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3answers
6k views

Image and Kernel of a Matrix Transformation

So I had a couple of questions about a matrix problem. What I'm given is... Consider a linear transformation $T: \mathbb R^5 \to \mathbb R^4$ defined by $T( \overrightarrow{x} )=A\overrightarrow{x}$, ...
2
votes
1answer
103 views

Find rotation that maps a point to its target

I have a 3D point that is rotated about the $x$-axis and after that about the $y$-axis. I know the result of this transformation. Is there an analytical way to compute the rotation angles? $$ ...
1
vote
1answer
454 views

What is the Matrix corresponding to a Linear Transformation

Given $T: P_2 \rightarrow P_3$ defined by: $T(at^2 + bt +c) = (a-b+c)t^3 + (-a + 3b - 2c)t^2 +(-a-b)t +(2b-c)$ What is the corresponding Matrix of $T$? This is what I have: First I rewrite the ...
0
votes
1answer
106 views

Legendre transform of log moment function

Here is something I do not understand for my lecture notes. The lemma is this. Let $\mu$ be a probability measure on $R$, and $\Lambda^*_\mu$ is the Legendre transform of $\mu$. $\Lambda_\mu^*\geq ...
2
votes
1answer
160 views

Find basis for $\ker T$ with $T:P_2 \to P_2: T(p(x)) = p(x) + p(-x)$

$T: P_2 \to P_2$ defined by $T(p(x)) = p(x) + p(-x)$. Find basis for $\ker T$ Here is my solution: $$p(x) = ax^2 + bx + c$$ $$p(x) + p(-x) = 0 \to 2ax^2+2c =0$$ So, a = c = 0. So, basis of $\ker T$ ...
0
votes
1answer
101 views

Basic Math: Simplifying $w=z^2+3z$

The question is: Consider the transformation $$w=z^2+3z$$ from the $z$-plane where $z=x+iy$ and $w=u+iv$. Determine the image of the line $y=1$ and $x=3$ in the $w$-plane My attempt at a solution: ...
2
votes
0answers
70 views

bound on Hilbert transform

Consider $\widehat{Tf(\xi)}=m(\xi)\hat{f}(\xi)$, where $m(\xi)=(1-\vert\xi\vert)1_{[-1,1]}$, i.e. $T$ is the operation of taking Fourier transform and multiplying with the function $m(\xi)$. I am ...
2
votes
1answer
201 views

Linear Algebra Question ( rank of matrix )

Let $\bf A$ be an $m \times n$ matrix. If $\bf P$ and $\bf Q$ are invertible $m \times m$ and $n \times n$ matrices, respectively prove $\operatorname{rank}(\mathbf{PA}) = ...
0
votes
1answer
89 views

Linear transformations and range

Let $A$ be an $m \times n$ matrix. Suppose that the matrix equation $AX = Y$ is consistent for any $Y$ that is an element of $R^m$ . (a) What is the range of $T_A$? Justify your answer. (b) Under the ...
1
vote
1answer
191 views

How to solve/transform/simplify an equation by a simple algorithm?

MathePower provides an form. There you can input a formula (1st input field) and a variable to release (2nd input field) and it will output a simplified version of that formula. I want to write a ...
1
vote
1answer
109 views

Creating a 3D surface from 2D graphs

So I have two sets of equations: $\mathcal{A}$ = \begin{equation} \{ f(y_{0},x), \, f(y_{1},x) , \;... \;, f(y_{n},x) \} \end{equation} $\mathcal{B}$ = \begin{equation} \{ g(y,x_{0}), ...
2
votes
2answers
48 views

Scaling range of $n$ numbers to $m$ numbers

I have $n$ values ($n \approx 10^8$) ... I want to plot these, where the $i$-th value is plotted at point $(x,y)$, $x=i$ and $y=$value$[i]$. Note: The values do NOT follow any pattern or function. ...
4
votes
2answers
355 views

How to solve an overdetermined system of point mappings via rotation and translation

I have a set of points in one coordinate system $P_1, \ldots, P_n$ and their corresponding points in another coordinate system $Q_1, \ldots , Q_n$. All points are in $\mathbb{R}^3$. I'm looking for a ...
1
vote
1answer
41 views

Multi 3D Screen to POV

See, im a simple hardcore Programmer. In fact i wrote some 3D-Programs till now. I had nothing to do this weekend, so i made a 3D-based filemanager. My desktop is multihead so the point-of-view of ...
2
votes
1answer
160 views

Reflecting an exponential function over a y = 3 line.

