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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30 views

Calculate rotation and translation of object from corresponding points. NOT affine transformation

I have measured 4 3D points X and corresponding 4 3D points Xp after rotation and translation of object. From equation ...
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47 views

Nullity and rank of a linear transformation

Let $T:V\to W$ be a linear transformation, where $$T=d^2/{dx}^2,\\V=\{f(x): f \text{ polynomial of degree}\leq n\},\\W=\{f(x): f \text{ polynomial of degree}\leq n-2\}.$$ What is the nullity of ...
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103 views

Matrix of transform rotation

Im trying to create matrix which rotates vector. I have $\vec{g}=(g_1,g_2,g_3);\:g_1\in\mathbb{R},g_2\in\mathbb{R},g_3\in\mathbb{R}$ - it represents gravitation. And $\vec{o}=(o_1,o_2,o_3)$ is vector ...
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19 views

Deriving the $F_3$ type generating function in Hamiltonian formulation

I'm working on some practice questions and I am a bit confused with this one: Generating functions of the type $F_1(q,Q)$ satisfy the condition: $$pdq-PdQ = dF_1$$ Starting from this condition ...
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1answer
31 views

Conformal mapping of nonsimply connected domains

The question asks: Map the complement of the arc $|z|=1$, $y\geq 0$ on the outside of the unit circle so that the points at $\infty$ correspond to each other. How would you construct such conformal ...
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1answer
19 views

If S and T are transformattion mappings, what is [ST]?

S and T are transformation mappings, what does [ST] and [TS] mean? Does it mean transform via S and then apply T to the result and vice versa?
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31 views

Unimodularity of spin transformation

Consider a spin transformation \begin{equation} \zeta \to \tilde\zeta=\frac{\alpha\zeta+\beta}{\gamma\zeta+\delta} \end{equation} with all quantities being complex. It is said that without loss of ...
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238 views

Multiple integral variable substitution using Jacobian matrix and matrix rotations

Question: By an appropriate choice of new variables evaluate the integral $\int\int_R(x^2+y^2)\,dx\,dy$ over the interior of the square bounded by $y=\pm x$ and $y=\pm (x-2)$. I sketched the square ...
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1answer
84 views

Linear Algebra - - Linear transformation

The matrix $$ A=\left[\begin{array}{ccc} 1 & -a & a \\ -1 & a & a+2 \\ 1 & 2a+3 & -3a-4 \end{array}\right], $$ where $a \in \mathbb{R}$, represents a linear transformation $T: ...
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1answer
50 views

Why does the discrete cosine transform as matrix multiplication work this way?

I have read that the DCT can be computed as a matrix multiplication. The 8x8 DCT matrix is: $D=\frac{1}{2}\left[\matrix{ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & ...
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1answer
16 views

Let $X_1$ and $X_2$ be independent $n(0,1)$ random variables. Find the pdf of $(X_1-X_2)^2/2$.

I understand that $(X_1-X_2)/\sqrt2)$ ~ $n(0,1)$ since it is a linear combination of $X_1 $ and $X_2$ and hence $(X_1-X_2)^2/2$ ~ $\chi^2_1$. I'm having trouble on how to prove/show this ...
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76 views

Convert hermitian matrix to symmetric

Is there some simple transformation (or a simple way to find it) which would convert any given Hermitian matrix $A$ to a symmetric matrix $B$ with the same spectrum as that of $A$ (so I guess that ...
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21 views

Transformation of binomial distribution

If X ~ binomial(10, 1/3), find the pmf of Y, where Y = X^2. If my understanding is correct, pmf p_y(m) where Y is some function g(X) and m is number of successes desired is equal to p_x(g^-1(y)). ...
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1answer
227 views

Subspaces, transformation matrices exercise

I have trouble understanding the following exercise so I would really appreciate any help you could give me: Let $k$ be a non zero vector in $\mathbb R^n$, written in standard basis. Let $H$ be ...
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2answers
47 views

Find invariant points, how to express using parameter

I have a matrix $$\begin{pmatrix}0&-1\\1&2\end{pmatrix}$$ where I have to find the invariant points for a transformation using this matrix. I have no problem working through to two ...
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2answers
30 views

inverse of similarity transformation

If $S$ is a similiarity transformation, i.e. there exists $c>0$, such that $$ \lvert S(x)-S(y)\rvert = c\lvert x-y\rvert. $$ Then, apparently, we have that $$ \big\lvert ...
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2answers
91 views

