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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2answers
282 views

How do I detect if a 4x4 transformation matrix contains reflection?

We currently check if the determinant of the upper left 3x3 values is negative to detect reflection in a 4x4 transformation matrix but we are unsure that it works in all cases (any arbitrary 3D ...
0
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1answer
61 views

Can anyone explain relationship between “onto” and “columns are independent” ?

I remember reading this statement before. It is as follows. Transformation is onto if and only if columns are linearly independnet Transformation is one-to-one if and only if rows are independent ...
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1answer
37 views

Find a matrix representing a given linear transformation [duplicate]

$T(X) = [\{x_1-x_2+x_3\}, \{0+x_2-x_3\}, \{0+0+0\}]$ is a linear transformation from $\mathbb R^3$ to $\mathbb R^3$. Find a matrix $A$ such that $T(x) = A(x)$ Can anyone point me in the right ...
0
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2answers
191 views

what is the difference between linear transformation and affine transformation?

Recently, I am struglling with the difference between linear transformation and affine transformation. Are they the same ? I found an interesting question on the difference between the functions. But ...
2
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1answer
62 views

Easy little triangle configuration

One of the four shapes is not needed to make the shape in the first pic. Which one? Once again, is it just noticing some properties? Or are there any other logical ways of figuring it out? I ...
2
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1answer
27 views

Proving: $\exists n\le \dim_F(V):V=\Im (T^n)\oplus \ker(T^n)$

Let $V$ above $\mathbb{F}$ and let $T:V\to V$. Prove there is a $n\le \dim_\mathbb{F} V$ such that $V=\Im(T^n)\oplus \ker(T^n)$ Now I know that for all $k$: $\ker (T^{k}) \subseteq \ker(T^{k+1})$ ...
0
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1answer
91 views

Can There Be A Dilation That Maps Parallelogram B to Parallelogram A?

There are 2 parallelograms, A and B. They have the same angle measures. Both have 2 sides that measure 6 units. Parallelogram As 2nd set of parallel lines are longer than the 2nd set of parallelogram ...
0
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3answers
58 views

If the transformation is not onto, does that mean that it is not one to one?

If transformation T: V -> V and it is not onto, then nullity is not 0 So, it seems like it is not one-to-one when it is not onto. And, If transformation is onto, is it one to one? because nullity ...
0
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1answer
51 views

What does it mean "$X \mapsto AX$ is a surjective mapping from $\mathbb{R}^n$ to $\mathbb{R}^n$?

question is If square matrix $A$ has determinant $1$, then $X \mapsto AX$ is a surjective mapping from $\mathbb{R}^n$ to $\mathbb{R}^n$. What does $X \mapsto AX$ mean?? is it equivalent to say ...
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2answers
218 views

How to rewrite double sum in matrix operation?

I have a double sum $\sum_{i=1,j=1}^n \alpha_i \alpha_j y_i y_j(x_i,x_j),\ x_i \in R^{d},\ y_i \in R,\ \alpha_i \in R $ How it can be rewritten in terms of vectors and matrices operations?
0
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1answer
77 views

Graph shifting, compression, and stretch

Given $f(x)$, sketch $p(x) = (1/2)f(2x-6)-3$. I can't put the graph here. You can just tell me the order of transformation of the graph. What i did by myself is horizontal compressing (using $2x$ in ...
1
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2answers
342 views

Prove two commutative linear transformations on a vector space over an algebraically closed field can be simultaneously triangularized

Prove two commutative linear transformations on a finite-dimensional vector space $V$ over an algebraically closed field can be simultaneously triangularized. It is equivalent to show if $AB=BA$, ...
1
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1answer
35 views

Prove that $S(W)$ is Invariant subspace

Let $S, T: V\to V$ such that $ST=TS$. Let $W\subseteq V$. Prove that if $W$ is invariant subspace of $T$ then also $S(W)$ is invariant subspace of $T$. Let $w\in W$. $$T(S(w)) = S(T(w)) = S(w')$$ ...
0
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1answer
218 views

Prove the rank of the direct sum of two linear transformations (on finite-dimensional vector spaces) is the sum of their ranks.

I would like to show the rank of the direct sum of two linear transformations (on finite-dimensional vector spaces) is the sum of their ranks. Definition: Let $M$ and $N$ be any two vector spaces, ...
0
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1answer
30 views

Transform $f(x)$ to time based function $f(t)$

I have a function of $(x,y)$ , for example $y = mx +c$, And I also have a function for velocity in time manner, for example $v = 2t$ Basically I want to draw $f(x)$ in some delta time $t_0 - t_1$ ...
1
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1answer
241 views

Z transform piecewise function

I have this piecewise function: $$x(n)= \left\{ \begin{array}{lcc} 1 & 0 \leq n \leq m \\ \\ 0, &\mbox{ for the rest} \\ \\ \end{array} ...
1
vote
1answer
116 views

Know if a 4x4 matrix is a composition of rotations and translations (quaternions)

I am using quaternions to describe 3D transformations. A position in space is representated by a (x,y,z,1) vector, and a transformation by a 4x4 matrix, following quaternions logics as far as I could ...
0
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1answer
47 views

what is the rank of the multiplication transformation $A$ if $AX=PX$ and $P$ has rank $m$?