How would you write the equation of $f(x) = 4^x$ that reflects over the line $y = 3$? I've put in $f(x) = 3 + 4^{-x}$ which I thought was the right answer, but it isn't. Thanks in advance!
1
vote
1answer
132 views

Transformation of Quadric Surfaces

Is there a transformation $T: \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ such that a hyperboloid of one-sheet can be mapped to a hyperboloid of two-sheets using such transformation?
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2answers
58 views

Multivariate Random Variables

$f(x,y) = {2\over 5}(2x+3y) \quad for\quad 0<x,y<1 $ and we want to know the distribution of $2X+3Y$ I did it in a very lousy way which is let $ U=2X+3Y ,\; V=X$ Then have $\;f_{U,V}(u,v)$ ...
0
votes
1answer
360 views

Finding reflection of a matrix

To 2 decimal places, what is the value of the lower-right entry in the reflection matrix $Q_a $if a = 1.05? Not even sure where to begin, is there a formula? This is what I could find in my textbook ...
4
votes
1answer
113 views

How do you create a “stretch” transformation while keeping volume constant?

A stretch transformation can be represented as: $$ \begin{bmatrix} k & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$ However, this changes the volume of any object which ...
2
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0answers
245 views

Understanding Discrete Cosine Transformation

I'm currently working on some software and a key component is 2D DCT. But my question is more general, as I'm trying to understand the DCT in general, let's say from engineers point of view. For ...
0
votes
2answers
152 views

Map 2D points inside a closed curve to unit disk

Hy everyone, I have a set of 2D cartesian points (x,y coordinates) lying inside an arbitrary closed contour , something like this: arbitrary_closed_contour by 'arbitrary' I mean that the closed ...
0
votes
1answer
342 views

Composite transformation expresing as single transformation

This is the composite transorfmation that I have this is the working that I did, but something tells me this might not be the right answer, P.S can some with more points add composite ...
0
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0answers
88 views

Riccati equation transformation

Is possible to transform this Riccati equation into a linear differantial one? Thank you. $$ y=y_1+\frac{1}{z} $$
1
vote
1answer
51 views

Coordinate transformation to get even function

Suppose I have the function $$f(y)=2y^4-5y^3+3y^2,$$ with zeroes $y=0$ (2x), $y=1$, $y=3/2$, which I only need on the part of the domain $0\le y\le 1$. Is there a transformation $y\rightarrow y'$, ...
5
votes
2answers
465 views

Are Legendre transforms of non-convex functions useful?

Do Legendre transforms have any applications that do not appeal to convexity? What is the intuitive interpretation of the Legendre transform of a non-convex function?
0
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2answers
2k views

Linear Transformation with 2x2 Matrix Basis

The question asks: Find the "coordinates" of $v=\begin{bmatrix} -2 & -2 \\ -2 & 4 \end{bmatrix}$ relative to the ordered basis, $F=(f_1, f_2, f_3, f_4)$ where $f_1 = \begin{bmatrix} 1 & ...
0
votes
1answer
74 views

Matrix representation of transformation in ordered bases

An example question asks me to determine $[T]_{\beta}^\gamma$ where $\beta,\ \gamma$ are standard ordered bases of $\mathbb{R}^n$ and $\mathbb{R}^m$ respectively, of $$T_1: \mathbb{R}^n \rightarrow ...
3
votes
1answer
389 views

Find Möbius transformation that send Re(z)=Im(z) to a circle and the real axis to itself

Problem 3.3.7d in Complex Variables, 2nd edition, by Stephen D. Fisher. Find a linear fractional transformation $T$ that maps the real axis onto itself and the line $y=x$ onto the circle ...
2
votes
1answer
71 views

Semigroup question

I am looking for the technical term for an element of a transformation semigroup that sends everything to one state. The best term I could think up was filter. For those that don't know a ...
0
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2answers
201 views

3D transformation between two polylines problem

Say I have 2 separate objects. One is a line defined by two points, the other is a polyline defined by three points. Line 1 consists of the set of two points: $a=(0,0,0)$ and b=$(0,0,1)$ Line 2 ...
3
votes
1answer
735 views

Transformation matrix to go from one vector to another

I've two vectors $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$. How to find transformation matrix for transform from a to b?
1
vote
2answers
909 views

Formula transformation

Is it possible to transform this equation to give R? $$y=x\left[\frac{\left(1+\frac{R}{12}\right)^{12\times{25}}}{\frac{R}{12}}-1\right]$$
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1answer
2k views

Function transformation order of operations

I am reviewing for a midterm for Pre-Calculus and I am trying to understand the concept of function transformation: Let's say I am given a function $f$ with the domain in the interval of $[1,5]$ and ...
4
votes
1answer
136 views

Is there a geometric argument that the Legendre transform of a convex function is convex?

