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Questions tagged [linear-transformations]

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V → W between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. (Def: http://en.m.wikipedia.org/wiki/Linear_map)

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How to understand root subspace in linear algebra

According to the definition of the root subspace,there exists an exponent k that$(\mathcal A-\lambda\epsilon)^k v=0 $,but how to confirm this exponent?If it is an n*n matrix,what happens when the ...
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Frobenius series solutions and asymptotic

Consider the equation below which is an eigenvalue problem I am studying: $$ \label{left} \lambda(v-v'')+c\left(v''-v+be^\xi v+(1-b)e^\xi v'-e^\xi v''\right)'=0. $$ Restriction on the parameters: $\...
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Interpretation of vectors in dual forms - in matrix equation, and in linear combination of vectors

While a matrix equation $A \vec x=\vec b$ identifies $\vec x$ and $\vec b$ as two vectors, its equivalent form as linear combinations of vectors ${x_1} \vec {a_1} + {x_2} \vec {a_2} = \vec b$ reveals ...
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If there any plane curve whose critical points' curvature are invariant by linear transformation?

I was studying if the curvature of $f$: $$ f(x) = \frac{ax}{b+x} $$ can have the critical points located at the same vertical than this other $g$ curve: $$ g(x) = \frac{ax}{b+x} + cx = f(x) + cx $$ ...
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Matrix representation of Fractional Linear Transformation and the Identity Matrix?

For $x \in \mathbb{R}$, define the fractional linear transformation of $x$ as $f(x)$ where: $$f(x) = \frac{ax + b}{cx+d}$$ Then $f(x)$ has a matrix representation in $\mathbb{R}^2$ of $F$ where: ...
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Is there an easy way to remove scale from a squared linear transformation matrix

Given a linear transformation matrix $A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix}$, I know that one can use SVD or QR decomposition ...
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Coordinate transformation in ODE's with a unit step function

Consider the following general set of 2 ODE's $$\dot{x}=\Theta(\dot{x} )f_1(x,y)+(1-\Theta(\dot{x}))f_2(x,y)$$ $$\dot{y}=(1-\Theta(\dot{y}))g_1(x,y)+\Theta(\dot{y})g_2(x,y)$$ where $\Theta(x)$ is ...
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If two operators with all eigenvalues real numbers commute then both are triangularizable!

Let $A,B:E\to E$, where $E$ is a finite dimensional vector space, be two linear operators such that all of the eigenvalues are real numbers. If $AB=BA$, prove that there exists a basis in which ...
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Rotation of vector by rotation matrix

Assume the following expression $$ \begin{bmatrix} a_1^* \\ a_2^* \end{bmatrix} = \begin{bmatrix} \cos(45) & - \sin(45) \\ \sin(45) & \cos(45) \end{bmatrix} \begin{bmatrix} a_1 \\ a_2 \end{...
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How to determine the base of $\ker\phi$ for polynomial function?

Given is a base defined as $$B:=(x\mapsto1,x\mapsto x,x\mapsto x^2,x\mapsto x^3 ,x\mapsto x^4)$$ A set V defined as $$V:= \{ f: \mathbb{R} \mapsto \mathbb{R}\ |\ \exists\ {a_0},...{a_4} \in \mathbb{R}\...
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Find Bilinear Transformation that maps the points $-i, 0, 2+i$ from $z$ plane on to the points $0, -2i, 4$ of the $ω-$plane?

The solution is supposed to be $w = \frac{2(z+i)}{z-1}$ Also, is it Okay to ask such questions here? It's not homework, it's the last few sets of problems I can't get on my own. Some belong to ...
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Find Bilinear Transformation that maps the points 0, 1, 2 from z plane on to the points 1, $\frac{1}{2}, \frac{1}{3}$ of the ω-plane [on hold]

Can someone get an answer of .. $$ω = \frac{1}{z+1}$$ I have tried it 4 times. No luck...
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How to find basis of linear subspace $V$ when $V$ contains all polynomials with the degree up to 4?

I have been given the following definition $V := \{ f : \mathbb{R} \to \mathbb{R} \mid \textrm{ there are } a_0,\ldots,a_4 \in \mathbb{R}$ and $f(x) = \sum_{i=0}^4 a_i x^i$ for all $x \in \mathbb{R}\}$...
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Find a Linear Transformation with fixed Kernel and Image.

Find a linear transformation $T : \mathbb{R}^4 \rightarrow \mathbb{R}^4$ such that $$ \operatorname{Ker} (T) = [(1, 0, 0, 1) ; (-1, 0, 0, 1)] $$ $$ \text{Image}(T) = \operatorname{Im}(T) = [(...
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Transformation matrix maps pair of intersecting straight to pair of intersecting straight lines [on hold]

In my Computer graphics mid-term exam, it was asked to prove, Transformation matrix maps pair of intersecting straight lines to intersecting straight lines. Here, transformation matrix is taken in ...
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1answer
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Why is any rightinverse to T injective? The linear transformation $T$: $\mathbb{R}^5 \longrightarrow \mathbb{R}^4$

Could use some help with this. The linear transformation $T$: $\mathbb{R}^5 \longrightarrow \mathbb{R}^4$ is given by $$ T \left[\begin{matrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \end{matrix}\...
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Transformation of region

Question:- Show that the image of the set $ S = {z \in \mathbb{C} | Im(z) \geq 0, |z|\geq 1} $ under the map $w = u+\iota v = z+ \frac{1}{z}$ is the upper half plane of $v\geq 0$ My approach Let $ z= ...
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How to define pivot columns?

