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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Linear transformation of vector

I have computer graphics class and i had something like that on lecture: $$ \begin{bmatrix} \overrightarrow{b1} & \overrightarrow{b2} & \overrightarrow{b3} \end{bmatrix} \begin{bmatrix} c1\\ ...
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0answers
25 views

What is a transformation that can't have shearing called?

What is a transformation called when it can have separate scaling for x and y, rotation, and translation, but it cannot have shearing or scaling AFTER rotation? Basically if this transformation is ...
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1answer
29 views

coordinate transformation and tensor

A 2 dimensional Euclidean space is represented by two different coordinate systems: the Cartesian system $(x_1,x_2)$ and an alternative system $(\xi^1,\xi^2)$ where ...
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0answers
20 views

Linear Algebra - verification of my answer, basis for $ImT$

I'd like to verify this answer, because I think that the answer in my book is incorrect. I'll be very glad if someone could tell me, if the basis I found for $ImT$ is correct. Let : $T:R^3 ...
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1answer
22 views

Matrix transformation conserving the “positive semi-definite” aspect

Let's say I have two covariance matrices $A$ and $B$ (so they're both positive semi-definite), What kind of transformations can I apply on either one of them or both without loosing the ...
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2answers
36 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 $KerS=\{0\}$ ...
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0answers
7 views

Composition of a rotation and a homothetic transformation of different centers?

Consider the rotation $r_{\Omega,\alpha}$ of center $\Omega$ and angle $\alpha$. Furthermore let $h_{\lambda,S}$ be the homothetic transformation of center $S\neq \Omega$ and ratio $\lambda$. What ...
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1answer
15 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 ...
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1answer
4 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 ...
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1answer
163 views

Fourier transform for a 2D curve

I am stuck on the following problem about a Fourier transform of a 2D curve: I have to calculate the Fourier transform (using 1D complex FT) (and the opposite of it) for a 2D curve z(t). The curve is ...
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1answer
24 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 ...
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1answer
18 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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0answers
15 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 ...
2
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1answer
444 views

Exponential Function Shifts

I have some confusion about shifts concerning exponential functions. I can best describe my question with an example. Take $y = e^{-(x-3)}$. This graph has a reflection over the $y$-axis and is ...
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0answers
18 views

Study the caracteristics of the transformation $f=r\circ t \circ h$.

Let $OABC$ be a square with $(\vec{OA},\vec{OC})=\frac{\pi}{2}$. Let $r$ be the rotation of center $B$ and angle $\alpha=\frac{\pi}{2}$, $t$ the translation of vector $\vec{CA}$, $h$ the homothetic ...
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2answers
346 views
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0answers
14 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
14 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
48 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
17 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
41 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
31 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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1answer
31 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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2answers
22 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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0answers
17 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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0answers
15 views

Is this transformation affine? [on hold]

When transforming coordinates from $P(x,y)$ to $P'(u,v)$ using equations $u=x*y\mid a$ and $v=x/y\mid b$, where $a\mid b$ means that $a$ divides $b$, i.e. $b$ is a multiple of $a$, is this ...
4
votes
1answer
37 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 ...
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 ...
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2answers
45 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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2answers
512 views

Approximate a polynomial function using a sum of sine waves

I have a polynomial function which I need to approximate by a sum of sine waves with constant amplitude along a given domain. From what I hear, this might be a good time to make use of Fourier ...
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0answers
47 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
63 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
23 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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2answers
58 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
19 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
40 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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1answer
28 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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2answers
40 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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1answer
18 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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0answers
18 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. ...
2
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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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0answers
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What makes the Gaussian kernel so magical for PCA, and in general? [migrated]

I was reading about kernel PCA (1 2 3) with Gaussian and polynomial kernels. How does the Gaussian kernel separate seemingly any sort of nonlinear data exceptionally well? Please give an intuitive ...
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1answer
47 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
2k views

Negative & Positive Shear Factor

My question relates to constructional geometry & matrices aren't to be involved in the solution because stated Math level is up to O Levels... The figure below shows shear with y=3 as invariant ...
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1answer
24 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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1answer
56 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 ...
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2answers
63 views

For general non-symmetric square matrices is there a matrix norm that is invariant under similarity transformations?

I think that there is no similarity-invariant matrix norm for general matrices. But are there similarity invariant norms for special types of matrices (e.g. for matrices whose eigevalues are different ...
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21 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
513 views

Finding reflection of a matrix

To 2 decimal places, what is the value of the lower-right entry in the reflection matrix $Q_a $if a = 1.05? Not even sure where to begin, is there a formula? This is what I could find in my textbook ...
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1answer
37 views

how to find triangular point from a side

i have two triangles. Say a , b , c and p, q, r and the projection of the abc to pqr a - > p b - > q c - > r here known point values are a b c p q and r unknown. $\overline{PR}=\overline{AC}$ ...