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), ...

learn more… | top users | synonyms

0
votes
1answer
43 views

Proof : If $S,T:V \rightarrow V$ (V is finite) and $KerS=\{0\}$ then $Im(TS)=Im(T)$

I have this problem : Proof : If $S,T:V \rightarrow V$ (V is finite) and $KerS=\{0\}$ then $Im(TS)=Im(T)$ My solution : Let $v \in ImT$ exist $w \in V$ such that $T(w)=v$. Since $KerS=\{0\}$ ...
0
votes
1answer
32 views

Composition of translation and rotation is a rotation, but what is its center?

Consider the rotation $r_{\Omega,\alpha}$ of center $\Omega$ and angle $\alpha$.Furthermore let $t_{\vec{v}}$ be the translation by vector $\vec{v}$. Then $$t_{\vec{v}}\circ ...
0
votes
1answer
8 views

Bounds of a Bivariate Function

I am given that $h(x, y) = \frac{x}{(x+y)}$ , $x > 0$ , and $y > 0$. I am supposed to deduce that the bounds for $h(x, y)$ are $0 < h(x, y) < 1$, but I do not understand how to arrive at ...
2
votes
1answer
243 views

What is the difference between coordinates transformation and change of coordinates?

In the context on 3D computer graphics, what is the difference between coordinates transformation and change of coordinates? It can just be a matter of notation, but my book makes a clear distinction ...
0
votes
0answers
16 views

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 ...
0
votes
0answers
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 $\xi\in\mathbb{R}^n$, ...
1
vote
0answers
30 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|] ...
3
votes
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 ...
2
votes
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 ...
1
vote
0answers
46 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 ...
0
votes
3answers
67 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 ...
0
votes
2answers
38 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\\ ...
1
vote
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 ...
0
votes
0answers
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 ...
2
votes
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 ...
4
votes
1answer
56 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 ...
0
votes
2answers
87 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 ...
0
votes
0answers
92 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 ...
0
votes
1answer
83 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 ...
0
votes
1answer
121 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, ...
-1
votes
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 ...
0
votes
1answer
24 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 ...
1
vote
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 ...
2
votes
2answers
80 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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
0answers
21 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. ...
1
vote
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 ...
2
votes
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) ...
0
votes
1answer
61 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 ...
1
vote
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 ...
0
votes
0answers
25 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} ...
0
votes
1answer
127 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
votes
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 ...
0
votes
2answers
48 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) ...
0
votes
2answers
52 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 ...
7
votes
2answers
119 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 ...
1
vote
1answer
64 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 ...
0
votes
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 ...
1
vote
1answer
53 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 ...
-2
votes
1answer
39 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 ...
0
votes
0answers
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 ...
0
votes
0answers
18 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 ...
5
votes
1answer
53 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 ...
0
votes
0answers
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 ...
1
vote
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 ...
0
votes
2answers
59 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 ...
0
votes
0answers
25 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 ...
2
votes
1answer
60 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
votes
1answer
59 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 ...