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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Show: $s_A\circ s=s\circ s_B \iff$ $d$ is the bisector of $[AB]$

Consider two distinct points $A$ and $B$ of the plane space. Let $d$ be a line different from $(AB)$. Denote $s_A$ (resp. $s_B$) the central symmetry of center $A$ (resp. $B$). Denote $s$ the symmetry ...
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15 views

Equality with fourier transform

I have problem with the following equality where the Fourier transform appears: Assume that $u_1,u_2:\mathbb{R}^n\to\mathbb{C}$ are Schwartz functions. Prove that for any $\xi\in\mathbb{R}^n$, ...
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0answers
24 views

probability integral transformation and distribution of P= P[ |T| <= |t|] .

The task is to find the distribution of P. where , P=P[ |T| <= |t|]. (T is a continuous random variable with PDF f(t)). now , I tried to make the following two arguments : 1.P= P[ |T| <= |t|] ...
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2answers
58 views

Why do a set of continuous transformations form a manifold?

I am reading Sean Caroll's book on GR, and he defines manifolds to be "a space that may be curved and have a complicated topology, but in local regions looks just like R$^n$. Here by "looks like" we ...
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1answer
22 views

Expressing the inverse of $C(x) = (\langle x,a_i \rangle )$

Assume we have the following linear transformation: $$C(x) = \tilde{x} = \left( \begin{array}{c} \langle x, a_1 \rangle\\ \vdots \\ \langle x, a_k \rangle\\ \vdots \\ \langle x, a_n \rangle ...
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0answers
45 views

Is there an interesting interpretation of the ROWS of an affine transform matrix?

Context: I have a question about affine transform matrices in 3-space. Matrices are 4x4, with the right-most column being the translation, and the bottom row being [0,0,0,1]. In discussions I read ...
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3answers
57 views

Find the transformation matrix that has kernel the span of two vectors

Find a linear map $T : R^4\to R^3$ whose kernel is generated by $v=(1, 2, 3, 4)$, $w=(0, 1, 0, 1)$ This question has been answered but I don't understand the workings. In particular I don't ...
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2answers
32 views

Find all Linear Transformations with specified Image and Kernel [closed]

$T: R^3 -> R^3$ Find all the linear transformations such that: The vectors $$v = \left( \begin{array}{c} 1\\ -1\\ 0\\ \end{array} \right)$$ $$w = \left( \begin{array}{c} 0\\ 1\\ 1\\ ...
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1answer
35 views

Understanding transformation as algebraic structure

I am confused about the following structure, and would be very thankful if somebody could give me a hint. Let $\mathbb{S}$ be a set with n elements $\mathbb{S}=\{a_1, a_2, ..., a_n\}$, and $(x,y) \in ...
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20 views

Transformation Matrix with respect to a basis. [duplicate]

I have a question regarding transformation matrices with regard to a basis. Lets say that there is a basis $B = \{v_1,v_2,v_3\}$ There is a formula that says that $[A]_B = C^{-1}AC$ where $C$ is a ...
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1answer
33 views

Is $f: F \to R, \ (a_j)_{j \in \mathbb N} \mapsto \sum_{j \in \mathbb N} \ a_j $ bijective and find the inverse function!

$F$ is the set of the sequences in $\mathbb C$ and $R$ is the set of the series in $\mathbb C$. $f: F \to R, \ (a_j)_{j \in \mathbb N} \mapsto \sum_{j \in \mathbb N} \ a_j $ Now $\sum_{j \in \mathbb ...
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1answer
55 views

Fourier transform of $f(x) = \chi\cos^n(\pi x)$

I ran across an abandon post from 2013 where the OP has no work shown but just a problem statement. The OP was last seen May 2013 so I doubt they will be returning to edit their post with relevant ...
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2answers
78 views

Proving the image of a parallelogram is a parallelogram after a linear transformation.

Let T be an invertible linear transformation from R2 to R2. Let P be a parallelogram in R2 with one vertex at the origin. Is the image of P a parallelogram? How would I go about finding this out? I ...
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0answers
80 views

Geometry question: translating a rectangle according to a specific rule

Please take a look at the figure below. I have two line segments: a, which goes from point A to point B, and b, which goes from point B to point C. Each line defines a rectangle, which has width d and ...
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1answer
79 views

How to translate to a specific point with rotational transformation.

Basically I have two rectangles. ABCD and EFGH EFGH is rotated around it's centre point (X) ABCD has centre point (W) I also know for the sake of this example that EFGH is rotated counter clockwise ...
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1answer
92 views

Reflection of a vector across a line

For homework, I need to find the reflection of the vector <1,1,1> over the line defined by all the scalar multiples of <2,1,2>. I tried looking at the other questions about similar topics here, ...
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1answer
19 views

Finding system of equations such that

The question is: In $\mathbb{R}^4$ plane $V$ is given, $V=\mathrm{span}(\alpha_1,\alpha_2)$ where $\alpha_1=[1,3,4,1]$, $\alpha_2=[1,2,2,3] $ a) Find the formula for isomorphism ...
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1answer
23 views

