Transformation has many meanings in mathematics. If using this tag, add another tag related to the object being transformed. If there is a tag for your specific kind of transformations, use that one instead: e.g., (laplace-transform), (fourier-analysis), (z-transform), (integral-transforms), ...

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126 views

How to apply perspective transform to Bezier curve?

I found that both Bezier curves and B-splines are described with a formula $p(t)=\sum\limits_{i=0}^d B^i_m p_i$ but in the case of B-splines $B^i_m$ are B-spline blending functions, while for Bezier ...
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71 views

Is the linear transformation $T(f(t))=t(f(t))$ from $P$ to $P$ an isomorphism?

Is the linear transformation $T(f(t))=t(f(t))$ from $P$ to $P$ an isomorphism? I can say it is since: The dimensions of domain and codomain are equal and $\text{ker}(T)$ is the function ...
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45 views

What's the difference between these two transformations of functions?

I'm about to graph the transformation of a function, but in this problem I encountered something new. The function transformation looks like this: y=12(f(x)+2) Thing is, I've never seen the f ...
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126 views

What is a Sub Formula and What is a Maximal Sub Formula in Propositional Logic

What is a Sub formula of a Propositional Formula? Suppose I have a formula C or -C Then what are the sub formulas of this and what is the maximal sub formula of this Propositional Formula. I am a bit ...
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2answers
123 views

How to efficiently encode this?

I have 5 ring oscillators whose frequencies are f1, f2, ..., f5. Each ring oscillator (RO) has 5 inverters. For each RO, I just randomly pick 3 inverters out of 5 inverters. For example, in RO1, I ...
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1answer
101 views

Dimension of Image of Composition of Linear Transformations

Take two linear transformations T from V to W and S from W to U. I want to show that the dimension of the image of their composite SoT from V to U is 'smaller' than or equal to the dimension of the ...
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1answer
87 views

Images of Lines

I'm studying for this exam and one of the questions I am stuck on is: Find the image of the line $$3x-y+1 = 0$$ under the transformation $$z \mapsto \frac{2}{z+1}$$ So I know I have to convert the ...
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0answers
18 views

How transpose of a matrix helps in making better sense of the data

The transpose of a matrix is obtained by flipping it about its diagonal. What is a practical scenario where we gain better insight into a set of data points by transposing it?
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1answer
96 views

Prove that det(M) = det(N)

Let T:V --> V be a linear transformation, and let B and C be two bases for V. Let M be the matrix of T with respect to the basis B, and let N be the matrix of T with respect to the basis C. Prove that ...
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1answer
27 views

Vector Spaces V and W with linear transformation T: V --> W statements

I'm in a little slump with this question. I have a general idea, but I don't know exactly which theorem to pair them up with because I think that it may be too simple. Here is the question: For ...
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1answer
130 views

Historical reason to define a matrix vector product the way it is

What is the reason why we defined a matrix vector product (a transformation) this way: $$\begin{pmatrix} a_1 & a_2 \\ a_3 & a_4 \\ \end{pmatrix}\cdot ...
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41 views

Given $f_{X,Y}(x,y)$, what is the pdf of $Z=XY$?

I approached the problem as following: $f_{Z}(z) = \int_{0}^{\infty} f_{X,Y}(X=\frac{z}{y},Y=y)dy$ However, according to the textbook, the problem should be solved as below: $f_{Z}(z) = ...
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0answers
18 views

Can this transform be rewritten as a more standard integral transformation?

Here is the transformation pair I've been working with. $\hat{f}(n)=\displaystyle\lim_{a\to1}\sum_{j=0}^{\lfloor\log_a n\rfloor}(-1)^j\binom{k}{j}a^j f( a^{-j} n)$ ...
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2answers
380 views

Find the matrix A of the linear transformation T(M)

I know that if I substitute the first matrix for $T(M)$ I see what T does to each of the basis vectors. I don't understand how that creates a $3\times 3$ matrix though. I was looking at this ...
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132 views

Transformations of the complex plane

I was reading my text and just had some questions about transformations: (1) Are all line preserving transformations linear transformations? Why? I want to say yes... but I feel like the answer is ...
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1answer
294 views

Complex Numbers and Transformations

If a transformation t acts by rotating every point of the plane around the origin by $\pi/5$ clockwise and then proceeds to translate it by vector $v$ = $(1,2)$. How do I describe this ...
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1answer
143 views

Rotation of 2D polar graph in a 3D space along some fixed axis?