Consider the vector space consisting of all linear transformations on a vector space $V$, and let $A$ be the (left) multiplication transformation that sends each transformation $X$ on $V$ onto $PX$, ...
1
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1answer
58 views

Under what conditions is the operator $Ax = [x, y]x'$ a projection?

Suppose that $V$ is a vector space, $x'$ is a vector in $V$, and $y$ is a linear functional on $V$; write $Ax = [x, y]x'$ for every $x \in V$. Under what conditions on $x'$ and $y$ is $A$ a ...
0
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2answers
47 views

Prove that a subset of a linearly independent set is a linearly independent set

Let $S$ be a linearly independent subset of a finite dimensional space $V$. Let $S_1 \subset S$, then prove that $S_1$ is linearly independent. I have looked all through my textbook, but I have ...
0
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1answer
63 views

When doing 3D rotations my angle flips 180 degrees

I'm implementing 3D rotations for a set of 3D circles. To do that I'm using the parametric equation as described in http://demonstrations.wolfram.com/ParametricEquationOfACircleIn3D/. It works as ...
5
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1answer
189 views

Prove spatial velocity identity - screw theory

This question involves a proof regarding coordinate transformations of velocities of screw motions. This comes from "A Mathematical Introduction to Robotic Manipulation" (the text is available for ...
0
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1answer
27 views

What kind of a matrix has a unitary diagonalizing matrix?

Suppose $D = P^{-1} A P$. When is $P$ unitary? In other words, what kind of a matrix $A$ should be, such that $D=P^{\dagger}AP$? i.e. what are the conditions a matrix must have to be able to ...
2
votes
1answer
39 views

If $D=P^{-1}AP$, then $f(D)=P^{-1}f(A)P$?

Suppose I have a diagonalizable matrix $A$, such that $D = P^{-1}AP$ Can I apply an element-wise function $f$ and expect that $f(D)=P^{-1}f(A)P$, assuming $f$ is not a linear transofrmation? Or in ...
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1answer
33 views

going from one unit vector basis to another unit vector basis

Somehow I am confused about this. Say I start with Spherical coordinate $(r,\theta,\phi)$, and I want to find expression $\hat{\phi}$ in terms of $\{\hat{x}, \hat{y}, \hat{z}\}$. At first I thought ...
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1answer
35 views

Transformation Definition

Let $A$ and $B$ be two $n\times n$ matrices that have no eigenvalues in common. Let $T$ be the transformation $$ T(S):=AS-SB $$ that maps the $n\times n$ matrices, $M_n$, to the $M_n$. Can we ...
1
vote
1answer
58 views

Transforming Vectors

Let $T$ be the linear transformation from $\mathbb{R}^3$ to $\mathbb R^3$ that reflects every vector about the $xy$-plane and then triples its length. How do I find the matrix for $T$?
0
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1answer
57 views

Given two linear transformations, find the preimage of a given point for the composite transformation

If someone could run quickly through the theory and methods on this it would be hugely appreciated. Thank you. Let $f: \Bbb R^2 → \Bbb R^2$ be reflection in the line $y = x$ and let $g: \Bbb ...
0
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1answer
63 views

Tricky change-of-basis transformation problem

I have absolutely no idea what to do here because of the $\sin(x).$ Let $V = \text{Span}\left\{x, x^3, \sin(x) \right\}$, and consider the basis for $V$ given by $\beta = \left\{x-2x^3, x^3+\sin(x), ...
0
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1answer
43 views

How to find the Fourier Transform of the form $\frac{cos(2\pi t)}{t^2}$?

I'm having trouble on figuring out how find the Fourier Transform of the following function, and I'm not allowed to use the straight up definition of the Fourier Transform but rather use it's ...
2
votes
1answer
3k views

Does the Fourier Transform exist for f(t) = 1/t?

My professor says that the following function has a Fourier Transform: $$f(t) = \frac{1}{\pi t}$$ He said that all I have to do is apply some of the Fourier Transform properties and not the direct ...
1
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1answer
96 views

Linear Transformation and Matrices

I have been studying linear algebra for a while now, and I still can't understand the basic concept of linear transformation and the easy ''translation'' of them the matrices. I understand that every ...
0
votes
1answer
17 views

Clarification of a transformation step (probably very simple).

