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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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Question about symmetric linear transformation

In the real inner product space $V$, a linear transformation $A$ is a symmetric transformation if $\langle A\alpha,\beta\rangle =\langle\alpha,A\beta\rangle$ In my book it says a linear ...
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2answers
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Finding the characteristic polynomial of the $T: \mathcal{M}_{n} (\mathbb{R}) \to \mathcal{M}_{n} (\mathbb{R})$ given by $T(M)=M^{\text{tr}}$

Here's a problem from Larry Smith's Linear Algebra textbook: Let $\mathcal{M}_{n} (\mathbb{R})$ be the set of real matrices of order $n \times n$. Let $T: \mathcal{M}_{n} (\mathbb{R}) \to \...
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1answer
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Find kernel and image of linear tranformation from matrix

Given following linear transformation in matrix space 2x2: $$T(X) = \begin{pmatrix} 1 & 7 \\ 7 & 2018 \end{pmatrix}X - X\begin{pmatrix} 1 & 7\\ 7 & 2018 \end{pmatrix}$$ How to find ...
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linear transformation between two bases for the polynomial space $P_2[x]$

$B=\{1,1+x,1+x+x^2\}, C=\{1+x,2x,x^2-1\}$ $B$ and $A$ are both bases for the polynomial space $P_2[x]$. $T: P_2[x] \to P_2[x]$ $$[T]B = \begin{pmatrix} 3 & 3 & 3\\ -2 & -...
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Representing a transformation's matrix as an inner product

Let $V$ be an inner product space, $T: V \to V$ a linear map, and $A=M(T,P,P)$ for an orthonormal basis $P=[v_1,...,v_n]$ (where $M(T,P,Q)$ is the matrix representation of $T$ with respect to $P$ and $...
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Matrices linear algebra linear transformations [on hold]

Prove that following statements are equivalent for A€M(C). 1. A is invertible 2. Coloumns of A are linearly independent. 3. The rows of A are linearly independent.
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For A,B subsets of a Normed Vector Space, A closed, B Compact, Show A - B Is Closed [duplicate]

Statement of the problem: Let $E$ be a Normed Vector Space over the real numbers. Let $A, B$ be subsets of $E$ such that: $A$ and $B$ are non-empty, $A \cap B = \emptyset $. Assume $A$ is closed and ...
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1answer
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Deeper understanding of the adjoint of a linear operator

My undergraduate classes in Q.M describes the adjoint of a linear operator purely as a mathematical formality. At this point, I'd like a deeper and heuristic understanding of it. My questions are ...
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2answers
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For some linear transformation T, if $T(u_0)=w$ and $T(v)=0$ then $T(u)=w$ if and only if $u=u_0+v$ proof inquiry

Let $T : V → W$ be a linear transformation. Let $w ∈ W$ and let $u_0 ∈ V$ satisfy $$T(u_0) = w$$ Show that $u ∈ V$ is a solution of the equation $T(u) = w$ if and only if $u = u_0 + v$, where $T(v) =...
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Uniqueness of a linear map on a basis of a vector space

From Linear Algebra Done Right, 3rd edition, by Sheldon Axler: Suppose $v_1, \ldots, v_n$ is a basis of $V$, and $w_1, \ldots, w_n \in W$. Then there exists a unique linear map $T: V \to W$ such ...
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1answer
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Find charateristic polynomial for linear operator

I have a linear operator $L:P_2(\mathbb{R})\to P_2(\mathbb{R})$ where $L(\alpha+\beta X)=(3\alpha+2\beta)+(\alpha+2\beta)X, \alpha ,\beta\in\mathbb{R}$, I want to find the following for $L$: The ...
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Group theory : representation Matrix of generating element for families of functions?

