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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How can I show that the kernel of $f-id_V$ is equal to the image of $f$?

How can I show that $\ker (f-id_V)=\Im f$ given that $f:V\longrightarrow V$ is a linear transformation such that $f\circ f=f$? Thank you.
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1answer
6k views

“Well defined” function - What does it mean?

What does it mean for a function to be well-defined? I encountered with this term in an excersice asking to check if a linear transformation is well-defined.
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580 views

Proof of the affine property of normal distribution for a landscape matrix

The widely used/mentioned/assumed affine property of multivariate normal distributions says that: Given a random vector $x \in R^N$ with a multivariate normal distribution -- $x \sim N_x(\mu_x, ...
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228 views

Proving a subspace under a linear transformation by the closure of standard addition and scalar multiplication

$T(x,y,z)= (3x-2y, -2x+3y, 5z)$ be a linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^3$ Show that $A= \{(u,v,z) \in \mathbb{R}^3~|~(u,v,w)=T(x,y,z)\}$ for some $(x,y,z)$ in $\mathbb{R}^3$ is ...
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120 views

some corollaries of the rank - nullity theorem

Here is a problem which I encountered in linear algebra. I realized that it might be a corollary of the "rank - nullity theorem" but I don't know how to work with it. Hope you can help! Thank you! ...
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2answers
77 views

Using transformations and basis to find standard matrices

Let $A =\{(1,3), (2,5)\}$ be a basis of $\mathbb{R}^2$. Let $M =\left[\begin{array}{rr} 1 & -2\\ 3 & 0\end{array}\right]$ be the standard matrix for the linear transformation from ...
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1answer
92 views

Representation of Linear Transformation with respect to basis please helppp

Let $A = (1,3) (2,5)$ be a basis of $\mathbb{R}^2$. Let $M =\left[\begin{array}{rr} 1 & -2\\ 3 & 0\end{array}\right]$ be the standard matrix for the linear transformation from $\mathbb{R}^2$ ...
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64 views

Linear Algebra Transformation Question

Suppose 3x3 matrix A = [122,011,012](commas separating rows) And suppose T: R^3 -> R^3 be defined by T(x) = A(x) for every x in R^3 (a) Find A^-1 (Easy) (b) Suppose T^-1 be the inverse transformation ...
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0answers
44 views

Hyperbolic analogue of the Euclidian Reflection across a straight line

I have already solved that $\Phi(z)=z$ in the geodesic $g$, but I am stuck on this part of the problem: Let $g$ be a complete geodesic of $H^2$, which is a semicircle of radius $R$ centered at ...
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1answer
27 views

Fractional linear maps and geodesics.

Consider the fractional linear map $\Phi(z)=(az+b/cz+d)$ where $a$, $b$, $c$ $d$ are real numbers with $ad-bc=1$. Suppose in addition that $|a+d| > 2$ a) show that if $c \ne 0$, there exists ...
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1answer
32 views

How do I determine a vector x $\in \mathbb R^4$ that satisfies $T(x) = A$

$T$ is a linear transformation $A = \begin{bmatrix} 2\\4\\8\\ \end{bmatrix}$ How can you define a vector $ x \in \mathbb R^4 $ which satisfies $T(x) = A$ this is just a made up example. Forgive me ...
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1answer
58 views

Linear transformations… $T(\mathbf v)\ne0$, $T^2 - 0$… Prove $\mathbf v$ and $T(\mathbf v)$ are linearlly independent…

Suppose $T:\mathbb R^n \to\mathbb R^n$ is a linear transformation and suppose that $\mathbf v$ is a vector such that $T(\mathbf v) \ne 0$ but $T^2(\mathbf v) = 0$ (where $T^2 = T \circ T$) Prove ...
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2answers
46 views

Find the matrix of $T$ given :$ T([3, 5]) = [2, -1]$ and $T([1, 2]) = [3, 7]$ , $T$ is linear.