When you use Gaussian elimination to solve a homogeneous system of linear equations, you end up with "pivot variables" and "non-pivot variables". The non-pivot variables have the property that they ...
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35 views

Proving a matrix inequality given another inequality

Suppose that for the $2$-norm, we have $||A||_{2} < 1$. Show that $||I - A^{T}A||_{2} < 1.$ Assume $A$ is invertible. I don't know how to solve this problem. I'm studying for an exam. I know ...
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Composition of rotation and reflection

Suppose you find a spatial isometry and you want to classify it. You find out the only real eigenvalue is -1. Also, there is a unique fixed point c. In this type of cases i was told that we have a ...
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How to get a new matrix with the diagonal elements of another one, and zeros in the rest of the entries? [closed]

I'm NOT interested in how I can do this in MATLAB. I would like to know how, given a matrix $A\in\mathbb{R}^{n\times n}$, I can extract only the elements on its diagonal, put them in a new matrix $B$ (...
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If $\Sigma^{-1}=(A^{-1})^TA^{-1}$, then why does $|A^{-1}|=|\Sigma|^{-1/2}$?

In the derivation of the joint pdf of $f_\textbf{X}(\pmb{x})$, where $\textbf{X}=\pmb\mu+A\pmb Z$ and $\textbf{X}\sim~N_n(\pmb\mu,\Sigma)$, there is a step I do not understand. In particular, it is ...
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37 views

Canonical Jordan form contradiction

I am faced with the following problem: Given endomorphism $f$ whose characteristic polynomial is $$P_c(x) = (x+1)^{10} (x-1)^{10} x^{10}$$ and whose minimal polynomial is $$P_m (x) = (x+1)^5 (x-1)^...
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2answers
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Why is the matrix used to describe a linear transformation different if we use row vectors?

Lets say some linear transformation $T$ is defined by: $$T(x, y , z) = (x+2y, 2x+3y-4z, 4x-4y)$$ Now this can be expressed as $\begin{pmatrix}1&2&0\\2&3&-4\\4&-4&0\end{pmatrix}...
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1answer
41 views

General Gauss-Markov theorem [closed]

$Y=XB+u$ where $X$ is a non random $n\times k$ Matrix, $\textrm{rank}(X)=k, E(u)=0, E(uu')=\sigma^2\Omega$, How to form $(1)$ How to proof $(2)$ the general Gauss-Markov theorem?
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How is inverse of linear operator in $\mathbb{R}[X]$ defined?

Let $T \in \operatorname{End}(\mathbb{R}[X])$ How to think of inverse of $T$ ? I thought about it in two ways: $a)\quad \exists U\in \mathbb{R}[X]: Tf\cdot Uf = 1$, $b)\quad U\left(Tf\right) = f.$ ...
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Conjugation of $\left(\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right)$

I'm interested in the following question. Let $h=\begin{pmatrix} 0 & -1 \\ 1 & 0 \\ \end{pmatrix}$. This is an orthogonal map which is quite far away from the identity (say in the Frobenius ...
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1answer
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Difference between two Matrix representation of Linear Transformation

I have a linear transformation $$T(x_1,x_2,x_3) = (x_1-x_2+2x_3,2x_1+x_2,-x_1-2x_2+2x_3) $$ Assuming an ordered basis $\{u_1,u_2,u_3\}$ I get the linear transformation of the basis vectors as below: $...
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Question on bilineal and quadratic forms

My first question is: Let $\phi: V \rightarrow \mathbb{R}$ a quadratic form. We say that a vector $x$ is autoconjugate if its conjugate with its self, that is, $\phi(x)=0$. Does the set of all ...
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How to Create a Homography Matrix in Matlab from n Pairs of Points?

I'm trying to create a homography matrix using the procedure shown below within Matlab: However, I'm having trouble actually coding it. I don't want to use SVD if possible and I'm not allowed to use ...
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1answer
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Product of two Linear Operators

Let $S(e_n)=e_{n+1}$ and $T(e_n)=e_{n+2}$ be two linear operators on the Hilbert space $l_2(N)$, the space of all sequences $\sum_{1}^\infty |a_k|^2 < \infty$, and $\{e_n\}, n=0,1,2,...$ is the ...
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Proving a projection onto subspace if $P^2 = P$

Suppose that V is a vector space, and M is a subspace of V . A transformation $P : V \to V$ is called the projection of V onto M if (i) there exists a subspace N such that every vector v ∈ V can be ...
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1answer
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General Formula of a Linear Operator given its act on the Standard Orthonormal Basis

