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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Transformation Matrix Process

Can anyone break down this Transformation Matrix process for me after the characteristic polynomial?
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37 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 ...
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19 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 ...
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32 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 ...
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2answers
45 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 ...
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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 ...
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1answer
45 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 ...
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1answer
21 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 ...
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1answer
21 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 ...
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1answer
44 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 ...
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15 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 ...
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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 ...
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18 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 ...
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2answers
75 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 ...
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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 ...
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15 views

Inverse transformation of continous transformation is bounded

I am having a continous transformation: $f: \mathbb C \to \mathbb C $ with $B \subseteq \mathbb C $ bounded. Now I want to proove that $ A = f^{-1} (B)$ is bounded! How can I proove that this ...
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38 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 ...
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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 ...
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1answer
15 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 ...
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26 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 $V\ni v \ne 0$ ...
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19 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 ...
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2answers
53 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 ...
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48 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 ...
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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 ...
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1answer
18 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 ...
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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 ...
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33 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 ...
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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: ...
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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 ...
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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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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 ...
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1answer
31 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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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 ...
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2answers
37 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 ...
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1answer
27 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 ...
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1answer
40 views

transformations' rubik's cube problem [closed]

How many rubik cube moves do we need to go from the first pic to the second? (1) 1 (2) 2 (3) 3 (4) 4 I'm sure it's not 1, because one move can't do that. How can we even find it out if we don't ...
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1answer
22 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 $k$: $\ker (T^{k}) \subseteq \ker(T^{k+1})$ ...
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1answer
19 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 ...
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3answers
27 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 ...
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1answer
38 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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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\\ ...
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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 ...
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25 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?
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1answer
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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 ...
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2answers
48 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 $AB=BA$, ...
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22 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] ...
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1answer
21 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 $w\in W$. $$T(S(w)) = S(T(w)) = S(w')$$ ...
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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, ...
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1answer
22 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 $t_0 - t_1$ ...
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1answer
25 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} ...