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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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1answer
725 views

Show that T is a linear transformation (polynomial)

Let $P_2 $ be the space of polynomials of degree $\le$ 2, $ E=(1,t,t^2)$ and $ B=(1,1+t,t^2+1)$ be two bases for $ P_2$, and $ T:P_2 \to P_2 $ be the transformation $ T(p(t)) = p(t) - 2p'(t)$. a) ...
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15 views

Composition of linear mappings

Let $V$ be a vector space, $v,u∈V$, and let $T_1:V→V$ and $T_2:V→V$ be linear transformations such that $T_1(v)=5v+2u, T_1(u)=−2v−6u, T_2(v)=3v−6u, T_2(u)=5v+2u.$ Find the images of $v$ and $u$ ...
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Example of a non closed linear operator

I have just read that an operator $T$ is closed if, $\forall x_0\in D$, considered every sequence $x_n\rightarrow x_0$ such that $T x_n$ is convergent, we have that every $Tx_n\rightarrow Tx_0$. In a ...
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1answer
34 views

Find $\operatorname{ind}T$ where $T=\frac{d^2}{dx^2}$

Let $T$ be a linear and continuous operator defined as $$T=\frac{d^2}{dx^2}$$ Determine $\dim \ker T$ and $\dim \operatorname{coker}T$ in this two cases: $T: \mathscr{C}^2([a,b]) \longrightarrow \...
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Calculations on a field $\mathbb{Q}(\alpha)$ with min polynomial $P = x^3-2x^2+3x-5$

Suppose we have a field $\mathbb{Q}(\alpha)$ where the minimum polynomial of $\alpha$ is $P = x^3-2x^2+3x-5$ Prove that the mulitplication-by-$\alpha$ map $\phi$ : $\mathbb{Q}(\alpha) \rightarrow ...
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Finding the transformation matrix given an object and how it's transformed

I have a shape on a $2$ dimensional plane (axis are $x$ and $y$) and am told that the object stays the same with respect to the line $y=x$ and stretched by a factor of $2$ alone the line $y=-3x$. How ...
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28 views

Can every singular matrix be transformed into a diag matrix with only 0s and 1s along the diagonal by multiplication with an invertible matrix?

From linear algebra it is know that by choosing a "good" basis, that is multiplying a matrix with an invertible matrix $P$ from one side and with another invertible matrix $S^{-1}$ from the other side,...
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How do I prove $R(A)=(Ker(A^T))^\perp$?

Let $A:\mathbb R^m \to \mathbb R^n$ be a linear transformation. If $W$ be a subspace of $\mathbb R^n.$ define $$W^\perp=\{y\in \mathbb R^n|\langle x,y\rangle=0 \text{ for all } x\in W\}$$ Then which ...
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88 views

Interpreting some linear conditions

Let $K\geq 3$ and consider $K$ real numbers $\mu_1<\mu_2<...<\mu_K$. Let $\delta_j\equiv \mu_{j+1}-\mu_j$ $\forall j \in \{1,...,K-1\}$. For $h\in \{2,...,K-1\}$, consider the following set ...
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What does $T(v_i)$ mean in the definition of Linear Transformation for some $T$ and basis vector $v_i$

I am unsure what does $T(v_i)$ means as shown in pictures below (I will type out the gist of it too) Given a linear operator(transformation) $T$ on vector space $V$. Let a basis $B$ of $V$ be $B=\{...
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1answer
796 views

Finding the image and kernel of an orthogonal projection.