Composition of linear transformations different

In $\mathbb{R} ^3$ a base $A=\{\alpha_1,\alpha_2,\alpha_3\}$ and in $\mathbb{R}^2$, $B=\{\beta_1,\beta_2\}$ are given, where $\alpha_1=[1,1,1],\alpha_2=[1,1,0],\alpha_3=[1,0,0]$ and ...
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3answers
44 views

Isomorphism of linear map

Suppose we have an equation of linear transformation $\varphi : \mathbb{R}^4 \rightarrow \mathbb{R}^4$. How to show that such transformation is isomorphic and how to find inverse isomorphism ...
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2answers
78 views

Finding a Möbius Transformation given constraints

I am trying to solve this problem, but am running into very complicated solving, and think that there is a simpler approach that I am missing. Find a Möbius transformation $M(z)$ that satisfies ...
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1answer
20 views

Finding the matrix of linear transformation

What is the orthogonal projection on the line of equation $x = y$ of the point $\begin{pmatrix} 3 \\ -1 \end{pmatrix}$? Assume this is a linear transformation. The matrix for this linear ...
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2answers
50 views

How to define an affine transformation using 2 triangles?

I have $2$ triangles ($6$ dots) is a $2D$ plane. The points of the triangles are: a, b, c and x, y, z I would like to find a ...
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20 views

Surjective transformation between $ \# A =\# B $ is also injective?

I am having two non-empty, finite sets $A$ and $B$. They have the same cardinality $ \# A =\# B $. There is a function $f: A →B$. Now I want to proove that if $f$ is surjective, $f$ is injective too. ...
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1answer
29 views

Obtaining a Transformed Matrix

I have a matrix $$m = \begin{bmatrix} 0 & 2 & 1 & 4 & 3 \\ 1 & 0 & 3 & 2 & 4 \\ 3 & 1 & 0 & 2 & 4 \\ 4 & 3 & 1 & 0 & 2 \\ 4 & 3 ...
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1answer
15 views

Show that $T^n(x,y)=\left(x+n\alpha \mod 1, y+nx+\frac{n(n-1)}{2}\alpha \mod 1 \right)$

Let $\mathbb{T}^2=\mathbb{R}^2 / \mathbb{Z}^2$. Let $T: \mathbb{T}^2 \rightarrow \mathbb{T}^2$ be the transformation. Let $\alpha \in \mathbb{R}$. $$T(x,y)=\left(x+\alpha \mod 1, x+y \mod 1 \right) ...
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1answer
59 views

Find a matrix such that the image is equal to the solution space of a linear system of equations

x1 + 2x2 + x3 − x4 = 0 −x1 + 2x2 + x3 + x4 = 0 x1 + x3 = 0 Consider the following matrix A from the system of equations: $$A = \left(\begin{array}{crc} 1 & 2 & 1 & -1\\ -1 & 2 ...
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1answer
26 views

Functional equation for polynomials

While reading a chapter entitled "Functional equations for polynomials" in the book "Polynomials" by Victor Prasolov, he states that Every polynomial $f$ of degree $n+1$ satisfies the identity ...
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23 views

Normal forms of vector fields

Consider the $m$-parameter family defined by \begin{align} \left(\begin{array}{c}\dot{x}_1\\\dot{x}_2\end{array}\right)&=J\textbf{x}+ \left(\begin{array}{c} ...
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1answer
111 views

Transpose transformation matrix with respect to the base R2x2

I found the following transformation matrix dor the transpose of a 2x2 matrix in $R^{2x2}$ (vector space of the 2x2 matrices with real numbers as elements). \begin{bmatrix} 1 & 0 & 0 ...
2
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1answer
28 views

Transformations of diffeomorphism $f(z)=e^{i\alpha}z+\overline{z}^3+z^2\overline{z}$ that eliminates $\bar z^3$

Find a transformation of the form $z=w+a\overline{w}^3$ such that $$f(z)=e^{i\alpha}z+\overline{z}^3+z^2\overline{z}$$ where $\alpha\neq2\pi p/q,\ q=1,2,3,4,$ becomes ...
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2answers
47 views

Determine the image of the strip $S$ consisting of all points $z$ with $\frac{-\pi}{2}\lt Re(z) \lt \frac{\pi}{2}$ and $Im(z)>0$ under $w=i\sin z$

$\color{green}{\text{transformation is}\space w=i\sin z}$ $$w=i\sin z = i\sin(x+iy)=\frac{1}{2}\left(e^{ix-y}-e^{-(ix-y)}\right)=-\cos(x)\sinh(y)+i\sin(x)\cosh(y)$$ $\therefore u = -\cos(x)\sinh(y) ...
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2answers
47 views

Find the region in the w-plane to which the line y = 1 is transformed by $\frac{1}{z}$

I tried to do the following: $$w=\frac{1}{z}=\frac{x-iy}{x^2+y^2}$$ $\implies u = \frac{x}{x^2+y^2} and\space v = \frac{-y}{x^2+y^2}$ $\color{green}{need\space to\space transform\space the\space ...
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110 views

Transform polygons into one another?

I am aware that there must be no standard way to achieve this, but I don't know what has been done so far. I feel like I'm missing keywords to investigate further. I have any two 2D polygons $a$ and ...
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1answer
63 views

What's the exact definition of symmetry?

