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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://...

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Existence of $v$ such that $f_i(v) = \delta_{i j} $

Let $f_1,...,f_m \in V^*$ be a linearly independent family ,and F a field. Show that for each $1\leq j\leq m$ there is a $v\in V$ such that $ f_i (v)= \delta_{i j} $ for any $1\leq j\leq m$ I ...
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Prove $S=aT$ for some $a\in F$

Let $V$ be a vector space over a field $F$. Let $S,T\colon V\to F$ be linear transformations. Assume that $N(T)\subseteq N(S)$. Show that there exists $a\in F$ such that $S=aT$. $N(T)$ and $N(S)$ are ...
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What is the structure of M2(R)/W. Where W={g(x)/(x^2+1).f(x), f(x) ∈ R[x]} [on hold]

I know that R[x]/W is a copy of R2. How to show this
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26 views

Differentiation Operator is not a bounded operator for polynomials

If you consider the space of all polynomials on [0,1] (defined as $P_{[0,1]}$ as subspace of $C_{[0,1]}$) then the differentiation operator is not a linear bounded operator on this space. Why is that? ...
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If $L$ is a linear operator, then $\text{rank}(L^k) = \text{rank}(L^{k+1})$ for some $k$ and $V = \text{ker}(L^k) \oplus \text{ran}(L^k)$.

I am trying to prove the following theorem: If $L:V \to V$ is a linear operator and $V$ is a finite dimensional vector space, then $\text{rank}(L^k) = \text{rank}(L^{k+1})$ for some $k$ and $V = \...
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What does the theorem of Linear maps and basis of domain actually mean?

"Suppose $v_1...v_n$ is a basis for $V$ and $w_1...w_n \in W$. Then there exists a unique linear map $T: V \implies W$ such that: $T(v_j) = w_j$ for each $j = 1...n$." I understand that because $v_1.....
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Column Space after Left-Multiplication by an Invertible Matrix

Suppose that $A$ is an $n\times n$ positive definite matrix and that $B$ is a full-rank $n\times p$ matrix, with $p<n$. Is it the case that the column space of $A^{-1}B$ coincides with the column ...
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Fourier analysis, complex conjugate and orthogonality

To obtain the magnitude of a particular frequency contained in a periodic signal, you take the inner product of the signal’s function with the (analyzing) basis function corresponding to such ...
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How to write matrix form for an eigenvalue problem

I have an eigenvalue problem to solve; for the coupled linear equations $\lambda A_n=-\alpha_nA_n+A_{n+1}+A_{n-1}-\beta_nB_n$ $\lambda B_n=+\alpha_nB_n-B_{n+1}-B_{n-1}+\beta^\ast_nA_n$ I want to ...
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Given a field F, let V=Mn×n(F) be the vector space of square matrices, and W⊆V the subspace of symmetric matrices A of trace zero [on hold]

Given a field F, let V=Mn×n(F) be the vector space of square matrices, and W⊆V the subspace of symmetric matrices A of trace zero. That is, the entries of A satisfy Aij=Aji for all i,j, and A11+...
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$u,v,w$ are distinct vectors show the following is also a basis [on hold]

Let $u,v,w$ be distinct vectors of vector space over $\mathbb C$, such that $\{u,v,w\}$ is a basis of $V$. Show that $\{u−(1+i)v, u+v+w, −2iu\}$ is also a basis.
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How is transformation of function space to a vector field space gradient?

From S.L Linear Algebra book: Let $U$ be an open subset of $R^3$ , and let $V$ be the vector space of differentiable functions on $U$. Let $V'$ be the vector space of vector fields on $U$. Then ...
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A simple modular transformation of an integral

I want to perform modular $S$ and $T$ transformations for a period integral. The $S$ transformation takes $\sigma \to -1/\sigma$ and the $T$ transformation takes $\sigma \to \sigma + 1$. Here, $\sigma ...
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2D line rotation?

I am trying to solve a sample question. I think the answer for this is "a", but I am not sure. I wonder if my answer is correct, what is correct answer? and I need explaination.
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Scalars on a one dimensional field with inner product and non orthogonal basis.