I am trying to find the image and kernel of the orthogonal projection onto the plane x+2y+3z=0. The following is how I solved the problem, but it appears to be incorrect. What is wrong with my ...
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791 views

linear transformation on lebesgue measure theory

How to show linear image of lebesgue measurable set is lebesgue measurable in $\mathbb{R}^n$? E.g. $m(T(A))=|det T| m(A)$? Continuous image of measurable set may not be measurable
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1answer
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Describing invariant subspaces from characteristic polynomial and minimal polynomial

I am working on the following Linear Algebra problem: (a) Suppose $T: \mathbb{R}^4 \longrightarrow \mathbb{R}^4$ is a linear transformation with characteristic polynomial $x^2(x-1)^2$. Describe the ...
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2answers
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Equation of unit circle under linear transformation - can't understand role of inverse matrix

In the course I'm following (topic is Geometric Transformations, NOT linear algebra yet) they introduce a way to find the equation of the unit circle under some invertible linear transformation $f(C)$ ...
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7 views

What is the transformation matrix?

I have the following problem: "In $\mathbb{R^2}$ a basis is given $a=(a_1,a_2)$ where: $a_1=(1,-1)$ $a_2=(0,1)$ For $f:\mathbb{R^2} \rightarrow \mathbb{R^2}$ it is known: $f(a_1)=-6\cdot a_1$ $f(...
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51 views

Invariant polynomials of under some linear transformations

I am looking for the polynomials invariant under two linear transformations. That is, if $x\in\mathbb{R}^4$ and given two sets of linear transformations $f$ and $g$, I am looking for the invariant ...
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How to prove an equation with $\ker$ and Im

Let f and g be two endomorphisms of E. How to prove that : $$ f(\ker{g \circ f}) = \rm{Im}~f \cap \ker g~.$$ It is obvious that $\ker f \subset \ker{g \circ f} $.
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Matrix product to keep desired elements

Is there a matrix based operation that, given two matrices of type below, transforms $$ \mathbf{e}=\begin{pmatrix} 0 & 1 & 1\\ 1 & 0 & 0\\ 1 & 0 & 0 \end{pmatrix}, \mathbf{A}=...
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What is the geometric interpretation of the action of a degenerate real skew-symmetric matrix?

As discussed in What is the geometrical action of a skew-symmetric matrix on an arbitrary vector?, an arbitrary real skew-symmetric matrix can be brought into the block diagonal form $$\begin{bmatrix} ...
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Jordan Normal form for matrix A with characteristic $\chi_{A} = (x-a)^5$, minimal polynomial $m_A = (x-a)^4$.

Let $B= A-aI_{5 \times5}$. As $ null(B^4)=5$ we have $null(B^3)=4, null(B^2)=3, null(B)=2$. I now try to construct a Jordan basis for $ \mathbb{F^5}$ by the same method used in the proof of the Jordan ...
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Linear operator between infinite dimensional vector spaces

I found (in a book) an example of a linear operator $T$ between infinite dimensional vector spaces (in this case Hilbert spaces) $$T:H\rightarrow H',$$ whose image does not coincide with the entire ...
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1answer
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Check in each case whether the given map is a monomorphism/epimorphism/isomorphism.

Let $V$ be a $3$-dimensional vector space over $\mathbb R$ with basis $\{v_1,v_2,v_3\}$. Check in each case whether the given map is a monomorphism/epimorphism/isomorphism. (Recall that it is enough ...
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Finding the dimension of a set of linear applications

Let $V,W$ be vector spaces with finite dimension over a field $K$ , let $A\subseteq V, B\subseteq W$ be vector subspaces. Denote $\hom(V,W)$ the vector space of all linear applications from $V$ to $W$,...
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(principal) sub matrix in the language of linear transformation

So we know that every matrix is a linear transformation given some basis. What can we say then about the submatrix (more precisely, principal submatrix) of a given matrix? Is there a nice description ...
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Concept/Terminology for extending a linear transformation to any number of dimensions

Let's say I have a n-gon $P$ in the plane, whose vertices $x_1,x_2,...,x_n$ are expressed by the vector $P=[x_1, x_2,\cdots,x_n]$. I want a linear transformation $A$ that when applied to $P$ gives the ...
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1answer
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Find matrix for linear transformation L(M) = transpose(M) in the given basis B of 3x3 magic-squares

I already found a basis B for 3x3 magic-squares, but I am unsure of where to start on part b. Finding the transformation matrix for $L(M) = \text{transpose}(M)$ in said basis.
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Passive transformation by antiunitary operator and orientation of the complex plane.

