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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If $T,S \in L(V)$ are positive operators, how can I show that $TS$ is self-adjoint?

If we let $V$ be a finite dim. real/ complex inner product space, and $T \in L(V)$ and $S \in L(V)$ we let be positive operators, how can I prove that $TS$ is self-adjoint? I tried to decompose $TS ...
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113 views

Describing Cartesian transformations in Cylindrical Polar Coordinates

I have a question about converting functions defined in Cartesian coordinates to a cylindrical polar system. The particular coordinate transformation that I'm reading about is: \begin{equation} ...
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71 views

Linear Algebra - Understanding how to determine if a transformation is linear

I'm new to linear transformations in linear algebra and I can't quit understand how to find out if a transformation is linear. Any help would be much appreciated! a) $T:\mathbb R^3 \to\mathbb R^2$ ...
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16 views

How to name this transformation?

Let the transformation be $T:\mathbf s\to \mathbf s'$, where both $\mathbf s$ and $\mathbf s'$ are of the form $(s_0, s_1, s_2, ..., s_n)$, and $s_i'=a_is_i$ for each $s_i$ in $\mathbf s$ and its ...
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1answer
106 views

Big axis of an ellipse

I drew a circle in this square and I transformed them (view in 3d). How to find the angle between the big axis and $y$ or $x$ axis? Blue plane rotated by 36° around $y$ axis; azimut is 30° and ...
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47 views

$T$, $S$ are lineary dependent $\Leftrightarrow$ $[T]_B$, $[S]_B$ are lineary dependent.

EDIT: I see in the comments that my question is not clear enough so I will explain: if I want to check whether $T$, $S$ are linearly independent or not, I will just pick an easy to work with, basis ...
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439 views

How to compute a linear transformation which carries the circle $|z|=2$ into $|z+1|=1$?

Find the linear transformation which carries the circle $|z|=2$ into $|z+1|=1$, the point $-2$ into the origin, and the origin into $i$. In order to find the linear transformation will I use the ...
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33 views

Book recommendation for transformations.

Can I please get recommendations for books/notes on transformations? The topics covered should be Affine transformations Projective transformations Transformations in Euclidean space etc A ...
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69 views

Re-arrange expression to transformation form

$$\frac{6x-5}{3x+1}$$ How do you write this in the form $$\frac{b}{x+c} + a$$ I know how to find a (2) by asymptote theory, but I don't know how to re-arrange to find B.
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62 views

Linear Algebra Matrix Transformation Question

Can someone please help me out with this question. If a nonzero matrix $A$ is transformed from $\mathbb{R}^3$ to $\mathbb{R}^2$, then the null space of $A$ must be a one dimensional (sub)space of ...
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64 views

Rectangular transformation into Polar coordinates

I was working with a simple transformation of rectangular coordinates - symmetry around the y-axis, i.e. $$f(x,y) = (x, -y)$$ I wanted to express the identical concept in polar coordinates. After ...
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100 views

Algorithm to determine matrix equivalence

I'm a physicist who's not particularly good at linear algebra so please accept my apologies if this is standard textbook stuff that I'm just unaware of. I have two real rectangular matrices $A_{mxn} ...
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2answers
705 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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1answer
26 views

Generate function from discrete data (time-series)

How to transform discrete data into continous function ? I am working extensively with time series data and I would like to reduce amount of data in our frontend application. It would be cool to ...
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1answer
158 views

Transforming a circle to get a parabola

On http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/geometry/geo-tran.html I am unable to understand the following point Obviously, this transformation sends (x,y,w)=(1,0,1) to (x',y',w') = ...
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149 views

Rotation formalisms in three dimensions

I'm little bit confused. The Rotations are described by various means Direction Cosines Matrix (DCM); Euler Angles; Euler Axis/Angle; Quaternion. What is the difference between them. How I can ...
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314 views

Describing transformations using base vectors

So I just learned that we can describe vector transformations of shapes using base vectors, where the base vector I = $$ \begin{pmatrix} 1 \\ 0 \\ \end{pmatrix} $$ and J=$$ ...
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181 views

Prove that the following matrices cannot represent the linear transformation $T$ in ANY basis

$T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ defined as $T(x,y,z) = (2x,z,y)$ is a linear transformation. I need to prove that the following matrices cannot represent $T$ in ANY basis: ...
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72 views

What is/are the algebraic equation(s) for transforming a unit square into a specific parallelogram?