I know that T(v) = v' = Av , where v is the vector, v' is the image, and A is the matrix of transformation So I've set the two images (v') equal to the matrix [S sub1, S sub 2] What does [S sub1, S ...
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1answer
185 views

Rotate a triangle specified by vectors around its center

Well, I know that, in order to rotate a triangle specified by three vectors in $R^2$ we just rotate each vector in the same angle, and to do this we apply the rotation matrix in ...
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4answers
110 views

Proving the standard matrix U of T to be orthogonal

So my class is getting into orthogonality, however, our reading assignments haven't been touching on transformations. I have this proof problem that I cannot seem to get around. Does anyone have any ...
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3answers
35 views

Linear transformations and eigenvalues [duplicate]

Let $T: \mathbb C^n \rightarrow \mathbb C^n$ be linear. Let $\beta$ and $\gamma$ be any two ordered bases. Prove that the eigenvalues of $[T]_\beta$ and $[T]_\gamma$ are the same. Can anyone provide ...
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1answer
205 views

Definition of the Hamiltonian via Legendre transform.

In my book of classical mechanics (Mathematical methods of classical mechanics by V.I. Arnold), the Hamiltonian is introduced in this way (my translation): Let us consider the system of equations ...
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1answer
75 views

Möbius Transformation of Triangles

I understand that Möbius transformations are angle preserving transformations. Knowing this, my professor asked us to think about how the image of equilateral triangle is not an equilateral triangle ...
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1answer
105 views

Question on transformations

Two efficiency experts take independent measurements Y1 and Y2 on the length of time workers take to complete a certain task. Each measurement is assumed to have the density function given by f(y) = ...
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1answer
60 views

Transformations

The length of time that a machine operates without failure is denoted by X and the length of time to repair a failure is denoted by Y. After a repair is made, the machine is assumed to operate like a ...
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1answer
51 views

Why is this transformation true?

I have just a simple question i think. I tried to implement the $\chi^2$-test.I have this document where it is said on page 41, that I have to implement the test like this:$$\frac{\sum_{0\leq i < ...
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1answer
51 views

State true or false ?

if $Ax=b$ is consistent, then the solution set of $Ax=b$ is obtained by translating the solution set of $Ax=0$ is it true or false? or is it sometimes false and sometimes true?
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1answer
256 views

Prove scalar products are invariant under all orthogonal transormation

I wondering how to prove: That scalar products are invariant under all orthogonal transformation: $<\!x, y\!>\; =\;<\!Qx, Qy\!>$ which holds for all vector $x$,$y \in \Re^n$ and all ...
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2answers
114 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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1answer
129 views

Fourier transformation: Determining the axis

I need some help with the Fourier transformation of my data. My original data is a Distance VS Time: upon doing a Fourier Transform, I get the following: I understand that normally after a ...
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1answer
48 views

Transformations problem

$$ \mbox{Let}\ x\ \mbox{have pdf}\quad {\rm f}\left(x\right) = {n \choose x}p^{x}\left(1 - p\right)^{n-x} $$ for $x = 0,1,2,\ldots,n$ where $n$ is positive integer constant and $0 < p < 1$ is ...
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0answers
131 views

Python numpy issues with array dimensions

I'm supposed to implement householder transformation of a matrix A $\in R^{m \times n}$, with m $\ge$ n, i.e. multiply the matrix A with matrices so that it becomes an upper triangular matrix R. ...
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2answers
628 views

Are there non-affine matrices?

Matrices are useful for proving statements like The ratio between the areas of a parallelogram and the quadrilateral formed by joining their midpoints is $2$. The ratio between the volumes ...
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48 views

Find the joint distribution of f(x,y), given x and y are both poisson distribution.

I am not content with the description on text book that: u=x+y;v=y x~poisson;y~poisson;x and y are independent fuv(u,v)=fxy(u-v,v)=f(u-v)f(v); f(u-v)=poisson(u-v,sita); f(v)=poisson(v,lambda); ...
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1answer
62 views

Linear Transformations $T$ and $S$ and their Characteristic Polynomials

My friends and I cannot figure out this proof. We have part (a) done, but weren't not quite getting part(b). We think we need a change-of-basis equation. Any advice?
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1answer
68 views

$\mathscr{B}$-matrix of T and Characteristic Polynomial

I'm having a difficult time trying to figure out this proof problem. Any advice on first steps? Let A be an $n\times n$ matrix satisfying the matrix equation $A^{n} + ...
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0answers
82 views

Linear independent eigenvectors and eigenvalues

I have T as a linear transformation from V to V over the field F, V has dimension n. T has the maximum number n distinct eigenvalues, then show that there exists a basis of V consisting of ...
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1answer
77 views

If $f$ has a Laplace transform $F$ then $\lim_{s\to\infty}F(s)=0$?

