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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1answer
39 views

Find basis of an image

Let $ \psi : \mathbb{R}^4 \rightarrow \mathbb{R}^3$ be a linear transformation described by a formula $$\psi ([x_1,x_2,x_3,x_4])=[x_1+x_3+x_4, -x_2-x_4,x_1+x_2+x_3+2x_4].$$ Find basis of image ...
0
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0answers
11 views

How can I make this tangent function only appear once (or be spaced very widely)?

I only want the function to go from $x=5$ to whenever the function is 4.5 (in other words, when $y=4.5$). Is there any way to do this without specifying the domain? It has to have the shape of the ...
0
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0answers
18 views

Distribution invariant to orthogonal transformations

Let $X,Y$ be two real valued stochastic variables and define $$ \begin{pmatrix} \tilde{X}\\ \tilde{Y} \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} X\\ Y ...
5
votes
1answer
55 views

Is every symmetric matrix diagonalizable?

I know that Hermitian matrices are always diagonalizable and real symmetric matrices are real Hermitian matrices and therefore diagonalizable. But, it is always not the case that a symmetric matrix ...
0
votes
0answers
27 views

Searching for a constant transformation in $ \mathbb C$

I am having a continous transformation: $f: \mathbb C \to \mathbb C $ with a set $B \subseteq \mathbb C $, which is bounded. Now I want to proove that $ A = f^{-1} (B)$ is NOT bounded! I know it ...
1
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0answers
46 views

Prove that the continuous $f: \mathbb C \to \mathbb R$ has a global max and min

I am having this continuous transformation $f: \mathbb C \to \mathbb R$ and $\ f\ (\mathbb C)$ is bounded Now I have to prove that there are a global maximum and a global minimum. My thoughts: I ...
0
votes
2answers
62 views

Prove that a continuous inverse-transformation of $f: [0,1) \cup \{ 2 \} \to [0,1]$ exists

I am having this transformation $f: [0,1) \cup \{ 2 \} \to [0,1]$ $$f(x) = \begin{cases} x & x \neq 2 \\1 & x = 2 \end{cases}$$ I've already proved that it is continuous. Question: Is ...
0
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0answers
25 views

Approximation of a circle arc based on two lines, is this correct? How to rotate it afterwards?

Intro: We have two wheels that are rotated by something called T. I can tell T to rotate both wheels so much that they roll forward d cm (as shown in box A on the picture). I also have to keep track ...
2
votes
1answer
61 views

Can anyone check these true and false statements about linear algebra?

For any square matrix $A$, the image of $A^7$ is contained in the image of $A$ I think this question is asking If $A^7x=b$, then $b$ must be in $A$ with some vector $y$ such that $Ay=b$. It Seems ...
0
votes
1answer
59 views

Find the transformation that maps real axis to itself and imaginary axis to the circle $|w-\frac{1}{2}|=\frac{1}{2}$

Find the transformation that maps real axis to itself and imaginary axis to the circle $|w-\frac{1}{2}|=\frac{1}{2}$ What I did: $$z_{1}=0,z_{2}=i,z_{3}=\infty ...
0
votes
1answer
101 views

Transformation of coordinate axis to make matrix diagonal

Consider the matrix $$ A= \begin{bmatrix}1/8 & \frac{-5}{8\sqrt{3}} \\ \frac{-5}{8\sqrt{3}} & 11/8 \end{bmatrix} $$ which of the following transformations of the coordinate ...
3
votes
1answer
48 views

How to proove that a bijective transformation is NOT continous

I am having this transformation $f: \mathbb R \to \mathbb R$ $$f(x) = \begin{cases} x & x \in \mathbb R \setminus \mathbb Q \\x+1 & x \in \mathbb Q \end{cases}$$ I've already prooved ...
0
votes
0answers
22 views

Derivative Nullity for nonpolynomial spaces

One thing has been bothering me about derivatives, it's easy to explain nullity of a polynomial, since a term that is constant after n many derivatives will become zero at n+1 many derivatives. How ...
0
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0answers
52 views

How to determine roll-pitch-yaw parameters from homogeneous transformation matrix

We have a transformation matrix $T = \begin{pmatrix} cos(\theta_1) & sin(\theta_1) & 0 & l_1 cos(\theta_1) \\ sin(\theta_1) & -cos(\theta_1) & 0 & l_1 cos(\theta_1) \\ 0 ...
1
vote
1answer
36 views

Prove: the sum of simultaneously diagonalizable transformations is diagonalizable

Let $T, S$, linear transformations which are simultaneously diagonalizable. Prove that $T+S$ is diagonalizable. I need to rely on the the definition: $T,S$ are called simultaneously ...
3
votes
2answers
81 views

Is $f: [0,1[ \cup \{ 2 \} \to [0,1]$ continuous?

