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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Convert trapezoidal velocity profile into one with s-curve

I am an electronic engineer. I have created a system that contains a stepper motor. There is a speed table that specifies the speed values that are to be used to accelerate and deaccelerate the motor. ...
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Generate a uniformly sampled orthonormal matrix that 'rotates' $k$ vectors $x_0 \in \mathcal{R}^{n \times k}$ into $y_0 \in \mathcal{R}^{n \times k}$

We know that orthonormal matrices $H \in \mathcal{R}^{n \times n} $ are rotation matrices. Is there a general method to uniformly generate rotation matrices that can rotate a given set of vector $x_0 ...
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Solvability of a linear underdetermined system

For a linear system $$ M \alpha = \beta,\quad M \in \mathbb{C}^{n \times m}, \quad \alpha \in \mathbb{C}^{m\times 1}, \quad \beta \in \mathbb{C}^{n\times 1}, \quad m>n $$ with $\text{rank} M = p &...
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Antisymmetric Matrices and Orthogonality

My notes state: Given an orthogonal $n\times n$ matrix $A=I+pB$, where $I$ is the identity matrix, $p\ne 0$ is a real number and $B$ is an $n\times n$ matrix, then $B$ is skew-symmetric. I know that $$...
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Find all possible jordan normal forms of operator A

Find all possible jordan normal forms of operator A: $\mathbb{C}^7$ $\rightarrow $ $\mathbb{C}^7$ with following infos: rank$(A^3-A)^2 = 1$, $det(A-id)=12$,$rankA = 5$
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4 answers
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Linear transformations satisfying $T^3=T$ and $T^2\neq I$

I am looking for linear transformations $T:R^3\rightarrow R^3$, such that $T^3=T$ but $T^2\neq T$ and $T^2\neq I$. I've been playing around with this for a while, but can't see how any such ...
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How to scale a fat matrix to ensure its rows are orthonormal?

I have the following matrix $${\bf B} = \begin{bmatrix} 0&-4.2423&4.2423&1.4871\\ 1.6532&-1.2735& -1.2735&0.0024\\ 0 & -0.2805 & 0.2805 & -0.8823 \end{bmatrix}$$ I ...
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How to find a non-diagonalizable matrix $T \in L(\mathbb{C}^3)$ such that $6, 7$ are its eigenvalues?

I am currently reading Linear Algebra Done Right by Axler on my own without a teacher, and am stuck on the exercise in the title. My current progress is as follows: Suppose that $x, y, z$ is a basis ...
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Exercise 3.F.35 Show that $(\mathcal{P}(\mathbb{R}))'$ and $\mathbb{R}^\infty$ are isomorphic (Linear Algebra Done Right 3rd Edition by Sheldon Axler)

I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. 1.22 Example $\mathbb{F}^\infty$ is defined to be the set of all sequences of elements of $\mathbb{F}$: $$\mathbb{F}^\...
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Preserving Leading Eigenvalue while Shrinking Matrix

Suppose I have a graph adjacency matrix $G $ of size $n \times n$. By Perron-Frobenius, I am guaranteed to have a unique largest real eigenvalue $\lambda_{G,1}$ with corresponding eigenvector $u_{G,1}$...
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Proving that $\rm dim range \,ST \le \min (dim range\, S, dim range\, T)$.

$U,V$ are finite dimensional vector spaces. $W$ is a given vector space. Let $T\in L(U,V), S\in L(V,W)$ be linear transformations then it is to be shown that $\rm dimrange \,ST\le\min (dimrange\, T, ...
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Linear Transformation mapping linearly independent vectors onto a linearly dependent set

Let $T:\mathbb{R}^n \rightarrow \mathbb{R}^m$ be a linear transformation. Show that if $T$ maps two linearly independent vectors onto a linearly dependent set, then the equation $T($x$)=$0 has a ...
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Linear Transformation from $\mathbb{R}^6$ to $P_2(\mathbb{R})$

