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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mobius transformation form $M= B(0,1)\setminus\overline{B(1,\sqrt2)}$ to a sector.

I know that the intersections of the two circles need to be sent to $0$ and $\infty$ in order to get a sector $S = \{x+iy: x>0,0<y<x\}$ The intersections of the two balls were i and -i. So i ...
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The Composition of Two rotations

So far I rewrote the halfturns of d,c,b,a to halfturn (p,n)(m,l) where n=m because lines c and d are parallel so I can make ambiguous lines n and p parallel too. I also know that lines c,d can be ...
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Transformation that f(A+B)=f(A)+f(B) and f(AB) = f(A)f(B)

if $f:M_{n*n}(F) \rightarrow M_{m*m}(F) $ and f transformation identity to identity matrix and $f(AB) = f(A)f(B) , f(A+B)=f(A)+f(B)$. now we want to prove there is an integer like $k$ that $m = k*n$
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How principal component analysis ensures component orthogonality when using zero co-variance as the restriction to maximize variance?

I am currently learning the mathematics behind PCA and I found when PCA maximizes variance to find out the 2nd, 3rd, ... components, it uses zero co-variance as the restriction, as shown below, ...
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262 views

Transforming a function by a sequence geometric operations on its graph.

I am solving the following problem: Let $f(x) =\sqrt{x}$. Find a formula for a function $g$ whose graph is obtained from $f$ from the given sequence of transformations: shift right $3$ ...
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1answer
32 views

Find the image of the unit vector at the point $z_0=i$ under the function $f(z)=z^2+2z$

From this image find the rotation angle and the expansion factor. $z\in\Bbb C$ I am unsure how to find a way to accurately plot the image. I understand that the image should expand because the ...
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Prove that the map $f(z)=\frac{1}{z}$ sends any line onto either a line or a circle.

Show the cases in which the image is a line and the case in which the image is a circle. I understand that representing the equation of line, ($ax+by+c=0$ $a,b,c\in\Bbb R$ $a,b\neq0$ at the same ...
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1answer
33 views

Is centre of a circle remains invarient under Movius transformation?

Prove that the image of the center of a given circle is never the same as that of the circle image under a linear fractional map. Given some linear fractional map$f(z)=\frac{az+b}{cz+d}$, that is ...
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1answer
30 views

Linear Transformations

I have no idea how to work this question. Can someone offer some insight?
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48 views

How does $y=|x+3|+4$ become $y=\frac{1}{2}|2x+3|+4$ (compositions and translations)

Today, I had a test question that was bothering me because my friend and I had different answers to it. It's a grade 12 math question. It's telling us to explain the changes that were made to the ...
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4answers
70 views

Eigenvalues and Eigenspaces of a Projection

Let $P$ be the orthogonal projection onto a subspace $E \subset V$ ($V$ being an inner product space) with $\mathrm{dim(V)}=n$, $\mathrm{dim(E)}=r$. Obtain the eigenvalues and eigenspaces, along with ...
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Transformation of xy plane to polar coordinates. (What would be the bound of polar coordinate?)

I have a double integral $$\int_0^a \int_0^x (x^2+y^2)^{1/2} \operatorname d y \operatorname d x$$ So, I am double-integrating $r^2$ What would be the region of the polar coordinate..?
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1answer
25 views

Stereographic Projection from an Arbitrary Point

Let $p \in \mathbb{S}^{n}$, then the stereogaphic projection is a diffeomorpshim $h:\mathbb{S}^{n} \setminus \{p\} \to \mathbb{R}^{n-1}$. Suppose that $p$ is the 'north pole' ($p = (0,0,..,1)$), then ...
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1answer
31 views

Why is this laplace identity true $\int_{\Bbb{R}^+}\frac{f(t)}{t}\,dt = \int_{\Bbb{R}^+}\mathcal{L}\{f\}$?

I was wondering why this laplace identity is true? Does it follow from definition? $$\int_{\Bbb{R}^+}\frac{f(t)}{t}\,dt = \int_{\Bbb{R}^+}\mathcal{L}\{f\}$$ I'm trying to understand the first ...
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1answer
38 views

The Matrix of a Transformation from P2 to P1

I don't understand how T(1), T(x), and T(x^2) were found in the picture so I did it using another method I saw on StackExchange. (a + b)x - c => -c + (a + b)x + 0x^2 so the first row would be {0, 0, ...
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1answer
28 views

Line plane intersection

I have two planes in $\mathbb R^3$ as shown below: axes representation corrected after MvG's comment Each plane is a finite area, a rectangle with length and width $H_l, H_w$. Each plane has its ...
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32 views

inverse laplace tranform

I have a simple question, There are some functions f(t), g(t) and lets say F(s) and G(s) for the form of Laplace transform of f(t) and g(t), respectively. While I am solving differential equation ...
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0answers
25 views

How do I calculate 3D movement based on yaw, pitch and roll?

