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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Is this an invertible linear map?

$\DeclareMathOperator{\R}{\mathbb R} T:\R^2\rightarrow \R^3$ $T(x)= \begin{bmatrix}1&0\\0&1\\0 &0\end{bmatrix}x, \forall x\in \R^2$ I know that a linear map is invertible if and only if ...
1 vote
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Proving $f(V)=\mathrm{span}(f(\mathbf{v}_{1}),\dots, f(\mathbf{v}_{n})).$

I'm using the following definitions: Definition 1 (Linear Span) Let $(V,+,\cdot)$ be a vector space over a field $\mathbb{K}$. The linear span of a subset $X\subseteq V$ is defined as \begin{align*} \...
-3 votes
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Proof of $x_i x_j + y_i y_j + z_i z_j = \delta_{ij}$

Here $x_i$, $y_i$ and $z_i$ are arbitrary orthogonal unit vectors in three dimensions and $\delta_{ij}$ is the unit diagonal tensor. How to prove this identity when $x_i$, $y_i$ and $z_i$ are not the ...
1 vote
1 answer
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Estimate yaw given pitch, roll, and change in pitch and roll after rigid transformation

Given a gravity vector $g$ in a 3D coordinate frame $F$ we can find pitch $p$ and roll $r$ (Euler angles) of $F$ relative to $g$. Assume we apply a rigid transformation to $F$, sense a new gravity ...
-8 votes
1 answer
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Where do these coefficients come from when computing this linear map from its action on a basis?

Question: Find a linear transformation $T:M_2(\Bbb{R}) \to \Bbb{R}^3$ such that \begin{align*} T\left( \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \right) &= \begin{bmatrix} 1 \\ 2 \\ 0 \...
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Good books/resources for learning how to visualize Mobius transformations

Posts like this one have always baffled me. I have taken a few complex variables courses in my time and it seems that every time I take a course like this, either the textbook isn't great at ...
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A bilinear operator is continuous iff verifying $|| \phi (v;w)|| \leq M ||v|| ||w|| $

First I know that this question has all ready be asked for exemple here but for bilinear operator with only one variable here I want to show it for linear operator with two variables. Question: Prove ...
0 votes
2 answers
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Non-surjective linear operator that is open?

Is there any non-surjective linear operator that is open? By the open mapping theorem, $T:X \to Y$ is open iff $T$ is surjective. Is it possible then to have a non-surjective linear operator that is ...
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1 answer
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Prove If dim($V$) is even, Then there exists a linear transformation $T: V \rightarrow V$ such that Ker($T$) = Image($T$)

Prove If dim($V$) is even, then there exists a linear transformation $T: V \rightarrow V$ such that Ker($T$) = Image($T$) I'm having trouble trying to prove this statement. I've tried to use the Rank-...
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How to reverse the transformation of an affine transformation

Beginner question here... I'm a little unclear on how to go about reversing a transformation of a matrix from the format of Ax=b (or if it's possible for that matter). I started out with a 6x6 matrix <...
2 votes
1 answer
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How can i get animation of matrix transformations like this

I found these transformations in Khan Academy and I don't know how they created the transformation example videos! I want to create transformations in this way. https://www.khanacademy.org/math/...
2 votes
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Isomorphism between two general linear group.

If $V$ is a vector space over the field $F$, the general linear group of $V$, written $GL(V)$, is the set of all bijective linear transformations $V\to V$, together with functional composition as ...
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Ratio Scale transformation

I am in a pickle, and I would genuinely appreciate it if you could guide me. I spent 2 years to find a way to develop a ratio scale, and I did it; however, it cannot be used the way it is. The scale ...
0 votes
1 answer
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Dual basis problem on the space of polynomials $\mathbb{R}_{1}[x]$

Problem goes as follows: Let $\mathbb{R}_{1}[x]$ be the linear space of polynomials of degree $\leq 1$. Define the covectors $f_{1}, f_{2} \in (\mathbb{R}_{1}[x])^{*}$ (dual space), as: $$ f_{1}(p(x))...
3 votes
1 answer
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"Coordinate-free" minors, submatrices and cofactors.