Assume I have some sampling of a function $f(t)$ at points $t$: $$f(t_k) = d_k, \forall k \in \{1,\cdots,n\}$$ Assume we have vectors $\bf v_k$ which can be functions of $x_k$ and $d_k$, can we find ...
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There is no T-invariant subspace $U$ such that $\mathbb{R}^{3} = W\oplus U$

Could someone give me a suggestion to solve the following problem problem? PROBLEM. Let $T : \mathbb{R}^{3} \longrightarrow \mathbb{R}^{3}$ and $\beta$ a basis of $\mathbb{R}^{3}$ such that $$ \left[ ...
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1answer
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Dimension of kernel subspace of trace transformation

From S.L Linear Algebra: Let $V$ be the vector space of real $n \times n$ symmetric matrices. What is $\textrm{dim} \, V$? What is the dimension of the subspace $W$ consisting of those matrices ...
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3answers
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What two transformations are described by $A=\begin{bmatrix} 2 & 4 \\\ 1 & 2 \end{bmatrix}$

Apparently the following matrix $$A=\begin{bmatrix} 2 & 4 \\\ 1 & 2 \end{bmatrix}$$ is a composition of two linear transformations, $BC=A$. The objective of this exercise is to decompose $A$...
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1answer
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Show that a linear map is one-to-one if and only if it preserves linear independence.

From Jim Heffron's free textbook Linear Algebra: 3rd Edition, question 3.2.27 states: "Show that a linear map is one-to-one if and only if it preserves linear independence." The answer key, also ...
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3answers
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Will the product of a real valued matrix and its transpose always have a real eigenvector?

I'm trying to solve this really interesting problem, taken from "Berkeley Problmes in Mathematics" by Souza and Silva https://www.amazon.com/Berkeley-Problems-Mathematics-Problem-Books/dp/0387204296 ...
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1answer
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Help a robot avoid computing the matrix inverse using the Gram-Schmidt process

Imagine you have a robot whose position is recorded as $t_1, t_2, t_3, \dots$ in its coordinate frame. Check the visualization here. We can write down the robot's basis in our 2D image plane, i.e. ...
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How to find all the derivation of matrix algebra

Suppose F is a field, and $M_n(F)$ is a matrix algebra over F, denoted by V. If $\phi \in L(V)$, the set of all linear map between V, and $\phi$ satisfies Leibeniz's law: $\phi(AB) = \phi(A)B+A\phi(B)$...
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1answer
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all 2 dimensional invariant subspaces

How we can find all 2 dimensional invariant subspaces of \begin{pmatrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 8 \end{pmatrix} I know that there are at least 2 such subspaces, ...
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4answers
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Is it true that the eigenvalues of $A + B$ are the sum of some eigenvalue of $A$ and some eigenvalue of $B$?

Is it true that the eigenvalues of $A + B$ are the sum of some eigenvalue of $A$ and some eigenvalue of $B$? I'm taking a linear algebra class, and I recently learned about eigenvalues. I think that ...
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2answers
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Computing the matrix exponential for a Jordan matrix

How can I compute $e^{At}$ where $A = J_{3}(5)$? That is, $$A = \begin{pmatrix} 5 & 1 & 0 \\ 0 & 5 & 1 \\ 0 & 0 & 5 \end{pmatrix} $$ Using this, how can I write down a basis ...
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0answers
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Basis for $\mathcal{L}(E)$ formed by projections

Let $E$ be a finite dimensional vector space. Find a basis of $\mathcal{L}(E)$ formed by projections. a.k.a : endomorphisms p such that $p^2=p$. My solution was to let $(e_1,...,e_n)$ be a basis for ...
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5answers
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If $AB=I$ show that $A$ has full rank.

Let $A,B\in \mathbb R^{n\times n}$. Show that if $AB=I$ then $B$ has full rank. In fact, I show that $A$ has full rank, which is quite obvious but I really have difficulties to show that $B$ has full ...
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1answer
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What does “the expansion of a vector relative to a basis is unique” mean in this proof of the Fundamental Theorem of Linear Algebra?