Coordinates rotation by $120$ degree

If I have a point on a standard grid with coordinates say: $A_1=(1000,0)$ $A_2=(707,707)$ Is there a easy way to transfer this points to $\pm 120$ degrees from the origin $(0,0)$, and ...
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1answer
52 views

Odd polynominal

Let's define an odd polynominal be a polynominal which has odd degree, and ALL of its terms have odd exponential (except the constant), for example: $x^5+x^3+1$, or $x^7+2x^5+3x^3+4x+5$. We all know ...
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2answers
58 views

double integral over an arbitrary triangle

Assume we have an arbitrary triangle ABC in x-y plane and we want to integrate a function $f(x,y)$ over surface of this triangle as shown in fig. 1: We can define another coordination system [x' ...
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1answer
69 views

How to transfrom my equation to $Y=KX^2$

In general , $$\vec{C}(u)=\vec{a_0}+\vec{a_1} u+\vec{a_2} u^2$$ is a parabolic arc between the points $\vec{a_0}$ and $\vec{a_0} + \vec{a_1} + \vec{a_2}$. So I'd like to prove it by myself: My trial ...
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2answers
62 views

Why isn't $f(x)=\sqrt{2-x}$ reflected across the y-axis?

If I try to graph this function, it does not appear to reflect across the y-axis when it comes time to do the reflection. Rather, it is reflected around the point where the function begins on the ...
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24 views

General Tensor Assistance

Sorry if this is a stupid question, but it might help me grok things if I can connect from something that's intuitive to me. Consider a transformation from Cartesian coordinates to spherical ones: ...
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1answer
24 views

What is the new point reflected with a respect to fixed line through origin?

If I have $(x,y)$, what would be the reflected point with respect to fixed line through origin? does it depend on what line I have, If I have $x$ or $2x$ $3x$, they all pass through origin.
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16 views

How to implement the Integer Wavelet Transform for images?

I have a description of a wavelet transform, but I am unsure on how to implement the algorithm based on the information given: $A_i,_j = ((I_{2i,2j} + I_{2i+1,2j}) / 2 )_{floor}$ $V_i,_j = ...
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1answer
38 views

Surface area of transformed sphere

So if I have a sphere with center C and radius R and then apply one or more affine transformations (so any combination of rotating, scaling and translating), how would I go about finding the surface ...
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1answer
24 views

Find the Laplace transform of integral(from 0 to x) sin(2t) dt

Find the Laplace transform of $\int_0^x\,\sin\,(2t)\,dt$ So basically, $$\int_0^x\,\sin\,(2t)\,dt = -\frac{1}{2}(\cos\,(2x) - 1)$$ So $$\mathcal{L}\{\cos\,(2x)\} = \dfrac{s}{s^2 + 4}$$ ...
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19 views

How to calculate discrete cosine transform for a matrix

I have a 8x8 matrix and I want to calculate its discrete cosine transform (DCT-II). I have this formula but I don't know to use it with a matrix. In the French Wikipedia they gave an example for ...
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2answers
59 views

Prove that an element of the basis is an element of the Kernel after linear transformation

Let $T:R^4\rightarrow R^4$ and basis $B=(v_1,v_2,v_3,v_4)$. $$T(v_1)+T(v_2)=T(v_3)\; \text{ and } \; T(v_1)+T(v_3)=T(v_2)$$ Prove that $v_1\in Ker(T)$ What I wrote is: $$T(v_1)=T(v_3)-T(v_2)\; ...
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28 views

What is the proof/show that the post of linear transformation generated by LDA is at most k-1

What is the proof/show that the matrix $Sw$ generated by LDA is at most rank $p-k$, where $p$ is the dimension of the data and $k$ is the number of classes. LDA: ...
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49 views

Transformation Matrix project

My task is to find the Transformation Matrix, that projects, any point of the xy-plane, on the line $$ y = 4x$$ The solution should be: $$T=\pmatrix{0.06&0.235\\0.235&0.94}$$ But somehow i ...
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16 views

Performing a shift on a piecewise function

Following is a convolution of $y(t) = h(t)*x(t)$ $$y(t) = \cases{\frac{1}{2}e^{2} &\text{ if } t\ge 1 \cr\cr \frac{1}{2}e^{2t} &\text{ if }0 \lt t\lt 1 \cr\cr 0 &\text{ if } t\le 0}$$ ...
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1answer
49 views

How to find an unitary transformation of $A$ that minimize $(A'_{i,i}-1)^2$?