I am having this transformation $f: [0,1[ \cup \{ 2 \} \to [0,1]$ $$f(x) = \begin{cases} x & x \neq 2 \\1 & x = 2 \end{cases}$$ How can I prove that this transformation is continuous or ...
0
votes
0answers
16 views

Inverse of a discrete trasformation

I have defined the discrete transformation-like relationship: $$ Y(k)=\sum_{n=0}^{N-1} \frac{A(n)}{1+j \frac{w(k)}{p(n)}} $$ with $w(k)$ the k-th element of the vector $w$, $p(n)$ the n-th element ...
1
vote
3answers
39 views

No continuous transformation $f([a,b])= ]a,b[$

$ a,b\in\mathbb R$ with $a<b $. Now I want to show that there is NO continuous transformation $f: [a,b] \to \mathbb R $ with $f([a,b])= ]a,b[$ How can I proove that this transformation don't ...
0
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0answers
14 views

Problem of Closed linear transformation in Normed spaces [duplicate]

Let $X$ a normed space and let $A$ and $B$ be linear transformations such that $$X\subset D_A\rightarrow^{A} X \ \ \text{and} \ \ X\subset D_B\rightarrow^{B} X.$$ If $A$ and $B$ are closed, does it ...
0
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1answer
24 views

M22 → R Matrix Transformation Kernel

For a transformation such as this, how does one determine the form of the kernel? Is it simply making the right side equal to zero, solving for each individual variable, and then creating a matrix ...
0
votes
1answer
37 views

$p(x)$ divides the minimal polynomial iff $\exists v\ne 0: p(T)(v)=0$

Let $V$, a finite dimensional space. Let $T:V\to V$ a linear transformation. Show that $p(x)$, an irreducible polynomial divides $m_T$ (The minimal polynomial of $T$) iff there is a $V\ni v \ne 0$ ...
0
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1answer
45 views

Can anyone help me with “rotation matrix” and “Image of matrix”?

If A is a 3 by 3 matrix which gives a rotation about some line through the origin in R^3 , then columns of A form a basis of R^3 For any matrix A, the image of A^7 is contained in the image of A ...
1
vote
2answers
123 views

Prove that $T,S$ are simultaneously diagonalizable iff $TS=ST$. [duplicate]

Definition: We say that $S,T$ are simultaneously diagonalizable if there's a basis, $B$ which composed by eigen-vectores of both $T$ and $S$ Show that $S,T$ are simultaneously diagonalizable iff ...
0
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0answers
70 views

Can anyone help me with these true and false questions about linear algebra?

1 A system of real linear equations can have exactly two solutions. 2 If U and W are subspace of V, V=U+W (Finite dimension), then dimV is less than or equal to dimU+dimW 3.Every inner product ...
0
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0answers
34 views

Apply a rotation to Euler Axis angles

I have a camera orientation in world coordinates expressed in a vector containing cameras axis angles relative to the three world $x$, $y$, and $z$ axes (as example $(0,0, 0)$ would be a camera ...
0
votes
1answer
22 views

Questions about “onto” and “linear span of column”

If $T : V^6 \rightarrow V^4$ is a linear transformation And, It can not be one-to-one. Let $A$ be a matrix representation of $T$ Then $T$ is onto if and only if columns of $A$ span $V^4$ This ...
4
votes
2answers
42 views

Linear Algebra--searching a name for certain transformations

I am currently taking a Linear Algebra class in Spanish and having difficulty coming across the correct translation for what we are studying. I am looking at a question that asks for the rotation of ...
1
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2answers
42 views

Linear transformation ker and image

Let $\varphi\colon \mathbb{R}^4 \rightarrow \mathbb{R}^3$ be described by $\varphi(X)=AX$ where $A=\begin{pmatrix} 3 & 2 & 1 & 3 \\ 1 & 1 & 1 & 1 \\ 2 & 1 & 0 ...
0
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0answers
20 views

help me to check the following argument on a linear transformation

all, Just want to confirm the following argument: Assumption: we know that $S:=\{(x_1,x_2,x_3) : \|(x_1,x_2)\|_2 \leq (1+k)x_3\}$, for a given value $k$. By linear transformation: ...
1
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0answers
38 views

Transforming 1D Burger's Equation into infinitely many coupled ODE's

I've been working on the following problem but I can't justify my steps, would a savvy mathematician kindly tell me what, if any, violations I've made. Problem: Show Burger's equation can be written ...
0
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0answers
22 views

Express a 90 degree rotation matrix in terms of a 180 degree rotation matrix? (both anti-clockwise)

A = [-1 0 0 ] [ 0-1 0 ] [ 0 0 1 ] B = [0 -1 0 ] [ 1 0 0 ] [ 0 0 1 ] How can i represent B in terms of A?
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2answers
106 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
votes
1answer
39 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 ...
-2
votes
1answer
29 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
votes
2answers
80 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
votes
1answer
32 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
votes
1answer
24 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
votes
1answer
45 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
votes
3answers
35 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
votes
1answer
46 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 ...
0
votes
0answers
63 views

Represent $90$ degree clockwise rotation about the $z$-axis as a $3\times 3$ matrix

I honestly can't find anything regarding an issue I have with transformational matrices. I understand that this matrix: $$\begin{pmatrix} \cos 90&-\sin 90&0\\ \sin 90&\cos 90&0\\ ...
0
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0answers
14 views

Integration with change of variables for a transformed cube

Define the transformation $f(x,y,z) = (u,v,w)$ given by $u = x + \frac{1}{2}y^{2}, v = y + \frac{1}{2}z^{2}$ and $w = z + \frac{1}{2}x^{2}$ being injective on the cube $S = \left\{{(x,y,z)|0 \leq x ...
1
vote
2answers
56 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
votes
1answer
37 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 ...
0
votes
2answers
133 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 $AB=BA$, ...
0
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0answers
37 views

Transformation matrix of 2D image

I have 2D image with 256x256 pixels, while top left point is [0,0]. My task is to create transformation matrix, which will mirror this image by 10th row and then rotate it clockwise by the [5,5] ...
1
vote
1answer
26 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
votes
1answer
67 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
votes
1answer
26 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
vote
1answer
83 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} ...