Let $M$ be an $n \times n$ matrix, then the $(i,j)$-submatrix of $M$ is the $n-1 \times n-1$ matriz $M_{ij}$ given by removing the $i$-th line and the $j$-th column from $M$. The $(i,j)$-minor is $m_{...
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On the mathematics behind the Dyson Series

I've come across the Dyson Series solution of the Schrödinger Equation arising in the interaction picture when dealing with a time dependent Hamiltonian. Since then I've been looking for a rigorous ...
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Determine coordinates of functional in the dual basis of a given basis of $\mathbb{R}_{2}[x]$

The problem goes as follows. Consider the vector space of polynomials of degree $\leq$ 2, $\mathbb{R}_{2}[x]$ and the functional $ \phi(p(x)) = p(1) + p(-1) $. Determine the coordinates of $\phi$ in ...
2 votes
1 answer
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If the base function for the transformed function $f(x)= -4(3x)^2+5$ is $f(x)=x^2$, then: is $k=3$ or $k=9$?

If the base function for the transformed function $$f(x)= -4(3x)^2+5$$ is $f(x)=x^2$, then: is $k=3$ or is $k=9$? By comparing the transformed function to: $af(k(x-d))+c$, you can pinpoint the factors ...
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1 answer
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A mapping $T:\mathbb{R^n}\rightarrow \mathbb{R^m}$ is one-to-one if each vector in $\mathbb{R^n}$ maps onto a unique vector in $\mathbb{R^m}$.

I'm reading Linear Algebra and Its Applications by David C. Lay, Steven R. Lay and Judi J. McDonald. In section 1.9 exercise 24 there's a following statement d) A mapping $T:\mathbb{R^n}\rightarrow \...
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Find $Q'_j$ from $R_j=R'_j Q'_j$ where $R_j$ has zero determinant.

I'm writing my physics bachelor on the Raman scattering effect in solids. I'm trying to evaluate the scattering intensity response to varying polarization angle. This is the well known linear ...
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1 answer
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Linear operator - Exercise

Consider a linear operator as defined below $$S_N[u(x)] : = \int_\Bbb R \chi_{[-N,N]}(\xi)\hat u(\xi)e^{ix\xi}d\xi, \ u \in L^1(\Bbb R)$$ Prove that $S_n:L^1(\Bbb R) \to C^0_b(\Bbb R)$ is well ...
1 vote
1 answer
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Linear Transformation of a vector from $\Bbb R^2$ to $\Bbb R^3$

Consider $a = \{(0, 2), (2, -1)\}$ and $b = \{(1, 1, 0), (0, 0, -1), (1, 0, 1)\}$ Basis of $\Bbb R^2$ and $\Bbb R^3$. Let $$[S]^\alpha_\beta=\begin{bmatrix}2&0\\4&0\\0&-4\end{bmatrix}$$ ...
1 vote
1 answer
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Proving linear transformation is one to one if vectors in vector space are linearly independent

I have a linear transformation $F: U \to V$. Given that the vectors $\vec{v_1}...\vec{v_n} \in V$ are linearly independent I want to show that $F$ is one-to-one. Also $F(\vec{u_i}) = \vec{v_i}$ and $\...
0 votes
2 answers
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ABC is equilateral iff the midpoints of the sides and of the segments from vertix to centroid all lie on a circumference

I have come across a geometry problem that states: Let $ABC$ be an acute triangle. Let $AM$, $BN$ and $CL$ be the medians, which intersect in the centroid $G$. Let $M'$, $N'$ and $L'$ be the ...
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1 answer
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How to prove that all solutions to $Ax=b$ are $(x_0 + bv_1 +cv_2)$, where $A$ is a linear transformation from $\mathbb{R}^{N}$ to $\mathbb{R}^{N}$?

