# Tagged Questions

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### Transform gradient to reference element

Minimal example of the problem My attempt I think this is not a linear solution like $$\nabla u = \nabla A_K x + \nabla b_K$$ which must be wrong because $A_K$ is a ...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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' ...
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### 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 $r$.
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
### 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. ...
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 $T( \overrightarrow{x} )=A\overrightarrow{x}$, ...