I am trying to figure out what the transformation of the values {$x_1, x_2, x_3, x_4, x_5, x_6$} $\in \mathbb{R}^6$, becomes when transformed to $P_2(\mathbb{R})$. I understand that $P_2(\mathbb{R})$ ...
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Find a pair of linear transformations that do not commute

The problem statement is as follows: Suppose $\mathbb{F}$ is any field. Find a pair of linear transformations $S,T \in \mathcal{L}(\mathbb{F^{2}}, \mathbb{F^2})$ such that $ST \neq TS$ My attempt ...
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Checking whether differential operator maps vector space to itself

Given a vector space $V$ that is spanned by: $\{3^t, 2^{-t}\}$, can we conclude that the differential operator $\frac{d}{dt}$ maps $V$ onto itself? It is my understanding that by looking at the basis ...
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Give three distinct examples of linear functionals on $\mathbb{R}^{[0,1]}$ ("Linear Algebra Done Right 3rd Edition" by Sheldon Axler)

I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. Exercise 3.F.2 Give three distinct examples of linear functionals on $\mathbb{R}^{[0,1]}$. The only example which I ...
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Find a discontinuous linear transformation from the set of bounded real sequences to the reals

I am looking for a linear transformation T: B(N,R) -> R, with R the real numbers and B(N,R) the set of all bounded real sequences (with the sup norm), that is discontinuous. I first thought that ...
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1 answer
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How to find correspondent basis of a matrix associated to a linear transformation

Let's say I have a linear transformation $f: \mathbb{R^3} \rightarrow \mathbb{R^3}$ defined as: $f(x_1,x_2,x_3) = (x_1+x_2-x_3, \ 2x_1 - 3x_2 + 2x_3, \ 3x_1-2x_2 + x_3) $. How can I find $\beta_1, \...
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Is it possible that linear transform changes interval data to ratio?

I'm going through a stats intro and got puzzled by the concept of interval and ratio data. Celsius temps is an example of interval data that can not be multiplied and divided, because there is no true ...
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Does this implies $\mathcal{M}[T]_{\beta_1 \beta_2}$ is similar to the $\mathcal{M}[T]_{\beta_3 \beta_4}$?

$T\in\mathcal{L}(V) $ where $\dim(V) <\infty$ Consider $\beta_1, \beta_2, \beta_3, \beta_4$ four bases of $V$ . Does this implies the matrix $[T]_{\beta_1 }^{\beta_2}$ is similar to the matrix $[T]...
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Any linear transformation is a sum of a rotation and streching. So Is it possible to decompose any $A$ real matrix into such $R$ and $S$ components?

The Question Let $A \in \mathbb{R}^{n \times n}$ be the matrix of a linear transformation. I have learned that any linear transformation is either a rotation, a streching, or a mirroring (with the ...
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Is $\Lambda:X\longrightarrow\mathbb K$ continuous if $\{\Lambda(x_k)\}_{k\in\mathbb N}$ is bounded when $x_k \longrightarrow0$?

Let $X$ a normed space over $\mathbb K$ ($=\mathbb R$ or $\mathbb C$) and $\Lambda:X\longrightarrow\mathbb K$ a linear aplication. Prove that if $\{\Lambda(x_k)\}_{k\in\mathbb N}$ is bounded $\forall(...
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Given matrix associated to linear transformation find the basis it corresponds to

I know how to find the matrix associated to a linear transformation with respect to two given basis, but how can one find the basis given the matrix? Let's say we're given a linear transformation $T : ...
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Tensor product of two representations of $D_4$

Let $(\tau,\mathbb C^2)$ be the irreducible representation of $D_4$ by matrix multiplication, namely for every $v\in\mathbb{C}^2$: $$\begin{bmatrix}\tau(s)\end{bmatrix}v=\begin{bmatrix}-1 & 0\\ 0 &...
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How many vectors do I need to span the vector space for Velocity, Distance, Angles, Acceleration, Position, and Time?