The linear transformation $T:\mathbb{R}^N \to \mathbb{R}^N$ and is represented by the matrix $A$. A basis for the null space of $A$ consists of the vectors $v_1$ and $v_2$. Prove that if a particular ...
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Properties of a martrix which is a linear transformation on a subspace of $\mathbb{R^3}$

This is a multiple choice question which had four options and in the answer booklet it gives out of many others one particular answer regarding which I have some doubts. The question is as follows: &...
1 vote
1 answer
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Describe explicitly the linear transformation T from $F^2$ to $F^2$ such that $T\epsilon_1=(a,b),T\epsilon_2=(c,d)$

I have to describe explicitly the linear transformation T from $F^2$ to $F^2$ such that $T\epsilon_1=(a,b),T\epsilon_2=(c,d)$ My try: We know that $T\epsilon_1=(a,b),T\epsilon_2=(c,d)$ so let $\...
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2 answers
2k views

Question about change of basis and reflection about the line $y=2x$

In Friedberg et al. Linear Algebra there is an example that asks to find the matrix of the linear operator in the standard basis which reflects vectors of $\mathbb{R}^2$ across the line $y=2x$. ...
1 vote
2 answers
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$\{x \in V : Tx = c\} \neq \emptyset$ if and only if $\{x \in V : Tx = c\} = v + \text{null} \ T$

Exercise. Suppose $T \in \mathcal{L}(V,W)$ and $c \in W$. Prove that $\{x \in V : Tx = c\}$ is either the empty set or is a translate of $\text{null} \ T$. Source. Linear Algebra Done Right, Sheldon ...
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Does there exist a hadamard style transform over bits of digital numbers? Perhaps possible to be interpreted as $\mathbb Z^2$?

The Hadamard transform is well known in the information theory and signal processing communities and can be viewed in some sense as a discrete step version of the Discrete Fourier Transform. There ...
3 votes
1 answer
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Prove a set is linearly independent.

$\phi = \mathbb{V} \rightarrow \mathbb{V}$ is an operator satisfying $\phi^n = 0$ for some $n$ and $\phi^{n-1} \ne 0$ Let $v \in \mathbb{V}$ be a vector s.t. $\phi^{n-1} \ne 0$. Is the set {v, $\phi(v)...
8 votes
2 answers
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Subspaces with common images

Let $X$ and $Y$ be finite dimensional vector spaces over $\mathbb{C}$, and let $S,T:X\to Y$ be linear transformations. Is there a method for determining all subspaces $V\subseteq X$ such that $S(V)=T(...
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Given two linear transformations T1 and T2, show that range T1 = range T2 if and only if there is an invertible operator S such that T1=T2S.

I am working through Axler's Linear Algebra Done Right section 3D problem 5 which states "Suppose $V$ is finite-dimensional and $T_1, T_2 \in \mathcal{L}(V,W)$. Prove that range $T_1=$ range $T_2$...
4 votes
2 answers
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Munkres-Analysis on Manifolds: Theorem 20.1

I am studying Analysis on Manifolds by Munkres. I have a problem with a proof in section 20: It states that: Let $A$ be an $n$ by $n$ matrix. Let $h:\mathbb{R}^n\to \mathbb{R}^n$ be the linear ...
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1 answer
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trouble with a change of basis

I have two O.N sets $\{|e_i\rangle\}_{i=1}^r$ and $\{|e_i\rangle\}_{i=1}^r$ Then there is gotta be a change of basis matrix C, such that $|\tilde e_i\rangle = \sum_{j=1}^rc_{j,i}|e_j\rangle$ I was ...
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1 answer
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Linear transformation matrix given two vectors

I'm having trouble with this problem. Given two vectors $ \overrightarrow{v} = \left( \begin{matrix} v_1 \\ v_2 \end{matrix} \right) $ $ \overrightarrow{v'} = \left( \begin{matrix} v'_1 \\ v'_2 \end{...
5 votes
4 answers
611 views

What is the image of $x^{\rm T}Qx\le 1$ under a linear map $x \mapsto Cx$?