Does there exist some systematic way of rotating a 2-D polar graph $r=f(\theta)$ around some axis in a 3D space? For example: $f(\theta)=cos(\theta)$ in 2-D looks like: If we want to rotate the ...
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1answer
56 views

3D coordinate Transformation

I am currently trying to align two bodies which do not have similar sizes and shapes. But both of these two bodies share some keynodes(similar nodal position with 0.1% error difference). How can I ...
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1answer
66 views

If $f \in S(\mathbb R)$, can we say $\widehat{|f|} \in L^{1}(\mathbb R)$?

Let $f\in L^{1} (\mathbb R) := \{f:\mathbb R \rightarrow \mathbb C \ \text {measurable functions} : \int_{\mathbb R} | f(x)| dx < \infty \}$ and the Fourier transform of $f$, $\hat{f} (y) : = \int ...
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1answer
434 views

Finding Eigenvalues of Block Matrix

I have a block matrix of size $3N \times 3N$ of the form: $B = \left[\begin{array}{cccc} A & C & \dots & C\\ \vdots & A & \dots & C\\ C & \vdots & \ddots & ...
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1answer
39 views

Does the transformation $x=Pabc$, $y=Qab(1-c)$, $z=Ra(1-b)$ map a unit cube to a tetrahedron?

Does the transformation $x=Pabc$, $y=Qab(1-c)$, $z=Ra(1-b)$ map a unit cube in $abc$ coordinates to the tetrahedron with vertices $(P,0,0)$, $(0,Q,0)$, $(0,0,R)$ and $(0,0,0)$ in xyz coordinates? ...
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1answer
46 views

Please help on this matrix transformation problem

Let A be the matrix below and define a transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ by $T(U) = AU.$ For each of the vectors $B$ below, find a vector $U$ such that $T$ maps $U$ to $B$, if ...
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22 views

inequality related to transformations and inner products

Let $T$ be a bounded transformation from a hilbert space to itself. Suppose that if $||f||\leq 1$ and $||g||<1$ then $|\text{Re}(Tf,g)|\leq M$ where we are taking the real part of the inner ...
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35 views

Linear transformation diagonalization [duplicate]

I encountered this question and I need some assistance to solve it. $T$ is a linear transformation from $V \to V$ We are given that $T^2 = T$ Show that $T$ can be diagonalized.
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22 views

Transformations Difference

On what basis we depend when we choose the tool to transform to frequency domain, I can't distinguish in what case we use one of these transformations ( trig. fourier series, complex fourier series, ...
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1answer
164 views

Inversion of Circles

I'm studying for my exam and one of the questions I am stuck on is: Show that under inversion in the unit circle a circle with centre C and radius r inverts into a circle centre ...
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1answer
173 views

Geometry of Complex Numbers

Write down in the form ${Z}\rightarrow{AZ+B}$ the following transformations of the complex plane: (a) translation in the direction $(2,-3)$ (b) rotation about (0,1) through $\pi/4$ I know from my ...
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1answer
37 views

transform a formula for doubling time

I have a problem with transforming a formula for doubling time. It has two variables: increase (i) and number of periods (p). The original formula: $p = \frac{\log(2)}{\log (1+i)}$. When I ...
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2answers
210 views

Possibility of making diagonal elements of a square matrix 1,if matrix has only 0 or 1

Let $M$ be an $n \times n$ matrix with each entry equal to either $0$ or $1$. Let $m_{i,j}$ denote the entry in row $i$ and column $j$. A diagonal entry is one of the form $m_{i,i}$ for some $i$. ...
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1answer
67 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 ...
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2answers
72 views

Linear algebra and matrix.

prove or disprove : If A and B are 2 by 2 orthogonal matrices over R then A+B cannot be orthogonal. OR If S,T:R^2--->R^2 are orthogonal transformation then S+T is not an orthogonal transformation.
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1answer
52 views

Convert a matrix so it can be multiplied on the right side, instead of the left

Sometimes I come across examples of geometric transformations that multiply the transformation matrix on the left side of the vector and sometimes on the right. How do you modify a matrix so that ...
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1answer
49 views

Isomorphism between $\mathbb{F}^{m \times n}$ and $\mathcal{L}(V, W)$

Let $\mathbb{F}^{m \times n}$ be the vector space of all $m \times n$ matrices and let $\mathcal{L}(V, W)$ be the vector space of all linear maps from a vector space $V$ to a vector space $W$ ($V$ and ...
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61 views

Is $g:R^2\to R:(x,y)\mapsto xy$ a linear transformation?