I am trying to build intuition on Legendre transforms. Arnold's Mathematical Methods of Classical Mechanics has some nice geometric interpretations, but he does not provide a proof that the Legendre ...
2
votes
0answers
193 views

transformation of coordinate systems by rotation

I am trying to convert a set of coordinates from ECEF (Earth Center Earth Fixed) to ENU (East North Up). The operation is performed by applying a rotation matrix as shown in: ...
3
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1answer
194 views

Fitting Shape in Circle for Shape Classification

I need to classify arbitrary 2D shapes. The classification should be invariant to at least affine transform. To achieve this invariance, I decided to "normalize" each shape by fitting it to a unit ...
1
vote
1answer
78 views

Help in finding Jacobian

I have $$\begin{aligned}x_{1}&=r\sin(\theta_{1}),\\ x_{2}&=r\cos(\theta_{1})\sin(\theta_{2})\\ x_{3}&=r\cos(\theta_{1})\cos(\theta_{2}). \end{aligned} $$ I know how to compute the ...
1
vote
2answers
44 views

Optimally projecting a point onto a line whose orientation is known

I have a line $l$ starting at origin ending at 0,0,1 along the $z$ axis. $l$ is rotated $P$ degrees around the $x$-axis and then $Q$ degrees around the $y$-axis. So I have a new endpoint for the line. ...
1
vote
1answer
337 views

Translating Cubic with Algebra?

I'm having a little trouble figuring out how to translate $ax^3+bx^2+cx+d=y$ by vector $(1,1)$ using only algebra. If possible could someone give me a hand? An example: Translate ...
1
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1answer
71 views

Transforming a Continuous Function

My math is quite limited so please bear with me. I will get to the point: Is there a way to transform a continuous function into a bounded one? In essence I have a normalized Gaussian distribution ...
2
votes
2answers
170 views

A linear transformation $T$ is defined by $T(x_1, x_2)$ Find the image of the line that passes through the origin and point $(1,-1)$

I know the definition of a linear transformation, but I am not sure how to turn this word problem into a matrix to solve: $T(x_1, x_2) = (x_1-4x_2, 2x_1+x_2, x_1+2x_2)$ Find the image of the line ...
1
vote
3answers
707 views

Show that the transformation T defined by $T(x_1, x_2)\; = \;… $ is NOT linear.

I'm studying for a test, and I need help with this problem. I am not sure how to prove that this is not linear due to the notation. The comma is throwing me off. Show that the transformation $T$ ...
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1answer
50 views

Trouble with function transformation (Left and right)

I am reading this example in the book for Pre-Calculus and it is explaining how functions are shifted left or right using g(x)=f(x-1). Here is what it says in the book. Define a function g by g(x) = ...
4
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2answers
338 views

Transforming Differential Equation to a Kummer's Equation

I'm trying to transform an equation of the form $$ yw''(y) - [b - ay] w'(y) - [d + ey]w(y) = 0 $$ into the form of a Kummer's or confluent hypergeometric differential equation: $$ y w''(y) + [f - ...
0
votes
1answer
258 views

Rotation matrix for a set of points

I've got a set of $N$ points $p_1,\dots,p_N$ that all belong to a real object. Consequently, there are $N-1$ vectors $\vec{v}_i$ when $\vec{v}_i$ points from $p_1$ to $p_i$. Now, the object is ...
4
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1answer
127 views

Can a transformation matrix be expressed in terms of the vector to be transformed?

I'm currently learning linear algebra with my friend via an online course, and we have a disagreement that we would like settled. Upon learning that vectors can be projected onto lines by a simple ...
0
votes
1answer
89 views

Can the dimension of the image of a linear map “increase”?

Suppose we have a linear transformation $f: V \to V$. How is it possible that $\dim(\operatorname{im}(f \circ f))$ is larger than $\dim(\operatorname{im}(f))+\dim(\operatorname{im}(f)) - \dim(V)$? ...
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3answers
707 views

Geometric interpretation of linear transformation

I have a linear transformation, given by the following matrix $$ \begin{pmatrix} x_1\\ x_2 \end{pmatrix} \mapsto \begin{pmatrix} 2 & 2\\ -1 & -1\\ \end{pmatrix} \begin{pmatrix} x_1\\ x_2 ...
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0answers
120 views

Determine transformation from set of points

I have unknown perspective transformation matrix and unknown coordinates of the points in xy-space, but have coordinates of the points in uv-space and know that some points have the same distance ...
0
votes
1answer
138 views

Graph transformations.

This is the exact problem from the worksheet. Now I understand that it is giving the parent function for $f(x)$. The only formula I know of for transformations is $y=f(x)\rightarrow y=af(bx+c)+d$ ...