I am working on a Robot Arm simulation project and trying to get those 6 varibles, in my question, coming from the Robot Arm. The robot arm has 6 joints and each joint can run the motor inside to ...
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Proving that a product of maps is linear

I have two maps $\gamma, \beta: [0,1] \to M_n (\mathbb{R})$, both of which are continuous, and consider the product map $\gamma \beta: [0,1] \to M_n (\mathbb{R})$. I'm trying to show that $\gamma \...
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A similarity-like transfromation with left-invertible matrix.

Suppose $X\in \mathbb R^{m\times n}$ $(m>n)$ is a left-invertible matrix, and its left-inverse is $X^+=(X^\top X)^{-1}X^\top$ (so that $X^+X=I$). Now I have a matrix $A\in \mathbb R^{n\times n}$. ...
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How can one justify that $\|(I - \mu\mu^T)x\|_2 = \|(I - \mu\mu^T)\|_2\|x\|_2$ where $\mu$ is from the tangent-normal decomposition of the vector $x$?

Full-details about the context of the tangent-normal decomposition are present in this blogpost (search for tangent-normal decomposition). Let $x, \mu$ be $p$-dimensional vectors on the unit sphere. ...
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Determining if a set is a subspace of a vector space given a linear map

Let P be a vector space over R and let L : P → P be some linear map. Determine if the set S = {p ∈ P | L(p) = p} is a subspace of P What is the significance of the mention of the linear map in this ...
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Determining if L is a linear map, and checking if its injective or surjective

I was working through some examples from my textbook and got stuck on this question, was wondering if someone could help me understand how to do it. $L: M_{2\times 2}(\mathbb R) \to M_{2 \times 2}(\...
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Linear maps, linear applications, linear algebra [closed]

enter image description hereenter image description here 1: https://i.stack.imgur.comstrong text/YYNu8.jpg Hi there, I'm trying to solve this ex on linear maps, the result is similar to the book's one ...
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1 answer
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Determining if a linear transformation across abstract vector spaces is surjective given a non-square coefficient matrix

I have been having a bit of trouble with this problem. It can be viewed here, at the very bottom, question C23, the answer is provided but obviously not in enough detail. Here is the definition of the ...
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Why is a compact linear operator also a bounded linear operator?

I know that a linear operator is bounded iff continuous, with these definitions of "bounded" and "continuous": a linear operator T (on H Hilbert space) is bounded if ∃M>0: ||Tf|...
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The difference between ODE's such as $f(x,y,y',y'', ...,y^{(n)})=0$ and $yf(x,y,y',y'', ...,y^{(n)})=0$

I'm curious about the difference between an ODE such as $ax^2yy''+bxyy'+cy^2=0$ and $ax^2y''+bxy'+cy=0$, where a,b,c are real and a is nonzero. Solving them using the standard method for Cauchy-Euler ...
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(Non)Linear transformation [closed]

I was thinking of how prints on clothes get stretched while wearing them and if it is a linear transformation, so one can easily design a piece of cloth (e.g. a sock)such that the image will be ...
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Solving partial differential equation of more than 2 independent variables

If we have a function f(x,y,z) and have this equation $af_{x}+bf_{y}+cf_{z} = 0$ I have 2 questions :How do we transform the variables x,y,z to ξ,n,k to solve the differential equation? I mean $1)\...
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Intuition behind the definition of matrix similarity/equivalence? [duplicate]

Given two matrices $A$ and $B$, they are similar if: $$B=P^{-1}AP $$ Furthermore, if they are similar they are relative to the same linear transformation (equivalent). However the proof I've checked ...
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Expression for this matrix transformation that assigns replicas of main diagonal as columns?