In the following proof, what does "But the expansion of a vector relative to a basis is unique" mean? Let$$T:V\to W$$ be a linear transformation. Then $$dim(Im(T))+dim(ker(T))=dim(V)$$ $\mathbf{Proof:...
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1answer
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Lower Dimensional Volume under Transformation

it is a well known fact that if $K$ is a measurable set in $R^n$ (we can restrict to convex bodies if you like), and $T$ a linear transformation then $$|TK|=|detT||K|.$$ If $K$ is not $n-$dimensional ...
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1answer
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Finite Rank Operator in Normed Space, not necessarily Hilbert neither Banach

Suppose that $E$ and $F$ are normed spaces and $T:E \rightarrow F$ is a bounded linear operator. I NEED TO SHOW WHAT FOLLOWS: If there are $n\in \mathbb{N}, f_{1}, ..., f_{n}\in E^{\ast}$ (dual of $...
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Given a Jordan canonical basis, how to find out to which generalized eigenspace picked generalized eigenvector belongs

Suppose we have finite-dimensional linear operator $A:V\to V$ , that has eigenvalues $\lambda_1 ,\lambda_2, ... \lambda_n$ . It is known that we can decompose $V$ into direct sum of generalized ...
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1answer
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Which polynomial kernels are cyclic, and how to find a cyclic generator

Let $T$ be a $\Bbbk$-linear operator on a vector space $V$. Given $f\in \Bbbk[x]$ consider the operator $f(T)$ on $V$. Write $K(f)=\operatorname{Ker}f(T)$ and call such $T$-invariant subspaces ...
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1answer
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If $T \colon V \to V$ is a linear operation and $B$ is a basis for $V$, then $T$ is one-to-one $\iff$ $[T]_B$ is invertible.

If $T \colon V \to V$ is a linear operation and $B$ is a basis for $V$, then $T$ is one-to-one $\iff$ $[T]$$B$ is invertible. Moreover if the equivalent statements hold, $[T$$-1$$]$$B$ $=$ $[T]$$-1$$B$...
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1answer
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Find sum of Subspaces combination of constraints

Let $ S = \langle \{ x^2-1,x^2+1\} \rangle $ and $ T = \{p\in P_2(\mathbb{R}), p(x)= ax^2+bx+c : a+b=0,c=0\} $ $\color{black}{a) \ Find \ S+T}$ From the definition of Sum in subspaces, we take e.g $...
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1answer
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Prove that BA cannot be invertible (A and B are linear transformations). Give an example of AB which is invertible.

The exact question I am trying to solve is as follows: Let A: $\mathbb{R}^3\rightarrow\mathbb{R}^2$ and B: $\mathbb{R}^2\rightarrow\mathbb{R}^3$ be linear transformations, so BA: $\mathbb{R}^3\...
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1answer
69 views

How does the minimum distance between coordinates behave under change of basis?

$\renewcommand{\vec}[1]{\mathbf{#1}}$I came across this problem/idea quite some time ago and was unable to reach a conclusion. (The title of this questions may be misleading. I am open for suggestions....
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1answer
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The set of all invertible operators need not be dense

I want to show that the set of all invertible operators $\mathcal{G}(\ell^2)$ is not dense in $\mathcal{B}(\ell^2)$. Consider the right shift operator $T\in \mathcal{B}(\ell^2)$. We also know that $T\...
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1answer
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Let $T \colon V \to W$ be a linear transformation, and if $\dim(V) < \dim(W)$, T cannot be onto.

Let $V$, $W$, be two finite dimensional vector space. Prove that if $T \colon V \to W$ is a linear transformation from $V$ to $W$ and $\dim(V) < \dim(W)$, then $T$ is not onto. $$\dim(V) = \...
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2answers
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Find the column space of the matrix $ \left(\begin{smallmatrix} 1 & 0 & -1 & 0 & 1\\ 0 & 1 & -1 & 2 &0\\ \end{smallmatrix}\right) $