Let $V$ be a linear vector space with inner product, if $T$ is a operator on $V$, how can i prove that $Dim(Im (T))=1$ iff exists $u, w \neq 0$ in $V$, such that $T(v)=<v,u>w$ for all $v \in V$. ...
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How by using rotation matrix to relate polar $\frac{\partial}{\partial \rho}$ , $\frac{\partial}{\partial \phi}$ to Cartesian partial derivatives

How by using rotation matrix to relate the $\frac{\partial}{\partial \rho}$ and $\frac{\partial}{\partial \phi}$ to Cartesian partial derivatives? We do not want to use chain rule. The rotation ...
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35 views

Measuring the angle between skew coordinate axes using the skew coordinates

My application is that I have a cheap Chinese CNC machine where the X and Y axes are not quite orthogonal. They are close, but not quite there, and I'd like to measure the angle between the axes so as ...
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Linear Transformations and Matrices $T: R^2 \rightarrow R^3$

This is not really a homework but I am preparing for a midterm and I am stuck on a problem from the book. I have the solution but I can't understand it. I have to computer $[T]_\beta^\gamma$. ...
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$L(V)$ isomorphism $L(V*)$.

Let be $V$ a vector space with finite dimension over F. Show that $\mathscr{L}(V^{*})\cong \mathscr{L}(V)$. My attemp was try to show that $\mathscr{L}(V^{*})$ is isomorphic with the space $\matrix{M^...
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How can I prove that a linear map from a 1-dimensional space to itself is really just multiplication by some scalar?

Show that every linear map from a 1-dimensional vector space to itself is multiplication by some scalar. More precisely, prove that if the dimension of $V$ is equal to 1, and $T \in L(V,V)$, then ...
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Linear independence of finite dimentional vector space for a linear transform raised to a power

Let $V$ be a finite-dimensional vector space over $F, T∈L(V)$. Suppose 0 $\neq v∈V$ and m is a positive integer for which $T^{m-1} v \neq $ 0 but $T^{m} v= 0$. Show that S={$v, T v, . . . , ...
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Find invariant points under matrix transformation (degeneracy)

I have the matrix $$Q=\begin{bmatrix}-1&2\\0&1\\\end{bmatrix}$$ and want to find the invariant points. To do this, I solve the equation: $$\begin{bmatrix}-1&2\\0&1\\\end{...
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$A\in M_{m\times n}(k)$ is one one iff row rank is $n$

$A\in M_{m\times n}(k)$ is one one iff row rank is $n$. Here $k$ is a field. My Attempt 1): $A$ is $1-1$ $\iff$ker $A$ = $\{0_{m\times1}\}$ $\iff$All cols are linearly independant $\iff$Col Rank is $...
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Does it make sense to write $(T - \lambda I)v$ if $T$ is a linear transformation?

I am reading this page to try to understand Jordan form. $V$ is a finite-dimensional complex vector space, and until now $T$ has always represented an "operator", by which I guess they mean a linear ...
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2answers
26 views

Idempotent linear transformation from $V$ to $V$ is the direct sum of $\operatorname{range}(T)$ and $\operatorname{null}(T)$

The problem statement is: If $T\in\mathcal{L}(V)$ and $T^2=T$ (idempotent), prove that $V=\operatorname{range}(T)\oplus \operatorname{null}(T)$. I'm not exactly sure where to start with this ...
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About eigenvalues, matrix equation with non invertible matrices

Let A,B be two non-invertible square matrices of the same size. Would there be a general procedure for solving the equation $$ Ax = Bkx $$ for both $k$ and $x$? ($k$ is a scalar). For example, if ...
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Matrix Representation of Linear Transformation from R2x2 to R3

We have a linear transformation T : $\mathbb R^{2\times2} \rightarrow \mathbb R^{3}$ defined by $$T\left(\begin{bmatrix}a&b\\c&d\end{bmatrix}\right)=(a+b,2d,b).$$ Let A and B be the ordered ...
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given a system of N linear equations, Is there an algorithm that can find a solution that solves the most number of equations in this system

My apologies if this question makes no sense; I am trying to find an algorithm that can solve a linear system of equations. Unlike most problems like this- for this particular case, this algorithm ...
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1answer
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One-to-One and Nullspace

My linear algebra textbook by Friedberg states the following Theorem: Theorem: Let $V$ and $W$ be vector spaces, and let $T: V → W$ be linear. Then $T$ is one-to-one if and only if $N(T) =$ {$0$}....
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Calculating Null T and Range T for the linear transformation $T(x,y,z)=(x+2y-z,y+z,x+y-2z)$?

