jacobian of a transformation in cylindrical coordinates

In an area called transformation optics, they transform Maxwell equations from one space coordinate system to another, and then somehow obtain the properties of background material $(\epsilon , \mu)$ ...
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0answers
25 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} ...
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0answers
17 views

PDE equation conversion to parabolic PDE problem

Hello guys I am new here and I accept a serious problem with my exercising. Pronunciation says that I have to transform the dependent variable of this equation $\frac{\partial V}{\partial ...
0
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1answer
19 views

Show there's no ordered basis $E$ with the following conditions

Let $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$ such that: $$T\left( {\matrix{ x \cr y \cr } } \right) = \left( {\matrix{ 2 & 1 \cr 3 & 4 \cr } } \right)\left( {\matrix{ ...
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1answer
26 views

Equation of matrices

Let $V$, a 3d vector space above $F$. Let $T:V\rightarrow V$, linear transformation and $E$, an "ordered" basis such that: $$[ T ]_E = \left( \matrix{ 0 & 0 & a \cr 1 & 0 & ...
2
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0answers
15 views

Finding the matrix ${\left[ T \right]_E}$

Let the matrix ${\left[ T \right]_{B \to E}}$, the matrix where: $${\left[ T \right]_{B \to E}}{\left[ v \right]_E} = {\left[ {T(v)} \right]_B}$$ It's given that: $${\left[ T \right]_{B \to E}} = ...
2
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2answers
32 views

Inverse Z-tranform with a complex root

The z-transform of a signal is $$ X(z)=\frac{1}{z^2+z+1}$$ I attempted to solve for the the inverse z-transform by decomposing the denominator into complex roots, $\alpha$ and $\alpha^\ast$, to get ...
2
votes
0answers
72 views

Finding transformation from $T : \Bbb R^5 \rightarrow \Bbb R^4 $ …

Is there a Linear Transformation from $T : \Bbb R^5 \rightarrow \Bbb R^4 $ so $$\operatorname{Ker}T = \{( x,y,z,t,w) \in \Bbb R^5 \; | \; x = 2y, \text{ and, } z = 2t = 3w\}$$ if so find an example of ...
1
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1answer
66 views

Calculating the adjustment translation to be applied after rotating and scaling so that operations pivot about a given point.

I have a matrix for transforming an image into a target frame. The matrix is a function of a scale, $s$ rotation angle, $\theta$, and a translation that is applied after rotating, $tx, ty$. The ...
0
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0answers
12 views

When is $c_1 \cdot f(g(x+c_2)) = f'(x)g(x)$?

We are allowed to pick and $c_1, c_2$ that helps make this question easier. So when is $$c_1 \cdot f(g(x+c_2)) = f'(x)g(x) \tag{1}$$ Also, separately, I'm wondering: $$c_1 \cdot f(g(x+c_2)) = ...
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4answers
201 views

What functions have the property that $\frac{d}{dx}f(x) = c \cdot f(x+1)$?

If we are allowed to pick any real-valued constant $c$ that helps, when does $$\frac{d}{dx}f(x) = c \cdot f(x+1)$$ In other words, when does the derivative of a function $f(x)$ equal some constant ...
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0answers
18 views

PCA - How to calculate the scores

I'm currently learning Principle component analysis and I have, so far calculated the Eigen values and vectors. Assume that I have the following: $$ E = \begin{pmatrix} 1 & 2\\ 3& 4 ...
1
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0answers
21 views

Linear Transformation - linear algebra question [duplicate]

$T:\mathbb{R}_2[x] \mapsto \mathbb{R}_2[x]$ s.t.: $$ \begin{array}{l} T(1) = 3+2x+4x^2, \\ T(x) = 2+2x^2, \\ T(x^2) = 4+2x+3x^2. \end{array} $$ Is there base $B$ of $\mathbb{R}_2[x]$ that $[T]_B = ...
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1answer
48 views

Impulse response and z transform question?

We have $g(k)=\{ [(1/5)^k]u(k)\text{ for $1 \le k\le3$ and $0$ for other }k\}$ The input is $x(k)=\delta(k) +3\delta(k-1)+ \delta(k-2) $ Using Z transform we have to find the output $y(k)$ and the ...
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1answer
55 views

What's the pdf of $Z=X^2 +2X$ if $X$ is a standard normal? [closed]

Le be $X$ distributed as a standard normal. What is the density function of $Z=X^2 +2X$? Thanks for your help
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2answers
40 views

Transformation of the points on a plane

How do I transform a point $(x,y,z)$ on plane $\Pi (ax + by + cz = 0)$ to a point $(x',y',z')$ on plane $\Phi(ax+by+cz+d=0)$? What matrix should I use? Here is a 2-D representation of what I'm ...
0
votes
1answer
21 views

Find linear map of a transformation without neither known transformed or transformation mitrix

Consider the linear map from $R^3 \rightarrow R^3$ which takes $\vec{e_1}$ to $\vec{a_1}=\begin{bmatrix} 1\\0\\-1\end{bmatrix}$, takes $\vec{e_2}$ to $\vec{a_2}=\begin{bmatrix}0\\1\\3\end{bmatrix}$, ...
1
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1answer
46 views

linear transformations with matrices $A, A^*$

Let $K$ be a field, $K\subseteq \Bbb C$. $V$ is a linear space over $K$, $\dim(V)=n(n\geq2)$. Choose ordered basis $\epsilon_1,\epsilon_2,\dotsc,\epsilon_n$ for $V$. $\bf A,B$ are two linear ...
0
votes
0answers
15 views

Generate rotations about X & Y axes between certain 3D vectors

Given a semi-arbitrary 3D vector (the z will always be positive for my purposes), how could I find rotation about the X and Y axis? Alternatively, how might I simplify an XYZ rotation to the X and Y ...
0
votes
1answer
37 views

Transforming a curve on an arc to a line

I have a function, actually a point cloud, (similar to a sine wave) on an arc with a known radius of curvature. I need to remove the curvature to regenerate the original function (or point cloud). ...