I am trying to find a general formula for a linear operator on a Hilbert space when its action on the standard orthonormal basis is known. I include my work below. Please tell me whether my solution ...
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1answer
74 views

Eigendecomposition of a matrix (what assumptions need to be made)

For the sake of brevity, I'm given a 3x3 symmetric matrix with real entries with no further information as to what the rows and columns encode. (eg. this not need to be the case but the columns may ...
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Orthogonal Transformation Matrices issue

I'm really confused. If $V$ is an orthogonal matrix that transforms matrix $U_1$ into matrix $U_2$ why is it that $U_2 = V^TU_1V$? With what I know, an orthogonal matrix is one such that $V^{-1} = V^...
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Looking for ways to transform time-series data recorded from object movement into equation describing the movement direction of the object

Looking for some time-series data transformation advice! I want to know what's the best way to transform data of 9-tuples time series data of IMU (Inertia Measurement Unit) sensor, recorded from a ...
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1answer
38 views

Showing isomorphism in transformation between matrix and bilinear map spaces

From Serge Lang Linear Algebra: Show that the association $A \rightarrow g_A$ is an isomorphism between the space of $m \times n$ matrices, and the space of bilinear maps of $\mathbb{K}^m \times ...
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1answer
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Find a basis for witch the matrix consists only of 0 and 1

Let V and W be any two finite dimensional vector spaces over a field F. Let T : V →W be a linear map. Prove that there are bases of V and W such that with respect to those bases T is represented by ...
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Encryption and Decryption of Alpahabets to Integers

I'll explain my case with an example. Say, I've 2 words, FOOTBALL FOOD Let f(FOOTBALL) = 845614, f(FOOD) = 74312 and f(FO) = 132 Is there any method to get f'(845614, 2) = 132 and f'(...
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1answer
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(When) is the bicommutant of a linear operator equal to the set of “simpler” operators?

Let $V$ be a $\Bbbk$-linear space and $T\in \mathrm{End}_\Bbbk(V)$. Definition. Say an operator $S\in \mathrm{End}_\Bbbk(V)$ is simpler than $T$ if every $T$-invariant (internal) direct sum ...
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1answer
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Average area of the shadow of a convex shape [closed]

What is the average area of the shadow of a convex shape taken over all possible orientations? If we take a sphere, its surface area is exactly 4 times the area of its shadow. How can it be ...
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1answer
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Sequences of Continuous Linear Operators between Banach Spaces.

Let $E, F$ be two Banach Spaces. Let $\{ T_{n} \}$ be a sequence of continuous linear operators from $E$ into $F$ such that: For all $x \in E: T_{n}x \rightarrow Tx,$ some limit in $F$. Then the ...
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236 views

Linear Algebra - Proving a projection given a linear transformation

Suppose that $V$ is a vector space, and $M$ is a subspace of $V$. A transformation $P:V \rightarrow V$ is called the projection of $V$ onto $M$ if (i) there exists a subspace $N$ such that every ...
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2answers
75 views

Linear Algebra - Proving a projection onto a subspace is a linear transformation

How do I prove that a projection onto a subspace is a linear transformation ? Given that V is a vector space, and M is a subspace of V. I know these two facts: i) There exists a subspace $N$ such ...
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1answer
26 views

Understanding the association of the bilinear form and matrix associated with it

From S.L Linear Algebra: Show that the association $A \rightarrow g_A$ is an isomorphism between the space of $m \times n$ matrices, and the space of bilinear maps of $\mathbb{K}^m \times \...
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27 views

Monotonic linear interpolation of monotonic data on a scattered grid

Let $x_1,\dots,x_n\in[0,1]^d$ and $y_1,\dots,y_n\in[0,1]$ satisfy: $$\forall i,j\in {1,\dots,n},\hspace{10pt} x_i \le x_j \rightarrow y_i \le y_j.$$ I seek to find a continuous function $f:[0,1]^d\...
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99 views

Linear Algebra - Invariant Subspaces in 2-dimensions

How do I find 2-dimensional subspaces that are invariant subspace of T ? $T\left(\left[ \begin{matrix} x\\ y\\ z\\ \end{matrix} \right]\right) = \left[\begin{matrix} 2y+z \\ -2x+4y+z \\ -2x+2y+3z \...
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1answer
28 views

Meaning of the determinant of the restriction of a linear map

Suppose $T : \mathbb{R}^n \to \mathbb{R}^n$ is a linear map and let $U \subset \mathbb{R}^n$ be a $d$-dimensional subspace where $0 < d < n$ and $\ker T = U$. I was wondering how to make sense ...
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What's the geometric interpretation of the square root of a matrix?

Question: If I have a matrix $A$, I know that its square root is a matrix that has the same eigenvectors as $A$ but its eigenvalues are the square roots of the eigenvalues of $A$. What does this ...
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Doubt on range-space of linear transformation

I have a linear transformation $T:V\rightarrow W$. I believe that if T is onto, then, range space of T is equal to W. My reasoning is that if T is onto, then every member of W has a pre-image. So, W ...