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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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1answer
30 views

geometric intuition behind an n-dimensional rotation matrix

How do I derive an n-dimensional rotation matrix from a geometric perspective? I have read on wikipedia that it preserves distance so that $Q^TQ = I$ but the explanation to be honest isn't very clear. ...
2
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1answer
26 views

Is this matrices exist, with this propertie

if $\begin{bmatrix} 1\\ 1\\ 0 \end{bmatrix}$, $\begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix}$ $\in R(A)$, and $\begin{bmatrix} 2\\ 5 \end{bmatrix}$, $\begin{...
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1answer
28 views

If differential operators are linear operators, what might it mean to act a differential operator to a function to its left?

Given a differential operator like the regular derivative, or grad or curl or div etc, it can act on a function to its right to yield a new function. Because it is linear, it is effectively like a an ...
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3answers
69 views

Find the matrix X such that $A\cdot X = X\cdot A^T$

I am working at some linear transformations and I need to know if it is possible given a square real matrix $A$, to find a real square invertible matrix $X$ such that $$ X^{-1} \cdot A \cdot X = A^T$$...
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1answer
18 views

Show that if $F(x)$ is a polynomial over $K$, then $F(A)=0$ if and only if $F(T_A)=0$.

Let $V$ be the space of $n\times n$ matrices over a field $K$. Let $A$ be an $n\times n$ matrix. Let $T_A: V\to V$ be the linear transformation given by $T_A(B)=AB$. Show that if $F(x)$ is a ...
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2answers
16 views

Skew coordinate system transformation matrix

I am currently working through problems in Susan Lea's Mathematics for Physicists. However I am stuck on a problem and the book is not known to errata. The problem is: A Skew (non orthogonal) ...
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1answer
26 views

Try to find some number if they exist, using eigenvalue

Are there any value $a,b,c,d,e,f \in R$ such that A=$\begin{bmatrix} 1& a& b \\ c& 1& d \\ e & f& 1 \end{bmatrix}$. and $A^{2}=2A$? Try to find that value but ...
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1answer
15 views

Linear operator, when it will be surjection and when it would not be?

Let $A\in M4$ and $p(\lambda)=\lambda^{3}(\lambda-1)$. Linear transformation $A:R^{4}\to R^{4}$ defined $A(x)=Ax$, for some matrices linear operator it is surjection, for some matrices is not. Now if ...
2
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1answer
33 views

Find the dual space and dual basis of basis $\beta$ in vector space $\mathbb{C^3}$

Find the dual basis $\beta^*$ of basis $\beta =\{v_1,v_2,v_3\}$in vector space $\mathbb C^3$ . where $v_1 = (1, 0, −1),v_2 = (1, 1, 1), v_3 = (2, 2, 0)$. I know that dual basis is the set of linearly ...
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1answer
19 views

Rational scalar multiplication in Linear transformation using addition property

Linear transformation: Map $T:V\to W$ is said to be Linear transformation when it satisfy property : 1) $L(u+v)=L(u)+L(v)\forall u,v \in V$ 2)$L(cv)=cL(v) \quad \forall c\in \mathbb{F}$ I know when ...
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0answers
20 views

Non-Orthogonal and Real Basis of Simultaneously Diagonalizable Real Symmetric Matrices

It is well known that for two real symmetric matrices $\left[A\right]$ and $\left[B\right]$, a basis $\left[P\right]$ can be found which simultaneously diagonalizes both of these matrices (under some ...
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0answers
8 views

Is a “reciprocal-triangular” matrix/linear transformation “interesting”?

Are linear transformations of the following form interesting enough to have a name & specific theorems? Form of the transformation I'm referring to: ...
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1answer
22 views

What is the difference between a *surjective* unitary operator and a *bijective* unitary transformations

The wikipedia page on unitary operators states they are surjective, whereas the page on unitary transformations states that they are bijective. What is the difference between them and when is it ...
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0answers
17 views

Geometrical interpretation of the trace? [duplicate]

A while back, I learnt about the geometrical interpretation of the determinant of a linear map, that given the exterior algebra of vector spaces $V$, and a linear map $\phi: V \to W$, then we can ...
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2answers
25 views

Writing a complex invertible matrix as the product of a real invertible matrix and a complex number. [on hold]

If $$A= \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right)$$ is a complex invertible matrix, how can I show that there are $\lambda \in \mathbb{C}$, and $\alpha, \beta, \gamma,...
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1answer
29 views

Significance of row reduction

I hadn't realized it until I read the most shocking line while studying linear algebra: "Row operations CAN change the column space of a Matrix." Basically, I have accustomed myself in viewing ...
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1answer
27 views

Confusion regarding transformation matrix

In Sakurai's Modern quantum mechanics it is said that the rotation matrix in three dimensions that changes one set of unit basis vectors $(x, y, z)$ into another set $(x' , y' , z' )$ can be written ...
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0answers
36 views

What is an improper affine space at infinity? Schouten: Tensor Analysis for Physicists

Edit to add: This clearly deals with projective geometry. If I figure it out well enough to post a satisfactory answer, I will do so. If someone else posts a decent answer before then, I will be ...
2
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1answer
52 views

Can you convert a non “normal” complex square matrix into a “normal” one?

