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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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
881 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
251 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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1answer
224 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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434 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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2answers
187 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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1answer
99 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
335 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
47 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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1answer
65 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
400 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
1k 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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0answers
418 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
58 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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1answer
775 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
210 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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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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3answers
61 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
88 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
41 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
53 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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463 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
187 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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1answer
19 views

continuing and injective transformation

Does exist continuing and injective transformation B->A ? Are they homeomorphic?
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1answer
1k 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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121 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
173 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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152 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 ...
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1answer
508 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
16 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
31 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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1answer
39 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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1answer
28 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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42 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
119 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
116 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 ...
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1answer
31 views

Linear Algebra Linear transformation Help

If $T:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is a linear transformation, then there exists a basis for $\mathbb{R}^n$ in which $T$ is diagonal. Is this true or false
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36 views

Calculating with transformation matrix

Given is the transformation of coordinates $ T_{AB} = \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix} $. 1.) What are the new coordinates for the vectors (1,0) and (0,1)? It should be: $ ...
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295 views

Help Deriving the Canonical Form of this Elliptic PDE

So I'm given the following PDE, for which I'm to derive the canonical form: $$u_{xx} + u_{xy} + u_{yy} = 0.$$ Clearly $A=1, B=1/2$ and $C=1$. Hence we have $\xi_x/\xi_y = -1/2 ...
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1answer
42 views

finding this linear transformation

i am following this guide: http://www.calpoly.edu/~brichert/teaching/oldclass/f2002217/handouts/goof.pdf my question is to find the linaer transformation that adheres to $T(1,1,1) = (1,1,1)$ ...
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1answer
26 views

Plane transformations

I need help in understanding how plane transformations work: for example, let $$A = \{(x,y) \in \mathbb{R}^2: x^2 + y^2 < 1\}$$ Now let's change coordinates like this: $$x = u^2 - v^2$$ $$y = ...
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347 views

Givens rotation of the following vector of 3 elements.

I have to find the givens rotation matrix that will transform the following vector $[1, 1, -1]^T$ to $[y, 1, 0]^T$ (basically to insert a $0$ on the third position without altering the second one). I ...
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1answer
81 views

Find rotation angle of given image

At first: our aim is to find the total transformation of left house to the right house. What I did it first is translating the house with the center to the origin. I already found out that the ...
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2answers
455 views

Books on geometric transformations and/or analytic geometry?

I've been looking to expand my knowledge in geometry as it's not covered in my undergraduate curriculum. For some reason I'm repelled by the classical approach (hopefully it will pass) as I feel it's ...
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4answers
110 views

Quadratics, transformations, and formulas

Two-part question. Feel free to answer just one part, or both (write which letter part you are answering) a) If the quadratic function $g(x)=a(x-h)^2+ k$ does not touch the $x$-axis, what can be ...
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2answers
36 views

Find the transformation.

I have to define (find?) the linear transformation $ f:\mathbb{R}^{3}\rightarrow \mathbb{R}^{2} \ \ \ where:$ $f(1,1,0)=(1,1)$ $f(0,2,-1)=(-1,0)$ $f(1,2,-1)=(0,2)$ How to achieve this? It is hard ...
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1k views

Find kernel and image of linear transformation.

I am given transformation : $f:R^3 \rightarrow R^2$ $ f(x,y,z)=(-x+y+z,x-y+z)$ I am requested to find kernel and image of this transformation. I am finding kernel: $ (-x+y+z,x-y+z)=(0,0 )$ ...