I'm creating a 3D game demo and I need to calculate the position of the player in the space (i.e. the player's x, y and z coordinates). I understand that this would be affected based on the camera ...
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1answer
20 views

Is a series that contains the index term a function of the same series without the index term?

Can it be shown that $U_{2} = \sum_{i=1}^{n} [i*g(Y_{i})]$ is a function of $U_{1}=\sum_{i=1}^{n} g(Y_{i})$ ? My intuition tells me that this is not true because of the changing (for lack of a ...
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changing variables and taking limits respectively

Say we have the equation $3\left[a\frac{d}{dx}+R\right]\frac{\alpha^{4}}{R^{2}}=R$ If we make the trasformation $\phi=\frac{\gamma}{R}$ and $a\rightarrow{a_{0}+\gamma{a_{1}}}$, where ...
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2answers
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Fractional Linear Transformation: Region between two circles to strip

I'm trying to find Fractional Linear Transformation (if one exists) that maps the region between the circles $\|z+1| = 1\}$ and $\{|z|=2\}$ to the region between the horizontal lines $Im(z) = 1$ and ...
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1answer
42 views

Prove the result of multiplying a complex number by (1 + i)

I know that if I multiply a complex number that has the form a + bi by the number (1 + i), this will rotate the vector that corresponds to the complex number by 45 degrees in the counterclockwise ...
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Reducing heat equation into nondimensional form

I want to get nondimensional form of heat equation $u_t=a(x,t)u_{xx}$. For the case of $a(x,t)=a(t)$, by setting $A(t)=\int_0^ta(\eta)d\eta$ and $t=\phi(\tau)$, where $\phi$ is the inverse mapping ...
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Make a conform map g that sends the unit disk to $A = \{w: \operatorname{arg}(w) \in(\frac{\pi}{4},\frac{3\pi}{4})\}$ such that $g(2i) = 0$

make a conform transformation g that sends the unit disk to $A = \{w: \operatorname{arg}(w) \in(\frac{\pi}{4},\frac{3\pi}{4})\}$ such that $g(2i)$. I actually solved it by taking the inverse of $f(x) ...
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38 views

Mapping A point from one 3D Coordinate System to Another 3D coordinate System with Euler Angles between the two systems given

Suppose I have a point in the green coordinate system, and I wish to describe it in reference to the orange coordinate system. I know the roll, pitch, and yaw of the green system with respect to the ...
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3 views

How to take dft of irregularly sampled function in k space?

I would like to take inverse dft of irregularly sampled complex function in k-space. I am just summing in a loop over the length of the k vector, but is quite slow. ...
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19 views

how can I find matrix of linear transformation for this specific problem

I am migrating my problem here with the hope of getting some assistance http://stackoverflow.com/questions/29029523/how-do-i-find-the-matrix-of-the-linear-transformation
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suppose $|a|<1$, show that $\frac{z-a}{1-\overline{a}z}$ is a mobius transformation that sends $B(0,1)$ to itself.

Suppose $|a|<1$, show that $f(x) = \frac{z-a}{1-\overline{a}z}$ is a mobius transformation that sends $B(0,1)$ to itself. To make such a mobius transformation i tried to send 3 points on the edge ...
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14 views

Which transformations preserve this?

Let $a,b,x,y,z\in \mathbb{Z}$ (with $a,b$ given) and consider the equation $a(x^2+y^2+z^2)=b(xy+yz+zx)$. Consider transformations taking $x$ to $px+p'y+p''z$, $y$ to $qx+q'y+q''z$ and $z$ to ...
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1answer
86 views

Which linear transformations preserve this?

Let $a,b,x,y\in \mathbb{Z}$ (with $a,b$ given) and consider the equation $a(x^2+y^2)=bxy$. Consider transformations taking $x$ to $px+p'y$ and $y$ to $qx+q'y$. For which integers $p,q,p',q'$ is it ...
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1answer
25 views

Matrix of $L(A)=A^{T}$ From $R^{2 \times 2} \rightarrow R^{2 \times 2}$

A bit of trouble with this question: Find the matrix of the linear transformation $L(A)=A^{T}$ From $R^{2 \times 2} \rightarrow R^{2 \times 2}$ with respect to the basis $\begin{bmatrix} ...
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1answer
33 views

Inverse Fourier Transform Proof

I am aware of how Fourier Transformation and Fast Fourier Transformation works, however I do not understand the logic of the inverse of FFT. Could someone explain why the inverse fourier ...
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1answer
42 views