I have recently tried to make sense of the concept on an antiunitary passive transformation on a complex Hilbert space $H = \mathbb{C}^N$. I still do not know whether the concept even makes sense ...
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1answer
31 views

A linear map $T: \mathbb{C}^7 \longrightarrow \mathbb{C}^7$ such that $T^2 + T + I$ is nilpotent?

I am trying to find a linear map $T: \mathbb{C}^7 \longrightarrow \mathbb{C}^7$ such that $T^2 + T + I$ is nilpotent. My idea was to find a $7 \times 7$ matrix $A$ with entries in $\mathbb{C}$ such ...
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1answer
15 views

How to solve linear equations with structured matrices?

Suppose $\mathbf{x}\in R^{m\times 1}$, $\mathbf{X} = [\mathbf{x}\, \mathbf{x}\, \cdots]^\top\in R^{nm\times 1}$ and $\mathbf{b}\in R^{nm\times 1}$ and $\mathbf{A}\in R^{nm\times nm}$. How to solve \...
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Is there a Deterministic Matrix with Restricted Isometry Property?

The Restricted Isometry Property (Low-Rank Matrices) Let $\mathcal{A}:\mathbb{R}^{n\times n}\rightarrow \mathbb{R}^m$ be a linear operator. The constant $\delta_r:=\delta_r(\mathcal{A})$ is defined ...
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1answer
49 views

Construct a vector so the operator inequality $\|Av\|_1 \le \|A\|_1 \|v\|_1$ is equal

First, we should show that for $A\in \mathbb{R}^{m\times n}$: $\|Av\|_1 \le \|A\|_1 \|v\|_1$. $$(1)\quad\|Av||_1=\sum_{i=1}^m|(Av)_i| \le \sum_{i=1}^m \sum_{j=1}^n|A_{i j}||v_j|=\sum_{j=1}^n (\sum_{...
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9 views

Finding original transformation given the matrix transformation corresponding to bases

Let's say I have two bases $\mathcal{B} = \left\{x^{2}, x, 1\right\}$ and $\mathcal{S} = \left\{x^{2}+x, x-1, x+1\right\}$ of the set of all second-degree polynomials $\mathcal{P}_{2}$. Define a ...
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1answer
30 views

Prove that the vector space $V$ must be infinite dimensional

Suppose $V$ is a vector space and subspaces $U_1,U_2$ of $V$ are such that $U_1 \times U_2$ is isomorphic to $U_1+U_2$ but $U_1+U_2$ is not a direct sum. Prove that $V$ must be infinite dimensional. ...
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1answer
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Class of Fourier multiplier in $L^1$ is the class of Fourier transform of finite Borel measures. (Stein)

Hi. I am trying to prove an observation in Stein, singular integrals. Observation. $\mathcal{M}_{1}$ (class of Fourier multiplier in $L^1$) is the class of Fourier trasnforms of elements of $\mathcal{...
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how do I scale points so their average distance is sqrt(2)?

According to Multiple View Geometry in Computer Vision, Second Edition, pg 109, I need to Compute a similarity transformation T that takes points xi to a new set of points x˜i such that their ...
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2answers
29 views

Understanding the result of applying a linear transformation to a basis

There are two related concepts I do not grasp. A linear transformation is determined by its action on a basis. The action on a basis can be arbitrary. Can someone give insight into this?
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1answer
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Linear algera-Question on Isomorphism.

Can someone help me to solve the question of isomorphism I cannot gain an intuitive feel on it,because I cannot visualize $\mathbb C^3(\mathbb C)$.I am in problem with mainly the $2nd $ part of the ...
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1answer
23 views

Finding solutions of linear homogeneous system in integers and rationals.

I encountered a problem given by my linear algebra prof which asks to find integer solutions of $2x+3y+z=0$ .Also there was another problem in which I have to find rational solutions of the system $x+...
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1answer
16 views

Find a homogeneous linear equation in $4$ variables with $3$ given vectors as solution.