My question is basically the same as this, but I haven't found the answer given satisfying. The definition of symmetry I've come up with is this: Let $X \subseteq R^2$. A symmetry of $X$ is an ...
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0answers
23 views

Mobius transformations — $L^{-1} \times M\lbrace0,1,\infty\rbrace\times L$

$$L^{-1} \times M\lbrace0,1,\infty\rbrace\times L$$ I cannot seem to get the right answer when I multiple them out. $$(1-i)z \cdot (1-z) \cdot \frac{z}{1-i}$$ What do you get when you multiple ...
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1answer
50 views

Help understanding the output from Apache Commons Math's Discrete Fourier Transform

I'm using a discrete Fourier transform to translate a finite set of samples to the frequency domain. I'm trying to start with a very simple set, but am still getting confused. I'm starting with this ...
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1answer
38 views

Find basis of an image

Let $ \psi : \mathbb{R}^4 \rightarrow \mathbb{R}^3$ be a linear transformation described by a formula $$\psi ([x_1,x_2,x_3,x_4])=[x_1+x_3+x_4, -x_2-x_4,x_1+x_2+x_3+2x_4].$$ Find basis of image ...
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11 views

How can I make this tangent function only appear once (or be spaced very widely)?

I only want the function to go from $x=5$ to whenever the function is 4.5 (in other words, when $y=4.5$). Is there any way to do this without specifying the domain? It has to have the shape of the ...
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0answers
17 views

Distribution invariant to orthogonal transformations

Let $X,Y$ be two real valued stochastic variables and define $$ \begin{pmatrix} \tilde{X}\\ \tilde{Y} \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} X\\ Y ...
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1answer
51 views

Is every symmetric matrix diagonalizable?

I know that Hermitian matrices are always diagonalizable and real symmetric matrices are real Hermitian matrices and therefore diagonalizable. But, it is always not the case that a symmetric matrix ...
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27 views

Searching for a constant transformation in $ \mathbb C$

I am having a continous transformation: $f: \mathbb C \to \mathbb C $ with a set $B \subseteq \mathbb C $, which is bounded. Now I want to proove that $ A = f^{-1} (B)$ is NOT bounded! I know it ...
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0answers
46 views

Prove that the continuous $f: \mathbb C \to \mathbb R$ has a global max and min

I am having this continuous transformation $f: \mathbb C \to \mathbb R$ and $\ f\ (\mathbb C)$ is bounded Now I have to prove that there are a global maximum and a global minimum. My thoughts: I ...
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2answers
56 views

Prove that a continuous inverse-transformation of $f: [0,1) \cup \{ 2 \} \to [0,1]$ exists

I am having this transformation $f: [0,1) \cup \{ 2 \} \to [0,1]$ $$f(x) = \begin{cases} x & x \neq 2 \\1 & x = 2 \end{cases}$$ I've already proved that it is continuous. Question: Is ...
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23 views

Approximation of a circle arc based on two lines, is this correct? How to rotate it afterwards?

Intro: We have two wheels that are rotated by something called T. I can tell T to rotate both wheels so much that they roll forward d cm (as shown in box A on the picture). I also have to keep track ...
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1answer
58 views

Can anyone check these true and false statements about linear algebra?

For any square matrix $A$, the image of $A^7$ is contained in the image of $A$ I think this question is asking If $A^7x=b$, then $b$ must be in $A$ with some vector $y$ such that $Ay=b$. It Seems ...
0
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1answer
51 views

Find the transformation that maps real axis to itself and imaginary axis to the circle $|w-\frac{1}{2}|=\frac{1}{2}$

Find the transformation that maps real axis to itself and imaginary axis to the circle $|w-\frac{1}{2}|=\frac{1}{2}$ What I did: $$z_{1}=0,z_{2}=i,z_{3}=\infty ...
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1answer
43 views

Transformation of coordinate axis to make matrix diagonal

Consider the matrix $$ A= \begin{bmatrix}1/8 & \frac{-5}{8\sqrt{3}} \\ \frac{-5}{8\sqrt{3}} & 11/8 \end{bmatrix} $$ which of the following transformations of the coordinate ...
3
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1answer
48 views

How to proove that a bijective transformation is NOT continous

I am having this transformation $f: \mathbb R \to \mathbb R$ $$f(x) = \begin{cases} x & x \in \mathbb R \setminus \mathbb Q \\x+1 & x \in \mathbb Q \end{cases}$$ I've already prooved ...
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0answers
22 views

Derivative Nullity for nonpolynomial spaces

One thing has been bothering me about derivatives, it's easy to explain nullity of a polynomial, since a term that is constant after n many derivatives will become zero at n+1 many derivatives. How ...
0
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0answers
46 views

How to determine roll-pitch-yaw parameters from homogeneous transformation matrix

We have a transformation matrix $T = \begin{pmatrix} cos(\theta_1) & sin(\theta_1) & 0 & l_1 cos(\theta_1) \\ sin(\theta_1) & -cos(\theta_1) & 0 & l_1 cos(\theta_1) \\ 0 ...