Is there a way to find an unitary transformation $$ A'=U^+AU $$ that minimize: $$(A'_{i,i}-1)^2$$ In other words, the diagonal elements must be similar to one: $A'_{i,i} \approx 1$ Any hint? ...
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1answer
43 views

Linear transformation

Let there be a linear transformation $T:R^3\rightarrow R^2$ Is there a linear transformation so that: $Ker(T)=Span((1,2,1),(0,3,-1))$ and $Im(T)=Span((5,-7))$ Answer: $Dim(V)=Rank(T)+Null(T)=2+1=3$ ...
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13 views

Does logged data have to be transformed back to its original form before testing the accuracy?

I have run my data through a model in r, i ran ARIMA to forecast. The model forces a log transformation to be applied to the data. To test the accuracy of the fitted model formed by ARIMA would i need ...
0
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1answer
46 views

Why does the discrete cosine transform compact the information at the “low frequencies”?

I've been investigating about the discrete cosine transform. I think I understand the practical applications it has and how it is used in image/audio compression. I also know it is related with the ...
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14 views

Partial derivative of exponential of cosine transform of a vector

I am looking at the following partial derivative result that involves a cosine transform. Please refer to equation (6) and (9) in ("A unified framework of HMM adaptation with joint compensation of ...
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2answers
109 views

Writing a composite transformation as a matrix multiplication

I am confused about a question on matrix multiplication of a transformation. I have two matrices, P and Q as follows: $$P = \begin{pmatrix}\frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} ...
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2answers
79 views

How to find the equation of the graph reflected about a line?

Consider the graph of $y = e^x$ (a) Find the equation of the graph that results from reflecting about the line $y = 4$. (b) Find the equation of the graph that results from reflecting ...
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34 views

Finding transformation function for a distribution that looks like exponential

Suppose that we have two data sets, R and P. R is larger than or equal to ...
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32 views

Show that a transform is involutive

Let $\mathcal C$ be the class of continuous, nonnegative, not identically equal to zero, concave, positive homogeneous of first order functions from $\mathbb R^n_+ = \{ x \in \mathbb R^n \colon x \geq ...
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1answer
25 views

(Kleiner) transform preserves smoothness class

Consider the transform of nonnegative continuous concave positive homogenuous of first order function $f(x)$, $x \in \mathbb R^n_+$, $f \not\equiv 0$ given by $$ f^\times(y)= \inf \left\{ \left. ...
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1answer
31 views

Transformation of a surface normal

I'm taking a university level course in discrete geometrics and graphical programming, and I'm having trouble understanding this exercise. Let p be a point in R^3, n a surface normal, and M a ...
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12 views

Give the transformations of the following functions.

Give the transformations of the following 3 functions. Can you please give me at least 3 points to plot for each function(keeping the domain restriction in mind)? Also for rational function. Also ...
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16 views

help me find the gimbal locks

I have this transformation (x, y, z) |-> (x'', y'', z''). How can the gimbal locks be discerned and where are they? ...
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1answer
43 views

using Boltzman transformation to change PDE to ODE

using Boltzman transformation $\phi=z/\sqrt{t}$ to transform a patrial Problem: $$\frac{\partial \theta}{\partial t} =\frac{\partial }{\partial z} \left(D{\frac{\partial \theta}{\partial ...
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1answer
34 views

Laplace transform, Inverse Laplace transform

Let $(\mathcal{L}f)(s)$ be the Laplace transform of a piecewise continuous function $f(t)$ defined for $t\geq 0$. If $(\mathcal{L}f)(s)\geq 0$ for all $s\in\mathbb{R^+}$ does this imply that $f(t)\geq ...
0
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1answer
23 views

Linear transform $T$ such that $T(b^x)=b(b-1)^x$

The title pretty much says it all. I'm trying to find a linear transform, maybe a vague analog of a derivative, that has the property that if $f(x)=ab^x$, then $T(f)=ab(b-1)^x$, analogous to the ...
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1answer
48 views

Transforming a nonlinear system to a linear system

Suppose I have two points in $\mathbb{R}^2$ and I wish to find values of parameters $a$ and $b$ such that I obtain the power law $y=ax^b$ which goes through the two given points. I can solve the ...
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2k views

Finding the Dual Basis

Define the four vectors in $\mathbb{R}^4$ by $$v_1=\left( \begin{array}{ccc} 1 \\ 0 \\ 0 \\ 0 \end{array} \right), v_2=\left( \begin{array}{ccc} 1 \\ 1 \\ 0 \\ 0 \end{array} \right), v_3=\left( ...