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
41 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
13 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 ...
5
votes
1answer
86 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 ...
1
vote
0answers
56 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
68 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 ...
2
votes
1answer
70 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
88 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
143 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
59 views

How to prove that a bijective transformation is NOT continuous

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 ...
1
vote
1answer
40 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 ...
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
votes
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
votes
1answer
53 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
40 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
votes
1answer
57 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
246 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
votes
1answer
32 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
45 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
49 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 ...
1
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0answers
48 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
49 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?
-1
votes
2answers
219 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
54 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
36 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
161 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
46 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
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
votes
1answer
69 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
50 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
49 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
113 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\\ ...
1
vote
2answers
156 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
71 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
270 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
vote
1answer
31 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
169 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
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
vote
1answer
186 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
98 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
votes
1answer
46 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
vote
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
votes
2answers
45 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
votes
1answer
60 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
votes
1answer
183 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
votes
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 ...
1
vote
1answer
30 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 ...
1
vote
1answer
33 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 ...
0
votes
1answer
53 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 $T$?