Goal: To transform a unit square into a parallelogram in which (a) the diagonals are parallel to the unit square's diagonals, (b) the longest diagonal is equal in length to either of the unit square's ...
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2answers
212 views

Linear Algebra One to one and onto function

I was just wondering how I can tell if a function is onto. $\mathbf{R}^3\to\mathbf{R}^1$ Lets say the standard transformation matrix has vectors $\{1,0,0\}$, $\{0,1,0\}$, $\{0,0,0\}$. I know that this ...
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1answer
41 views

Linear transformation and linear subspaces

Let $T:V\rightarrow V$ bwe a linear transformation. Let $L \subset V$ be a linear subspace such that $L \cap \text{Ker}\,(T)=\{0 \}$. Prove that the image given by T of any linear independent ...
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63 views

Linear transformation of vector space - proof of statements

Let $T:V\rightarrow V$ be a linear transformation of vector space with finite dimension. Prove that the following statements are equivalent: $$1. \ \ V=Ker(T) \oplus Im(T) \\ 2. \ \ ...
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2answers
324 views

Combine a rotation matrix with transformation matrix in 3D (column-major style)

I'm trying to concatenate a rotation matrix with a 4x4 homogeneous transformation matrix with column-major convention. I want this rotation matrix to perform a rotation about the X axis (or YZ plane) ...
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2answers
785 views

Consider the trace map $M_n (\mathbb{R}) \to \mathbb{R}$. What is its kernel?

The map is the trace map. I.e, it takes any $n$ by $n$ matrix and associates to that matrix, a number of the form $\mathrm{Tr}(A) = \sum_{i=1}^n a_{ii}$, where $A \in M_n (\mathbb{R})$. I need to ...
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301 views

How to determine yaw-pitch-roll orientation by specifying a plane via 3 points?

[Note, this question is an attempt at rephrasing the one posted here, as it has not garnered any attention, unfortunately] Hello, Let's say you have three points in 3D space: A, B and C. Together, ...
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1answer
57 views

transformations of $\mathbb R^2$

Consider the transformation $(u,v)=f(x,y)=(x-y,xy)$. Demonstrate the effect of this transformation on the lines $x-y=\text{constant}$, $x+y=0$, and the curves $xy=\text{constant}$. In particular ...
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414 views

Finding the pre-image of a linear transformation

Let $T$, A linear transformation such that: $$T\left[ {\begin{array}{*{20}{c}} x_1 \\ x_2 \\ x_3 \end{array}} \right] = \left[ \begin{array}{*{20}{c}} 2x_1 - x_2 + 5x_3 \\ - 4x_1 ...
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1answer
143 views

Translating and scaling a line based on two grabbed points

Say there is a line segment going from 0 to 10, now imagine that point 7 and 8 are 'grabbed' and translated to respectively 6 and 11. The effect would this would be that the line segment get's scaled ...
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1answer
39 views

Real linear tranformation

When do we say that a transformation $T$ which takes the complex number field onto itself is real-linear? I need to know it for my homework but I can't seem to find the definition anywhere.
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67 views

Question on linear algebra mappings

If $T:R^m\to R^n$ is a linear transformation, show that there is a number $M$ such that $|T(h)|\leq M|h|$ for $h\in R^m$.
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51 views

Equality of two Determinants (transformation)

$det\begin{pmatrix} -\lambda & 1 & 1 & 1\\1 & -\lambda & 1 & 1\\1 & 1 & -\lambda & 1\\1 & 1 & 1 & -\lambda\\\end{pmatrix} = ...
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1answer
79 views

Equation for a sphere after a projective transformation?

I was wondering if there is a general equation for a sphere that has undergone a projective transformation (i.e, the sphere has been multiplied by a 4x4 homography matrix)? I attempted to multiple ...
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1answer
38 views

What is the difference between the terms transformation and map?