I've read the definition that "a complex square matrix ${\bf A}$ is normal if it commutes with its conjugate transpose". I've also read that "A is a normal matrix iff there exists a unitary matrix U ...
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0answers
16 views

RREF and determinant

Suppose $A=\begin{pmatrix}a&b&c\\d&e&f\\g&h&i \end{pmatrix}$ and $\det(A) = 1$. What does that tell us about the $\text{RREF}$ of the matrix $$B=\begin{pmatrix}a&b&c&...
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2answers
24 views

Linear Transformation of a Polynomial

I have an operation that takes $ax^2+bx+c$ to $cx^2+bx+a$. I need to find if this corresponds to a linear transformation from $R^3$ to $R^3$, and if so, its matrix. I know that $$ ax^2+bx+c = \begin{...
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1answer
37 views

Surjectivity and Orthogonal Complements with Quadratic Forms

My question is asking to clarify the final part of a proof proposed in the following link: Guarantees of a Quadratic Form for Two Related Matrices The question being: Suppose I have a matrix: $\...
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2answers
20 views

Given $T(v)=v-a\left<v,w_1\right>w_1-a\left<v,w_2\right>w_2$ where $a\in \mathbb R$ and $||w_1||=||w_2||=1, w_1\perp w_2$ Show that $T=T^*$

Given $T(v)=v-a\left<v,w_1\right>w_1-a\left<v,w_2\right>w_2$ where $a\in \mathbb R$ and $||w_1||=||w_2||=1, w_1\perp w_2$. Show that $T$ is self adjoint. I Think that completing $w_1,w_2$ ...
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1answer
29 views

What are all the angle-preserving $\mathbb{R}$-linear maps $T:\mathbb{C}\rightarrow\mathbb{C}$?

I know that linear maps of the form $T(z)=\alpha z,z\in\mathbb{C}$ preserve angles between two complex numbers. However, I know that every $\mathbb{R}$-linear map can be uniquely written as $T(z)=\...
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2answers
48 views

Prove that : There exists a vector $x$ such that $Mx = x$ , where $M$ is a Markov matrix [on hold]

Here's a proof that I found which looks pretty simple but I can't understand the last step. (A Markov matrix is a square matrix whose columns sum to one; $I$ is an identity matrix; $M^T$ and $I^T$ ...
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0answers
23 views

Back substitution and QR Factorisation via Householder

I'm having difficulties getting the same beta values from this OLS regression as I would without the QR decomposition. I believe it has to do with the shape of my Q block. I start as so: A =\begin{...
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1answer
23 views

Does one-to-one transformation means its matrix is regular?

Suppose that $T(v):=Av$ is one-to-one from $F^n$ to $F^n$ (A is a matrix) That is not surjective. Does that mean that $A$ is regular? On the one hand: $T$ is One-To-One $\implies\ker(T)=\{0\} \...
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2answers
48 views

Are linear subspaces of euclidian spaces closed [on hold]

Let $\mathcal{S}$ be a linear subspace of the Euclidean space $\mathbb{R}^N$. Is $\mathcal{S}$ necessarily closed?
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1answer
31 views

Finding the Jordan Form of a transformation defined by $T(X)=AX$ when $A,X \in M_{4\times 4}(\mathbb C)$

Given $A=\left(\matrix{0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&-1}\right)$ and define $T(X)=AX$. when $A,X \in M_{4\times 4}(\mathbb C)$ Find the Jordan ...
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1answer
35 views

Help in applying change of basis formula

I'm really stumped on this question, specifically regarding how to incorporate $T$ into the change of basis formula. Let $$ \Gamma = \left( \begin{bmatrix} 1 \\\\ 0 \end{bmatrix}, \begin{bmatrix} ...
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1answer
31 views

Question involving characteristic polynomial of a linear transformation

I was wanting some hints on a question and I have no idea how to approach this: Suppose $F$ is a field, $V$ is an $F$-vector space and $T: V \rightarrow V$ is a linear map. Suppose $p(x) \in F[x]$ ...
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1answer
46 views

For a matrix A, find a subspace of $R^3$ such that the function represented by A satisfies given properties.

This is a continuation of this question: Find real number $a$ such that matrix $A$ is NOT diagonalisable For the matrix $A =$ \begin{bmatrix}2&5&-1\\0&2&1\\-1&8&-1\end{bmatrix}...
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2answers
53 views

Looking for an exemple of $ \mathbb{Z} $ - linear application $ f : \mathbb{Z}^n \to \mathbb{Z}^m $ satisfying the following conditions :

I'm looking for : An exemple of $ \mathbb{Z} $ - linear application $ f : \mathbb{Z}^{ (\mathbb{N}) } \to \mathbb{Z}^{ (\mathbb{N}) } $ satisfying the following condition : $$ \exists y \in \mathbb{...
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1answer
24 views

prove: $||A-B||_{F}\geq||A-T(A)||_{F}$ as: given $A, B \in M_{n\times n}(\mathbb{R})$ , and $B$ is symmetric

given two matrices: $A, B \in M_{n\times n}(\mathbb{R})$ , let $B$ is a symmetric matrix ($B=B^{T}$), We'll define a linear map: $T:M_{n\times n}(\mathbb{R})\longrightarrow M_{n\times n}(\mathbb{R})$...
2
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1answer
48 views

What does it mean to say that the Laplacian commutes with translations?