Is monotony preserved under expectation?

let $X_1 \sim f_1(x)$ and $X_2 \sim f_2(x)$. Suppose we know that $\mu_1=E(X_1)<E(X_2)=\mu_2$ and let $\nu_1=E(\log(X_1))$ and $\nu_2=E(\log(X_2))$. Since $\log$ is monotonically increasing, my ...
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1answer
32 views

How to rotate an orientation (Euler angles)

If I have an orientation defined by Euler angles and I want to simulate a rotation of the coordinate system about the origin (doesn't matter to me how the rotation is specified), how would I get the ...
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1answer
29 views

Slice, projection, contour: A terminology question.

Consider a multivariate function, say $y=f(x_1,x_2,\dots,x_n)$, and suppose that $z=f(x_1,x_2,\dots,x_{n-1},g(x_1,x_2,\dots,x_{n-1}))$. What do we call $z$ with respect to $y$? Projection, level set, ...
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1answer
47 views

Computing range, null space, and matrix of a linear transformation

Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be defined by $(a_1, a_2, a_3) \mapsto (a_1, a_2, -a_1-a_2)$. I have to find $R(T), N(T)$ and a matrix that represents $T$. I know for my matrix that represents ...
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1answer
33 views

Find a surjective linear map with the following conditions

Find a surjective Linear map $ T: \mathbb{C}^{3} \rightarrow \mathbb{C}^{3} $ such that $ T(1,0,0) = (0,i,0) \space $ and $ T(0,i,0) = (0,0,1) $ We must also verify that $T$ is injective. I suppose ...
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Is this transformation surjective?

Consider the transformation $T:C_{\mathbb R} [0,1] \to \mathbb R$ defined by $T(f(t)) = \int_0^1 f(t)dt$. Is this transformation surjective? It would be enough to show that $$\mathbb{R} ...
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1answer
47 views

What does it mean for a matrix to change basis?

I understand what it means for vectors, i.e. $$ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} ...
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DPE problem invlolving Fourier transforms / partial eq.

Don't even know where to start with this question! would really appreciate some guidance.
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2answers
38 views

How does the Jacobian relate to sketch of x,y coordinates with u,v constant?

T is a non-linear transformation, with the following component functions: x = u/v, y = v On a sketch of the x-y plane, with u and v constant, how does the Jacobian, J = 1/v, relate to the sketch of y ...
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1answer
21 views

Multiplication order of rotation matrices

I have three 3D coordinate frames: O, A and B, as shown below. I want to know the rotation matrix RAB between A and B, that is the rotation that is required, with respect to the frame A, to move ...
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1answer
31 views

Change of Bases defined by a Plane

In the plane $V$ defined by the equation $x_1-2x_2+2x_3=0$, consider the basis $\mathcal{A} = (\vec{a_1}, \vec{a_2}) = ...
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Poisson transformations question

I have a poisson random variable Q. Q~Po($\lambda$) Y = $\frac{Q}{3}$ How would I write the probability mass function of the variable Y?
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1answer
48 views

Transformations between two coordinate systems on a rigid body

I have two coordinate frames, A and B, which are rigidly attached to each other on a body. This body then translates and rotates, such that A starts at A1, and moves to A2, and B starts at B1, and ...
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Fourier Tansform of derivative on Wolfram Alpha

If I'm not mistaken, the Fourier Transform of the nth order partial derivative of a function with respect to x, using the transform variable k is: (i*k)^n * [F(k)] so for the 1st order derivative ...
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1answer
29 views

$T(f(t))$ from “the space of all polynomials”

Let $T(f(t)) = (f(0), f(1), f(2), f(3),\cdots)$ from $P$ to $V$, where $P$ denotes the space of all polynomials. Is $T$ linear and if so, is $T$ an isomorphism? I feel like a counterexample is ...
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1answer
19 views

Hamiltonian, symplectic transformation

I am trying to understand symplectic transformations. Assume that $H(q,p)$ is a Hamiltonian and the corresponding Hamiltonian equations are given as, \begin{split} & \dot q = \frac{\partial ...
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20 views

Transfer Function of non-linear systems

I am trying to find an approximate transfer function of the following system using either Laplace or Fourier transform methods $$\frac{dy(t)}{dt} = k_1\times q_0\times x(t)-k1\times x(t)\times ...
0
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1answer
28 views

Proving that a transformation of a function gives a positive result

If $x$ is real and: $$p = \frac{3(x^2+1)}{2x-1}$$ Prove that: $$ p^2-3(p+3)\geq 0$$ I think this has something to do with equating the discriminant to $0$, but I'm not entirely sure I'd really ...