Let $Q$ be a real symmetric positive semidefinite $n \times n$ matrix. Consider a set $$ \Big\{ x \in \mathbb{R}^n \;\Big| \; x^{\rm T}Qx\le 1\Big\}, $$ which can be loosely described as an "...
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I proved any linear subspace of $\mathbb{R}^n$ is closed in $\mathbb{R}^n$ to prove $\overline{T}$ has measure zero. (Munkres "Analysis on Manifolds")

I am reading "Analysis on Manifolds" by James R. Munkres. Theorem 20.1. Let $A$ be an $n$ by $n$ matrix. Let $h:\mathbb{R}^n\to\mathbb{R}^n$ be the linear transformation $h(x)=A\cdot x$. ...
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Prove $\ker(A+T(A))\subseteq \ker(A)$ for $A\geq 0$ and $T$ positive linear map

Let $A\geq 0$ be a positive semi-definite complex matrix in $M_d(\mathbb{C})$. Let $T:M_d(\mathbb{C})\to M_d(\mathbb{C})$ be a positive linear map between $d\times d$ complex matrices, i.e., $A\geq 0\...
0 votes
1 answer
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Change of Basis in Game Development

I recently went back to re-learning some bits of linear algebra: change of basis. And, as an exercise, I decided to revisit a video game mechanic: portals. As this youtuber puts it, the math behind it ...
4 votes
2 answers
193 views

Find the symmetric matrix which maps one vector to another

I am searching a way to calculate the matrix $A\in \mathbb{R}^{3\times 3}$ for given two vectors $x,y \in \mathbb{R}^3$ such that $Ax = y$. I think normally I would need 3 pairs $(x_i, y_i)$ with $i =...
1 vote
1 answer
49 views

Injectivity of linear combinations of linearly independent invertible operators

Let $\mathbf{H}$ be an infinite dimensional, separable Hilbert space. Moreover, let $\{A_i\}_{i=1}^N$ be a set of linearly independent, invertible operators that act on $\mathbf{H}$. In other words, $...
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Theoretical question about finding the basis of an image.

Let's say that you calculate the basis of the kernel and it spans one vector. Then let's assume the rank of the matrix is two. Normally, you would choose the columns corresponding to the pivot points ...
0 votes
1 answer
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Is there a name for this vector space?

I've seen in some books of Analysis the notation $\mathcal{L}(\mathcal{L}(V), V) \cong \mathcal{L}_2(V)$ and I don't find anything about this vector space. What is it? Is there a name for it? Also, if ...
1 vote
0 answers
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Checking a proof about the structure of the matrix of a linear transformation

Artin's "Algebra" book (1st ed, page 114, Proposition 2.9b) claims that given any $m\times n$ matrix A, there're matrices $Q \in GL_m(F)$ and $P \in GL_n(F)$ so that $QAP^{-1}$ has the form: ...
1 vote
1 answer
226 views

What is the analogue of the cross ratio in higher dimensions and what role does it play in n-dimensional geometry?

I know the cross ratio is defined for four real collinear points and for four points in the complex plane. This is an important projective invariant for linear transformations. Is there an analog for ...
0 votes
1 answer
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For symmetric, non-singular, positive definite matrix $B$ and some unit vector $u$ show $u^T Bu \ge \frac{1}{ {\lVert B^{-1} \rVert} }$

For symmetric non-singular positive definite matrix $B$, and any unit vector ${\lVert u \rVert} = 1$, show that: \begin{gather*} u^T Bu \ge \frac{1}{ {\lVert B^{-1} \rVert} } \end{gather*} Since $B$...
0 votes
0 answers
25 views

Linear transformation of multivariate normal

A well known fact exists which is that if a multivariate normal distribution undergoes a linear transformation it's also multivariate normal. There are two proofs I have seen, If the transformation is ...
4 votes
1 answer
1k views

How to reduce the number of (overlapping) constraints in a linear program?

I am trying to solve a linear program with more than $7$ million constraints in around $5000$ variables. It could not be solved on my computer. In the constraints, there is overlap, e.g., $$ \begin{...
1 vote
1 answer
27 views

Right inverse of evaluation map from polynomial vector space

Say $E_1 : \mathbb{P}_3 \rightarrow \mathbb{R}$ by $f(x) \mapsto f(1)$. Is it sufficient to say there exists right inverse $S_R : \mathbb{R} \rightarrow \mathbb{P}_3 $ from the following? We see $\...

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