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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Can you give me a hint for the following?

I am a first year uni student learning about linear transformations I encountered the following question if $$T^2=T$$ such that T is from V to V proof or disproof that T is an Injective function , ...
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How to understand the differential is a linear map?

For a smooth function $f:\mathcal{E} \rightarrow R$, where $\mathcal{E}$ is a linear space. $Df(x): \mathcal{E}\rightarrow R$ is the differential of $f$ at $x$, that is, it is the linear map defined ...
Chuan Huang's user avatar
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Symmetric group representation $S_{3}$ to $\mathbb{C}^{3}$

Trying to find answers for this year french intern agregation (Algebra exam), i found this question about a proof that a particular group representation of $\mathcal{S}_{3}$ on vector space $\mathbb{C}...
Armand Jourdain's user avatar
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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*} \...
PAT's user avatar
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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 ...
Chuan Huang's user avatar
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Proof of $x_i x_j + y_i y_j + z_i z_j = \delta_{ij}$ [closed]

Here $x$, $y$ and $z$ 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 ...
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Where do these coefficients come from when computing this linear map from its action on a basis? [closed]

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 \...
thinking's user avatar
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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 ...
Grigor Hakobyan's user avatar
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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 ...
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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-...
fiends's user avatar
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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 <...
Kevin K's user avatar
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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 ...
juan19.99's user avatar
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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 ...
Tom's user avatar
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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 ...
Parham's user avatar
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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))...
Nick Johnson 's user avatar
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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_{...
Lucas Giraldi's user avatar
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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 ...
Alessandro Tassoni's user avatar
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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 ...
Nick Johnson 's user avatar
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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 ...
Stephen Makanga's user avatar
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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 ...
Eslam Aboelfadl's user avatar
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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 $$[S]^\alpha_\beta=\begin{bmatrix}2&0\\4&0\\0&-4\end{bmatrix}$$ ...
Janderson's user avatar
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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 ...
Turquoise Tilt's user avatar
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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 $\...
Akash Arjun's user avatar
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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 ...
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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: &...
Satish's user avatar
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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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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)...
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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 $T_2$...
Matthew S's user avatar
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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 ...
some_math_guy's user avatar
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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{...
Tomás's user avatar
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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 ...
mathreadler's user avatar
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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 $h(x)=A\cdot x$. ...
佐武五郎's user avatar
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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\...
Evangeline A. K. McDowell's user avatar
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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 ...
Daniel Marques's user avatar
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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, $...
Doofenshmert's user avatar
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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 ...
Newbie1000's user avatar
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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 ...
Paulo Estêvão's user avatar
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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 $B$...
clay's user avatar
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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 ...
maxical's user avatar
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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 $\...
isaac's user avatar
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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 ...
Francisco Sierra's user avatar
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Find the change of basis matrix so that the following is in Jordan Normal Form

Let the following matrix be given. Note that we are in the field consisting of five integers: F = (0, 1, 2, 3, 4) $A = \begin{bmatrix} 1 & 2 & 0\\ 3 & 2 & 1\\ 0 & 2 & 2 \end{...
Newbie1000's user avatar
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Transition matrices in $\mathbb{R}^n$: how to compute

I was reading "Elementary Linear Algebra Applications" by Howard Anton, and in section 4.6, Change of Basis, it talks about finding the transition matrix, $P$, from an old basis $B$ to to ...
SupersonicMan12's user avatar
1 vote
1 answer
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Continuous Linear Transformation Maps Open Set to Bounded Set

I want to prove that if a linear transformation maps some nonvoid open set of the domain space to a bounded set in the range space, then it is continuous. I tried to prove by contradiction that ...
Analysis Rookie's user avatar
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3 answers
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Symmetric property of symmetric matrices [closed]

Are symmetric matrices symmetric about a particular axis geometrically ? Or is the property defined only for the matrix representation of the linear transformation? If you consider a unit square, ...
Jay's user avatar
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Linear Algebra Doubt regarding matrix of transformation wrt basis B and C

Let $T$ be a linear transformation, T$:V\to W$, where $V$ and $W$ are vector spaces over a field $F$. Then the matrix of $T$ wrt the basis $B$ (of $V$) and $C$ (of $W$) is? Is it $C^{-1}T(B)$? This is ...
Shash Wemwal's user avatar
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Basis of $Hom(V,W)$ [duplicate]

Question: Let $V,W$ be vector spaces over $F$. Let $v_{1},...,v_{n}$ be a basis of $V$ and $w_{1},...,w_{n}$ be a basis of $W$. Find a related basis of $Hom(V,W)$ and give its dimension. I'm not ...
pseudobulbose's user avatar
2 votes
1 answer
104 views

Prove that $\Vert A \Vert \le \dfrac{1}{1-2\varepsilon} \sup_{x \in \mathcal{N},\ y\in \mathcal{M}} \langle Ax,y\rangle$

Problem. Let $A = (a_{ij})$, $1\le i \le m$, $1\le j \le n$ and $\varepsilon \in (0,1/2) $. Let $\mathcal{N}$ be an $\varepsilon$-net of $S^{n-1}$ and $\mathcal{M}$ be an $\varepsilon$-net of $S^{m-1}$...
Tung Nguyen's user avatar
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-1 votes
2 answers
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Having trouble finding an exact sequence. [closed]

Find an exact sequence: $\{0\} \rightarrow \mathbb{V} \xrightarrow{{\alpha}} \mathbb{U} \xrightarrow{{\phi}} W \rightarrow \{0\} $, such that $$\dim(\mathbb{V}) = \dim(\mathbb{U}) = \infty$$ $$\dim(\...
Avgustine's user avatar
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Let $T: F^2 \rightarrow F^2$ be the linear transformation defined by $T(a_1, a_2)=(a_1 + a_2, a_1)$.

So we have the question in title, and we have to prove that $T$ is onto and one-one (bijective). So first I try to prove injectivity by proving $T(0)=0$ or proving $T(x)=0\rightarrow x=0$: $$T(0,0)=(0+...
Kshitij Kumar's user avatar

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