A simple and quick question. Have been sitting over it for a while now but i can't get it right: Could someone just clarify how this transformation has been done? $$\frac{(v+1)^2 ...
0
votes
1answer
49 views

Null Space of Transformation

I am given that $V$ is n-dimensional vector space over $\mathbb{C}$ and $T \in L(V)$. And $T$ has least $m$ distinct nonzero eigenvalues. How do I show that $\text{null}(T^{n-m}) = ...
1
vote
1answer
54 views

the Fourier transform of a constant

How to calculate the Fourier transform of a constant without the aid of duality property? In other words, how do I calculate $$ \int_{-\infty}^{\infty}e^{-j\omega t}dt? $$
0
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1answer
43 views

What is the image of this mobius transformation

Consider the standard mapping $w=\frac{1}{z}$. What is the image of the "half" plane above the line whose imaginary part is $c$, for the three cases of $c\gt 0 , c=0 , c\lt 0$? For $c=0$ obviously ...
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0answers
33 views

Transformation for two different boundary functions in Stefan problem

Peace be upon on all of you, I have one-dimensional Stefan problem. Let say we have two boundary conditions of $u(t,s_{1}(t))=g_{1}(t)$ and $u(t,s_{2}(t))=g_{2}(t)$, where $u$ is temperature, $t$ is ...
0
votes
2answers
803 views

Inverse rotation euler angles

I have three angles representing a rotation (Pitch, roll and yaw). I need the inverse rotation (working on coordinate system transforms). What I do now is transforming these angle to a rotation matrix ...
1
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0answers
105 views

Dimension of image of a skew symmetric map is even

If $A$ is a skew-symmetric linear transformation on a finite-dimensional Euclidean space, then rank $\rho(A)$ of $A$ i.e., the dimension of image of $A$ is even. I am trying for a geometric proof of ...
0
votes
1answer
28 views

How to create own transformation change of variables

Evaluate the following integral using change of variables. Draw the original and new regions of integration. $$\int\int_{R} \frac{1}{x^2-y^2} dA$$ where R is bounded by the lines ...
1
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1answer
32 views

Composition of linear transformations that preserve angles

Given two invertible linear transformations T1,T2 in L(V) that preserve angles i.e. $\frac{(T(u), T(v))} {∥T(u)∥∥T(v)∥} = \frac{(u, v)} {∥u∥∥v∥} $. How can I show that T1T2 and T-1 also preserve ...
0
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3answers
83 views

Matrix with linear transformation with reflection

Find the matrix of the linear transformation A which is the reflection in the line $y = \sqrt{2}x$ with respect to the standard basis in $\mathbb{R^2}$. I Have no idea how to approach this problem... ...
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0answers
61 views

Help me generalize what this divisor transform does.

I have an algorithm which takes as input the series expansion of: $$\frac{-(1 + ax(-2 + x + ax))}{-1 + ax} \tag 1$$ or expressed differently: ...
0
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1answer
49 views

Dot product significance in vector transformation?

Suppose we multiply a 3 component vector by some 3x3 transformation matrix. Is it correct, then, to say the following about the transformed vector? Each component of the transformed vector is equal ...
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0answers
217 views

Conformal mapping of part of an annulus

I have a question about conformal mapping. I am wanting to map annuli to some other simple domain (probably rectangular). I have an image of my problem below In image A. we see a standard annulus ...
2
votes
1answer
263 views

Determine if the following function is one-to-one and/or onto

$T(x,y,z) = (xy,yz,xz)$ For one to one, I made $(x,y,z)=(u,v,w)$ and solved. $$xy=uv\to y=\frac{uv}{x}$$ $$\frac{uz}{x}=w$$ $$xz = uw \to x = u$$ $$uy = uv \to y = v$$ $$vz = vw \to z = w$$ So ...
0
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1answer
288 views

What is the correct change of variables to yield convexity in this nonlinear optimization problem?

$$ \text{min. } x/y \\ \text{s.t. } 2\leq x \leq 3 \\ x^2+y/z\leq \sqrt{y} \\ x/y=z^2 \\ x,y,z\geq 0 $$ To transform this problem into a nonlinear convex optimization problem, both the objective ...
0
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1answer
24 views

How can I find the density function of Z?

I am trying to find the density function for Z, this is what I am doing but I am not getting an appropiate function, I don´t know if there is something wrong with limits of the intregral. Or if this ...
0
votes
1answer
29 views

What is $f(T)$?

Let a linear transformation $T:\mathbb{R}^3\to \mathbb{R}^3$ defined as $T(v_1, v_2, v_3) = (v_1, v_3 - 2v_2, -v_3)$. Calculate $f(T)$ where $f(X) = -X^2 + 2 \in \mathbb{R}[X]$ I'm not so sure ...
2
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1answer
74 views

Transformation of a Random Variable

We have a random variable $x$ with p.d.f. $\sqrt{\dfrac{\theta}{\pi x}}\exp(-x\theta)$, $x>0$ and $\theta$ a positive parameter. We are required to show that $2\theta x$ has a $\chi^2$ ...