As I know, well-known functions that have Laplace transform vanishes at infinity. Because almost well-known functions has exponential form and the Laplace transform of function has exponential form ...
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1answer
195 views

Autocovariance function of a Poisson process transformation

here is the problem formulation: Let $\{N_t,t \ge 0\}$ follow a Poisson process with rate parameter $\lambda$ and let $A$ be a random variable with zero mean and unit variance, $A$ is independent of ...
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1answer
68 views

Take Laplace Transform of the integral J_0

I was just wondering how to use tables from Spiegal to solve $\int_0^\infty J_0(2\sqrt{ut}) J_0(u) du$ At the moment, I see similar transforms on page 244, but I don't actually know how to combine the ...
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2answers
64 views

Linear transformation of polynomials, given transformation output

I need to find a transformation of a polynomial, given the output of other polynomial calculations: If $T : P_1 \mapsto P_2$ is a linear transformation such that, $$ T(1 + 5x) = 1 - 2x ...
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1answer
200 views

An Unexpected Circle…

I played around with $$z=\frac{-1+e^{it}}{\phantom{-}2+e^{it}}$$ and found that, when I draw the real against the imaginary of $z$, it pretty much looks like a circle. But neither ${\frak{R}} z ...
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1answer
38 views

Coordinate transformation from unit ball to any ball

Hello If I have the unit ball $B_1(0)$ in $\mathbb{R}^n$ and want to transform it in any other ball $B_R(x^0)$ in $\mathbb{R}^n$, which transformation can I use? My idea is: ...
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1answer
114 views

How can I encode this?

Let say I have 7 integers: 1, 2, 3, 4, 5, 6, 7. Among the 7 integers, I choose 3 integers. For example, my choice is (1,2,3). Note1: The order of the integers in the choice doesn't matter. This means ...
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4answers
494 views

Relations between matrices and linear transformations, diagonalization. [duplicate]

This question bothered me for a while and hopefully someone can shed some light on the issue. A matrix $A$ is said to be diagonalizable if there is an invertible matrix $P$ and a diagonal matrix $D$ ...
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0answers
64 views

Transformed Laplace “solution space”

From my own knowledge I can tell that when we take the Laplace transformation of a function we are in essence transforming our f(t) into a F(s). I've looked at several Q/A here asking for the ...
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1answer
186 views

Is there a relationship between SVD and similarity transformation?

Is there a relationship between SVD and similarity transformation? I mean, in articles I read about the SVD method, I came across the equation $A=USV^T$ which seems to me like similarity ...
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1answer
155 views

How to apply perspective transform to Bezier curve?

I found that both Bezier curves and B-splines are described with a formula $p(t)=\sum\limits_{i=0}^d B^i_m p_i$ but in the case of B-splines $B^i_m$ are B-spline blending functions, while for Bezier ...
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Is the linear transformation $T(f(t))=t(f(t))$ from $P$ to $P$ an isomorphism?

Is the linear transformation $T(f(t))=t(f(t))$ from $P$ to $P$ an isomorphism? I can say it is since: The dimensions of domain and codomain are equal and $\text{ker}(T)$ is the function ...
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1answer
46 views

What's the difference between these two transformations of functions?

I'm about to graph the transformation of a function, but in this problem I encountered something new. The function transformation looks like this: y=12(f(x)+2) Thing is, I've never seen the f ...
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164 views

What is a Sub Formula and What is a Maximal Sub Formula in Propositional Logic

What is a Sub formula of a Propositional Formula? Suppose I have a formula C or -C Then what are the sub formulas of this and what is the maximal sub formula of this Propositional Formula. I am a bit ...
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2answers
139 views

How to efficiently encode this?

I have 5 ring oscillators whose frequencies are f1, f2, ..., f5. Each ring oscillator (RO) has 5 inverters. For each RO, I just randomly pick 3 inverters out of 5 inverters. For example, in RO1, I ...
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1answer
131 views

Dimension of Image of Composition of Linear Transformations

Take two linear transformations T from V to W and S from W to U. I want to show that the dimension of the image of their composite SoT from V to U is 'smaller' than or equal to the dimension of the ...
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1answer
92 views

Images of Lines

I'm studying for this exam and one of the questions I am stuck on is: Find the image of the line $$3x-y+1 = 0$$ under the transformation $$z \mapsto \frac{2}{z+1}$$ So I know I have to convert the ...
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How transpose of a matrix helps in making better sense of the data

The transpose of a matrix is obtained by flipping it about its diagonal. What is a practical scenario where we gain better insight into a set of data points by transposing it?