I have the following matrix $$ \begin{pmatrix} 1 & 0 & -1 & 0 & 1\\ 0 & 1 & -1 & 2 &0\\ \end{pmatrix} $$ And I am unsure as to how to write the column ...
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Vector orthogonalisation that keeps (as many as possible) zero elements

I have a matrix $\bf{A}$ of orthonormal vectors (columns). Most of these vectors have some elements which have a numerical value close to zero. For every column of $\bf{A}$ I go through all elements ...
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1answer
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Basis for that $T: \mathbb{R}^3 \to \mathbb{R}^3$ is in rational canonical form

Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation such that $T(x,y,z) = (x+y+z, x+y+z, x+y+z)$ Find a basis for $T$ such that your matrix$(A_T)$ is in rational canonical form. I ...
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1answer
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Does PCA always have to reduce dimensionality?

I came across this paper where the authors implement a regularized learning model to estimate the covariance matrix of a dataset. The authors say they "...propose a regularized form of Principal ...
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Proving row rank equals column rank of a matrix

I am studying Linear Algebra from the book of the same name by Larry Smith. I am confused with the way Smith proves that row rank of a matrix equals its column rank. Here are the things that he uses ...
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2answers
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How to determine the matrix T to find the eigenvalues [closed]

For linear map $T : V → V$ , find the eigenvalues, and for each eigenvalue λ find its algebraic and geometric multiplicities, and determine whether T is diagonalisable. $V$ is the vector space of ...
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1answer
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Prove that there exists an ordered basis $\gamma$ for which $[T^*]_\gamma$ has a column of $0$s.

$V$ is an $n$-dimensional vector space over a field $\mathbb{F}$. Assume that $T^*:V\rightarrow V$ is a linear operator on $V$ and $T^*$ is not an isomorphism. Prove that there exists an ordered basis ...
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2answers
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prove for T:V→V is linear, if surjective then T is bijective

I know that $T$ is surjective means $R(T)=V$, which means $\mathrm{Nullity}(T)=0$. But how can I show that $\mathrm{Nullity}(T)=0$ implies $T$ is injective? I know that if $f(x)=f(y)$ and $x=y$ ...
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0answers
51 views

About derivative of delta function - chain rule for delta function containing a function

I have a problem that relates to derivative of a delta function. The problem originates from a paper I was reading https://aip.scitation.org/doi/full/10.1063/1.2938860 In the paper, it is said that ...
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0answers
10 views

Reflecting x-axis over a line of tangent of a function

I learned about linear transformations of vectors and shapes during the linear algebra class. Amongst all the transformations, I saw the reflections over x and y axis by multiplying certain matrices. ...
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1answer
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2D Matrix Transformation Invariance

The transformation $\textbf{T}$ maps points $(x,y)$ of the plane into image points $(x', y')$ such that $$\begin{align*} x' &= 4x + 2y + 14 \\ y' &= 2x + 7y + 42 \end{align*}$$ Find the ...
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2answers
45 views

How to find the rotation angle and axis of rotation of linear transformation?

I need some help with this problem: We know that $T(x_1,x_2,x_3)=(x_2,x_3,x_1)$. We suspect that it is a rotation matrix, to be sure, we need to determine the rotation axis and the rotation angle. ...
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0answers
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How do I use a set of orthogonal vectors as a new coordinate system?

I'm dealing with a problem where I need to apply a coordinate transformation to a set of data, but am not sure how. This is probably very simple, but I want to make sure I do it right. Suppose that ...
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1answer
20 views

Coordinates to pixels

Let's say I have two geographical points $P_1=(512401.72N,0032120.17W)$ and $P_2=(512332.83N,0031948.64W)$. What would be the easiest way to scale line between the two so that it fit on a 800x800(...
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1answer
33 views

Show that $Γ$ is isomorphic to $L(V/W,V')$

Question: let $V$, $V'$ be vector spaces over field $K$ and $W$ be subspace of $V$ then show that $\Gamma = \{T\in L(V,V')\vert\forall w\in W: T(w) = 0\}$ is subspace of $L(V,V')$. Further show that ...