In English: Denoting by $ G $ and $ U $ as the linear transformation and the vector space, respectively, we have: $\\$ First, check if the zero vector $0=(0,0)\in U$ is also present in the ...
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1answer
57 views

How do i determine the charasteristic function of X^2?

I'm wondering how I kind show that the charasteristic function of $X^2$ given that $X\in N(0,1)$ is $\varphi_{X^2}(t)=\frac{1}{\sqrt{1-2it}}$. I have tried using the change of variables such that ...
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2answers
90 views

Integral transform with Dirac delta

Let $f,g: \mathbb{R}^n \to \mathbb{R}$. Let $\delta$ denote the Dirac delta function. How can I write the integral over $\mathbb{R}^n$ (on the left hand side) as an integral over $g^{-1}(0)$ $$ ...
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2answers
125 views

Intuition for Geometric Transformations

I've been making a lot of effort over the past few hours to gain some intuition into the art of geometric transformation but to little avail. I would really like to be able to look at a transformation ...
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40 views

$T: \mathbb{R}^n \to \mathbb{R}^m$ and relationship with dimensions of matrix

If $T(x) = Ax$. If $A$ is a $n \times m$ matrix. ($n$ = rows, $m$ = cols) Is $T: \mathbb{R}^n \to \mathbb{R}^m$ always the values of dimensions of the matrix $n \times m$? So in other words, the ...
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1answer
42 views

inverse transform of $Z(\omega) =\frac{a}{\alpha-i\omega}$

I am stuck at calculating the inverse transorm of $Z(\omega) =\frac{a}{\alpha-i\omega}$. Can someone help me please? thanks
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47 views

The Fourier Stieltjes transform is uniformly continuous

Let $G$ be a locally compact Abelian group and $\hat{G}$ be its dual group, that is the group of all complex functions $\gamma:G\to\mathbb C$ such that ...
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2answers
72 views

What does the notation f(t)1(t) signify?

I've come across this notation in a text on Control Theory, Modern Control Engineering by Katsuhiko Ogata, in a discussion of Laplace transforms. There is little more context I can give than that. ...
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207 views

Find a transformation from tetrahedron to cube in $R^3$ to calculate a triple integral?

I would like to calculate the triple integral of a function $f(x,y,z)$ over a region given by a tetrahedron with vertices $(0,0,0)$, $(a,0,0)$, $(0,b,0)$ and $(0,0,c)$. I am trying to do this by ...
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1answer
26 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?
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Can you transform any coordinate from any “space” to another “space” that's defined?

This question pertains to Matrix Transformations. So to provide an example, if I have 3D coordinates where $X = -1$ to $1$, $y = -1$ to $1$, $z = -1$ to $1$. They are "normalized" in my mind. Can I ...
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1answer
2k views

How to find the kernel of a linear transformation $T:\mathbb{R^3}\to\mathbb{R^3}$.

How to find the kernel of the linear transformation $T:\mathbb{R^3}\to\mathbb{R^3}$ given by $T(x,y,z)=(x,2y,0)$? I do not quite understand how to do this! and How to find the kernel of the ...
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1answer
57 views

What does “transform among themselves” mean?

I'm reading a script on atomic physics, and there's a chapter on irreducible tensors. I can't understand the meaning of "transform among themselves" in this context: An arbitrary rotation of the ...
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1answer
33 views

Does this transformation have an inverse?

Let $f(n)$ be a complex sequence. Then for prime $p$ define $\hat{f}(p) = \sum_{n = 1}^{\infty} a_n e^{-i 2 \pi n / p}$. Then let the transformation of sequences be $T$, i.e. $Tf = \hat{f}$. Is ...
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78 views

base transformation rule significance in finding big o notation

Recall the equivalence: $$m=b^k \implies k = log_bm$$ as well as the base transformation rule: $$log_am=(log_ab)(log_bm)$$ Are the following true or false? (a) $log_2n$ is $O(log_3n)$ (b) ...
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1answer
97 views

Properties of Linear Transformations?

A linear transformation, $$T: \Bbb{R}^m \rightarrow \Bbb{R}^n$$ is a function that has the following properties. $$T(\text {u} + \text v) = T(\text u) + T(\text v)$$ $$T(\text{kv}) = \text ...
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39 views

Stuck on finding rank of $T$ when $n=8$ of $T(A) = A - A^T$

$T: M_{n\times n}(F) \to M_{n\times n}(F)$ is a linear transformation. I know from rank-nullity that $\text{rank}(T) + \text{nullity}(T) = \dim(M_{n\times n}(F))$. I'm trying to find $N(T)$ and then ...