I am thinking on a question of how to find a linear homogeneous equation such that $(1,1,1,1)$,$(1,-1,-1,1)$ and $(2,3,3,2)$ are solutions of the equation.So far I think I am after a hyperplane in ...
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1answer
26 views

Graphical intuition for the eigenvalue of a linear map by looking at the unit sphere

Let's suppose we're in a finite dimensional normed real vector space $V$, where $\dim(V) = n$. Let's say we have a linear map $T: V \to V$, which transforms the unit sphere, $\{v \in V: ||v|| = 1\}$ ...
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1answer
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Need help regarding dimentions of vectors in matrix transformation

Suppose $T(x) = Ax$ ,let $A$ be a $3$x$4$ matrix filled with numbers (3 rows & 4 columns). How can I find dimensions of vectors that are inputs and also dimensions of outputs for the function $T(x)...
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1answer
32 views

How to realise the equality of a map in the quotient vector space

Suppose $$A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i\end{pmatrix},$$ and $$v = \begin{pmatrix}1 \\ 1 \\ 1\end{pmatrix}$$ is an eigenvector for $A$ with eigenvalue $...
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1answer
23 views

Why is the reflection of a vector on a line, twice the projection of the vector minus the vector? [duplicate]

I am working through a linear algebra text and I am doing exercises in a chapter dealing with linear transformations and linear functions. I am trying to come up with the transformation matrix for a ...
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0answers
22 views

Sign inconsistency in variation of scalar field $\delta\phi$

Let be $\phi$ a scalar field which depends for simplicity only on coordinates $x^\mu$ (i.e $\phi = \phi(x^\mu)$). Let's do an infinitesimal translation: $x'^\mu = x^\mu + \delta x^\mu$ Thus, the ...
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1answer
40 views

Please Help with Question about Linear Operators

The Question: Let $X$ and $Y$ be Banach spaces and $T: X \rightarrow Y$ an injective bounded linear operator. Show that if $R(T)$ is closed in Y, then $T^{-1} : R(T) \rightarrow X $ is bounded. My ...
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2answers
46 views

Finding a linear transformation with respect to basis

Let $T\colon \mathbb{R}^3 \to \mathbb{R}^{2x2}$ a linear transform given by \begin{equation} T(x,y,z) = \left({\begin{array}{cc} x + y + z & 2x -z \\ x-y+z & z-3y-2x \end{array}} \right), x,...
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1answer
49 views

Is $u \mapsto \|u\|_p$ a bounded linear functional on $L^p(\Omega)$?

Let $\Omega\subset \mathbb{R}^n$ be a bounded smooth domain and $1<p<\infty$. My question: is $F:L^p(\Omega) \rightarrow [0,\infty)$, $F(u)=\|u\|_p$ a bounded linear functional? I would say that ...
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1answer
16 views

Define rank of matrix by reduced row echelon form - well-defined?

In order to define the rank of a matrix, I want to use reduced row echelon form (rref). I have an ugly proof that the rref is unique in the following sense: If $A,B$ are in rref and $A=ZB$ with ...
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1answer
34 views

The relation of $\alpha$ having index of nilpotence $k>0$ and being (with $\sigma_{1}$) 1-1 and onto.

let V be a vector space over a field $F$and let $\alpha \in End(V)$ be nilpotent, having index of nilpotence $k>0.$Show that $\sigma_{1} + \alpha \in Aut(V).$ where $\sigma_{c}$ is defined as $\...
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1answer
29 views

Find the standard matrix of the linear operator T. Find the basis of the image of T, and find the basis of the kernel of T.

Let $T$ be a linear operator $\mathbb{R}^4\rightarrow\mathbb{R}^4$ such that $$ T\begin{bmatrix} 2\\1\\0\\0 \end{bmatrix}=\begin{bmatrix} 2\\1\\0\\0 \end{bmatrix},\; T\begin{bmatrix} 3\\1\\0\\0 \end{...