I am looking for an elegant way to make the following transformation of a matrix: \begin{equation} M := \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix} \...
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Invertible linear operators without eigenvalues [duplicate]

Suppose $V$ is a vector space over the complex numbers, and let $\alpha: V \mapsto V$ be an invertible linear operator. If the dimension of $V$ is finite, we know that $\alpha$ has (nonzero) ...
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Is it true that a non-linear norm preserving map $T:R^n \to R^n$ which is locally an isometry can be written as $e^{\sum{\lambda_i(\vec{x}) L_i}}$?

forgive me if my question is not completely rigorous or clear, I will try my best. I consider a non-linear norm preserving map $T:R^n\to R^n$ such that $T(0)=0$ and such that it can be locally ...
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Fourier transform on a multivariate function along a line

Given a function $f(\vec{x}) :\mathbb{R^3} \rightarrow \mathbb{R}$, it is always possible to compute the Fourier transform to compute of f, $F(\vec{\omega}): \mathbb{R^3} \rightarrow \mathbb{C}$. ...
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Trying to understand projection onto a plane, subspace of R^3, of a point b when there is no solution for Ax = b

I was following along some online classes for linear algebra and was doing fine until it came to projections onto subspaces and I really really could not understand the proof and logic behind the ...
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Estimation of the squared norm of the main diagonal of the jacobian matrix [closed]

If $ f:\mathbb{R}^{d}\rightarrow \mathbb{R}^{d}$ is a differentiable function and $x\in \mathbb{R}^{d}$, then, for the Jacobin matrix $J = \frac{df}{dx} \in \mathbb{R}^{d\times d}$, we define: $$ z=\...
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1 vote
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A bounded linear map that cannot be both surjective and injective

Let $X$ and $Y$ be two Banach spaces. Assume that $f$ is linear bounded mapping from $X$ to $Y$ such that for all $n$, there exists $a_n$ with $||a_n||=1$ and $||f(a_n)||=\frac{1}{n}.$ Prove that $f: ...
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3 votes
2 answers
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If $P_1$ and $P_2$ are orthogonal projectors, then $\mathrm{tr}(P_1 P_2) \leq \mathrm{rk}(P_1 P_2).$

Prove or provide a counterexample: If $P_1$ and $P_2$ are orthogonal projectors, then $\mathrm{tr}(P_1 P_2) \leq \mathrm{rk}(P_1 P_2),$ where $\mathrm{tr}$ denoted the trace and $\mathrm{rk}$ the ...
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Let $S:R^2→R^2$ be linear with $S^2=S$. Prove $S= 0, S=id_{R^2}$ or $∃B$, basis of $R^2$ such that $[S]^B_B=\begin{pmatrix}1&0 \\0&0 \end{pmatrix}$ [duplicate]

Can someone provide Hint to Prove this. Let $S: R^2 \rightarrow R^2$ be linear, such that $S^2 = S$. Prove that $S= 0$, $S=id_{R^2}$ or there exists B a basis of $R^2$ such that $[S]^B_B=\begin{...
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True or False: There is a $6\times 6$ matrix $A$ with $\text{Rank}(A)=4$ and $A^3 =0$

I understand how to do it if the question changed $A^3$ to $A^2$, because then you can just use the rank–nullity theorem. $\text{Rank}+\text{Nullity}=6$, $\text{Rank}=4$ so $\text{Nullity}=2$ so of ...
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A isomorphism between $\mathcal{L}(U,V)$ and $\mathcal{L}(U) \times \mathcal{L}(V)$.

Let $U$, $V$ be two Banach spaces and define the spaces $$ \mathcal{L}(U\times V) = \{T : U\times V \rightarrow U\times V : T \text{ is linear and bounded}\}, $$ $$ \mathcal{L}(U) = \{T : U \...
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2 votes
1 answer
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Prove whether the following transformation is linear or not.

I'm not sure if I'm going wrong with the proof, but I'm getting my answer as, the transformation is linear, whereas according to our teacher, it is not supposed to be linear. Q. $T(v_1, v_2) = (v_1, ...
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0 votes
1 answer
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Kernel and image of a nilpotent linear map

My question is as follows. Let $T : V \to V$ be a linear map of a finite dimensional vector space V. If $ker T = Im T$, then is $dim V$ even and $T^{2} = 0$? I believe this to be false since I have ...
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