In this explanation, $L$ is an operator that could be the Laplacian. However, I don't know what is $f$ and $T$. It later suggests that $T$ is a rotation, but what is $f$ and why does the relation $L(f\...
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1answer
24 views

linear transf.+charac. poly

Let V be the function space of all polynomials up to degree 2. Then define the following two linear transformations on $V$: For every $f(x)\in V$: $S(f(x))=x^2\cdot f(\frac{-1}{x})$ and $T(f(x))=f(...
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1answer
46 views

Visualize two linear transforms with same eigenvectors but different eigenvalues (real vs complex)

SUMMARY: A complex eigenvector is calculated from a $2 \times 2$ square matrix A. Expressed as a sum of real and imaginary parts $\lambda_{A} = \begin{bmatrix}a+c&0\\0&a-c\end{bmatrix} + \...
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1answer
43 views

For finite-dimensional vector space $X$, does $\phi: X \rightarrow \mathbb{R}$ continuous and bounded imply $\phi$ linear?

In a finite-dimensional vector space $X$, if $\phi: X \rightarrow \mathbb{R}$ is linear, then it can be shown to be continuous and bounded. Bounded in this context means that there exists an $M < \...
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2answers
147 views

Is Cross Product of two vectors a linear transformation? (Linear Algebra)

Background Information: I am studying linear algebra. For this question, I understand the definition of a vector in $$R^3 => v =(x,y,z)$$, and I know that A linear transformation between two ...
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1answer
51 views

Let $T: \mathbb{R}^4 \to \mathbb{R}^4$ be any linear transformation. Then how can I show that $T$ has a proper non zero invariant subspace. [duplicate]

Let $T: \mathbb{R}^4 \to \mathbb{R}^4$ be any linear transformation. Then how can I show that $T$ has a proper non zero invariant subspace. If $T$ has an eigen value then it is clear but if not then ...
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1answer
56 views

Find linear transformation from the strip $0<x<1$ to itself

Find the general form of the linear transformation from the strip $0<x<1$ to itself. Attempt: The linear transformation has the form $f(z)=az+b$. We need to find where $a$ and $b$ are. Let $...
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0answers
57 views

Determining eigenvalues of a linear transformation on a field extension via embedding into $\Bbb{C}$

I have been working on the following question: Suppose that $K$ is an extension of $\Bbb{Q}$ of degree $n$. Let $\sigma_1,\dots,\sigma_n:K\hookrightarrow\Bbb{C}$ be the distinct embeddings of $K$ ...
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1answer
33 views

Does centering the dependent variable and every independent variable change the estimated regressor?

In the linear regression model $Y=X\beta+u$ with $X=(\mathbf1, X_1, ...,X_p) \in\mathbb{R}^{n\times (p+1)}$ define $Y^*=Y-\bar{Y}$ and $X^*=(\mathbf0, X_1-\bar{X_1},...,X_p-\bar{X_p})$ Then \begin{...
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0answers
37 views

What is rank invariance?

I work in the area of soft matter and recently I have been reading a paper that talks about developing a new optimization algorithm for statistical physics problems. Turning statistical physics ...
0
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1answer
40 views

Does every map in dual have a predual in infinite-dimensional spaces?

Let $V$ be an infinite-dimensional vector space having a countable basis, let $V'$ be its dual space and let $R: V' \rightarrow V'$ be a linear map. Is it always true that there is a map $T: V \...
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1answer
49 views

show that $\operatorname{rank}(g\circ f) \leq \operatorname{rank}(f)+\operatorname{rank}(g)-n$

Let $E$ a vector space and $\dim(E)=n$ and let $f,g \in L(E)$ show that $\operatorname{rank}(f\circ g) \leq \operatorname{rank}(f)+\operatorname{rank}(g)-n$ I can see that $\operatorname{Ker}(g) \...
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0answers
30 views

Defining tensors as multilinear maps, without defining the dual space first

A $(p,q)$-tensor can be defined as a multilinear function in several ways, which are mostly equivalent: $$T:(V^*)^p\times V^q\to\mathbb R$$ or $$T:V^q\to V^p$$ or $$T:V^q\to \mathbb R\times V^p$$ ...
2
votes
0answers
47 views

Matching two coordinate systems with three vectors

This may be a bit long winded but I have not done linear algebra in a while and I want to make sure my math is correct. I am running two different computer simulations of proteins and I need to match ...
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2answers
25 views

Annihilator in dual space [closed]

Let U and W are subspaces of a vector space V. If U is subset of W, inh(W) is subset of inh(U). Is the Converse true? How?
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0answers
23 views

Projective Transformation between two matched 3D point sets

I have two point sets of about 800 3D-points that are matched. That means, I know the corresponding points in the two point clouds. Now I want to minimize the distance between the corresponding points ...