Recently I noticed the usage of map in my topology book, and got confused because I already saw the term transformation somewhere. Both have the form $T:a\to b$, so what is the difference? Are they ...
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2answers
50 views

$T: \mathbb R^n \to \mathbb R^n$, $\langle Tu,v\rangle=\langle u,T^*v\rangle$, is $T^*=T^t$ regardless of inner product?

Basic question in linear algebra here. $T$ is a linear transform from $\mathbb R^n$ to $\mathbb R^n$ defined by $T(v)=Av$, $A\in \mathrm{Mat}_n(\mathbb R)$. We are given some inner product $\langle ...
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2answers
188 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
136 views

Inverse Function + Reflection In Y-Axis

Not getting any of the answers in MC. Is the answer wrong, or have I done something wrong?
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18 views

continuing and injective transformation

Does exist continuing and injective transformation B->A ? Are they homeomorphic?
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1answer
699 views

Finding equation of the image under a linear transformation

The equation of C is $x^2 + y^2 =1 $ How do I find the equation of the curve $C'=f(C)$ This is the image of $C$ under the linear transformation $f$ represented by the matrix $A=\begin{bmatrix}2 & ...
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0answers
116 views

$3$D transformation matrix to $2$D matrix

I have a $3$D affine transformation $4\times 4$ matrix. I need to convert it (project) to a $2$D affine transformation $3\times 3$ matrix, which looks like this: $3$D rotations are irrelevant and ...
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1answer
121 views

Why a 2D Affine Transformation matrix is 3 by 3

The matrix which I get for Scaling , Shearing and Rotation are follows: Scale: Shear Rotation Why do we need Homogenous Co-ordinate to get the transformation matrix as listed below? (need a ...
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0answers
124 views

Get 2D coordinate transformation matrix based on points in a system and their angles in the other?

I'd like to get the parameters (rotation angle,$\Theta$, and translation coefficients, $x_0$ and $y_0$)) of a transformation for translating and rotating points in a coordinate system to another. As ...
5
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1answer
378 views

Jacobian of Fourier Transformation

I am trying to calculate the Jacobian determinate of the Fourier transform which I stumbled upon when studying the Path Integral in Quantum Field Theory. I know the answer should be $1$ but I don't ...
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1answer
14 views

when linear mapping keeps monotonicity of $L_2$ norm

Consider an arbitrary vector $\alpha$ from vector space $R^p$, a linear mapping $A: \alpha\rightarrow A\alpha$ transforms $\alpha$ to $A\alpha$ in space $R^q$. What condition should $A$ satisfy so ...
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1answer
22 views

Linear Transformation Problem Given 3 transformations

can anyone help me get started with this question. Right now I am guessing and checking which is not efficient. I figured out out that the transformation is (?,?,x-y-z) so far Let $T:\Bbb R^3\to ...
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34 views

Birational Transformations question

so I'm wondering, is there a birational transformation one can make to the equation $Y^2 = X^m + f_{m-1}X^{m-1} + ... + f_0$, where all $f_i \in \mathbb{Q}$ so it is of the form $Y^2 = X^m + ...
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25 views

Transform pdf in higher dimensions?

Seem to remember the following equation held: $f(u) = {dx\over du} f(x)$ if one is give the probability distribution of x and a relationship between x and u the pdf of u can be derived. Sorry can't ...
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40 views

What formula would you use to cast an average of several numbers into a smaller range?

Say I have four numbers ranging in value from 15-95. If I want to, for a simple example, say that if the average of the four numbers is 15 (lowest possible value), that would relate to 2 on a scale of ...
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1answer
113 views

Univariate and Matrix Representation of Affine Transformation

Let $\mathbb{F}$ be a finite field with $q$ elements and $\mathbb{E}$ an extension field of degree $n$ of $\mathbb{F}$. Let $S:\mathbb{F}^n\rightarrow \mathbb{F}^n$ be a affine transformation and ...
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2answers
88 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

How to find the values of transformstion and its center/axis? [duplicate]

I have system of two planes. Each plane is defined by three points, so I have their equations. One of these plane is stable, I can't perform any transformation. The second one is modifiable - rotation ...