I have a doubt regarding the calculation of range of a linear transformation. I will explain my doubt with an example. Suppose, $T:R^3 \to R^3 \ni$ $T(x,y,z)=(x+2y-z,y+z,x+y-2z)$ is a Linear ...
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1answer
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Map a Conic to the Normalized Image Plane

I have a problem with understanding a specific operation related to the mapping of an ellipse, captured by a camera, to a plane: According to this paper, given the calibration matrix $K$ of a camera, ...
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1answer
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Problem interpreting linear mapping

I have been practicing some basic linear algebra and came across this excercise: Compute the kernel of the mapping $f\colon \mathbb R^{2}\rightarrow \mathbb R, x \mapsto (2,-1)'x$, and draw a ...
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1answer
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Prove $A$ is a linear continuous operator, if $A(x+y) = Ax + Ay$. [closed]

Assume $A$ is a map from real normed space $E$ to $F$ such that $A(x+y) = Ax + Ay \ \forall \ x, y \in E$ and $A$ is bounded on unit ball $B(0,1) \subset E$. Prove $A$ is a linear continuous operator. ...
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Basis of the Dual Space of Polynomial Spaces

I have the following problem: Let $V=\mathcal{P}_n(\mathbb{R})$ be the vector space of polynomials of degree $\leq n$. Define $\alpha_k : V\to\mathbb{R}$ by $$ \alpha_k(p)=\int_{-1}^{1}t^kp(t)dt,\...
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Show that $L(O)=O$ if $L$ is a linear map from one vector space to another

From S.L linear algebra: Let $L:V \rightarrow W$ be a linear map from one vector space to another. Then show that $L(O)=O$. ($O$ is a null vector). There are two axioms for a linear map: $\...
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When and when isn't it permissible for matrix multiplication to a replacement for a linear transformation?

I've been told that what a matrix of $T$ actually represents$^*$ can be a bit complicated, and that, when a mapping is from $F^m \to F^n$, the matrix of $T$, when multiplied by an arbitrary vector in ...
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Relation of $f : V \rightarrow V$ and $Id_v$

(There's a slight typo in the question it should say $\Phi : \mathbb{F^n} \rightarrow V$ ) So in this question I can understand the matrix A, which just follows a commutative diagram. But I don't ...
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Show that $\forall x\in \Bbb{R}$ and for a fixed $a\in \Bbb{R},\;t_a= a+x$ is invertible

Let $a\in\Bbb{R}$ be fixed. Define \begin{align} t_a:\,&\Bbb{R}\to \Bbb{R}\\ &x\mapsto a+x\end{align} I want to show that $t_a$ is invertible. Below is my trial MY WORK Suppose $t_a v=0$, ...
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Bijective Linear Map

Hi, I'm studying the following proposition from Josept Muscat's Functional Analysis textbook: If $T:X \rightarrow Y$ is a bijective linear map, then $T^{-1}$ is linear, and is continuous when $c\|x\|...
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Matrix operator, basis in a Hilbert space.

a) Let $(\varphi_{\alpha})_{\alpha=1}^{N}$ and $(\psi_{\beta})_{\beta=1}^{M}$ be two orthonormal bases in a Hilbert space, $H$. Define an $N\times M$ matrix , $T_{\alpha \beta}=<\varphi_{\alpha},\...
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Non-invertible operators and function composition

Let $V$ be a finite dimension vector space on a field $F$, and $T\neq 0$ a non-invertible linear operator on $V$. Show that there exist $G$ and $H$ nonzero linear operators in $V$ such what $T\circ G =...
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1answer
19 views

Show that $\Omega(0)=I$

For each $r\geq0$, let $\Omega(r):C((-\infty,0])\rightarrow C((-\infty,0])$ be a map defined by $\Omega(r)(x(t))=x(t-r)$ for $t\leq0$. How can I prove that $\Omega (0)=I\ $? My attempt: If $r=0 \...
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$ax+by+c=0$ in computer graphics

I'm doing a course in computer graphics at the university. We are talked about this Lineare Equation: $ax+by+c=0$ for $2D$ and of course the one for $3D$. I don't really get it for what this ...
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Calculating the translation of an affine matrix so that it centres during scaling

I have this example of what I want to achieve. The second set of rects is where I want to get to. There are two sets of rects in the link above. The first set is where I currently am, I want to ...
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Dot products between unit vectors in two different coordinate systems

I have two coordinate systems. The first (coordinate system 1) is at some arbitrary location and the second (coordinate system 2) is identical but for a translation along the positive z-direction of ...
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2answers
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How many basis vectors are there in an eigenspace of dimension k?

If $T:V\to V$is a linear map and we know that $\lambda$ is an eigenvalue of $T$ and the eigenspace of T wrt $\lambda$ has dimension $k$ then does that mean there are $k$ linearly independent ...
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35 views

How to understand the convention of Euler angle?

We have 3 DOF for rotation in 3D space. So to describe an arbitrary rotation, we need to describe its 3 DOF. Euler angle does this by dividing a rotation in 3 steps, first rotate along the Z axis of ...
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Nonlinear distortion that maintains unit length of vectors

I have a head-mounted eye tracker that supplies me with unit length vectors which give me the direction of the gaze, but without calibration their reference rotation is arbitrary. To align them to the ...