Questions tagged [vector-spaces]

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-2
votes
0answers
29 views

are angles betwen a basis and a given vector correlated? [closed]

I was thinking about an orthogonal basis and some vector, and maybe, find some relation between the angles that form the basis with the given vector
0
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0answers
27 views

Generalized eigenvectors product

Let's consider a real square matrix $A$ with eigenvalues $\lambda_n$ and eigenvectors $\mathbb x_n$, i.e. $A \mathbb x_n = \lambda_n \mathbb x_n$. Suppose there are some generalized eigenvectors $\...
0
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1answer
39 views

Determinant of a block matrix with dissimilar elements [closed]

I am looking for an expression that convert calculation of given quadratic form into determinant of some block matrix. I see this in that form (that may be incorrect): $x^T A x = \begin{vmatrix} x^T ...
1
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2answers
232 views

A description of the kernel of projection map from the tensor algebra to the symmetric algebra $T(V)\to S(V)$

Let $V$ be a vector space over some arbitrary field. Let $T(V)$ and $S(V)$ be the tensor and symmetric algebras over $V$. We have the projection map $T(V)\to S(V)$, given by $x_1\otimes\cdots\otimes ...
0
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0answers
121 views

Sylow subgroups of orthogonal group

According to Wikipedia (current revision) the cardinality of $O(n,q)$ depends on the properties of the field we're working over. These are the results: We have the following formulas for the order ...
7
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1answer
356 views

Do the following binary vectors span $\mathbb{R}^n$?

Defining the binary vectors Let an ordered triple of natural numbers $(r, d, n)$ such that $0 \leq r < d \leq n$ be given. Consider the binary vector $v_{(r,d,n)} \in \mathbb{R}^n$ such that for ...
11
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1answer
430 views

(Barely) linearly independent vectors over $\mathbb{Z}/2\mathbb{Z}$

Let $V$ be a vector space over $\mathbb{Z}/2\mathbb{Z}$. Can there be a set $S$ of $2 n$ vectors in $V$ such that any $n$ vectors in $S$ span a space of dimension exactly $n-1$, but no $n$ vectors $...
0
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1answer
204 views

What are the eigenvalues and eigenvectors of a composition of two arbitrary linear transformations? [closed]

Let $S$ and $T$ be two linear transformations on $\mathbb{R}^n$. We can find the eigenvalues and eigenvectors of $S$ and $T$. I am trying to find out the relation(s) among the eigenvalues and ...
1
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0answers
17 views

Weird subspace/equality-constrained LP problem/variant of change-making problem

Assume that we have a set, $\mathscr{R}$ containing $m$-dimensional vectors. Solve $$\sum_{i=1}^n c_i\leq\delta$$ $$\text{subject to } \sum_{i=1}^n r_i c_i=x^\prime \text{ for all }x^\prime$$ where $0\...
2
votes
1answer
137 views

Neat/Approximate formula for maximum number of “almost orthogonal” vectors in a complex vector space?

In a $d$ dimensional vector space defined over $\mathbb{C}$, how do I calculate the largest number $N(\epsilon, d)$ of vectors $\{V_i\}$ which satisfies the following properties. Here $\epsilon$ is ...
2
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0answers
96 views

Recover unknown vectors with dot-product queries

Suppose there are $n$ unknown unit vectors in $\mathbb{R}^d$, $V=\{v_1,\ldots,v_n\}$, no two identical. Your task is to determine the vectors in $V$. The only tool at your disposal is to query a ...
1
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0answers
53 views

Function for unique volume element

This is an issue that I'm am trying to solve for a fine-tuning measure in particle physics, but it is purely mathematical. Consider three vectors $\{v_1, v_2, v_3\}$ in $\mathbb{R}^3$. I would like a ...
2
votes
1answer
123 views

The minimal volume of the intersection of two $\mathscr{l}_1$-ball in high dimension

We define $$B_1(r,c) = \{x\in\mathbb{R}^d : \Vert{}x-c\Vert_1 \le r\}$$ Now for arbitrary constant $r \ge s > 0$, given constant $\epsilon \in (r-s,r+s)$, considering $v \in \mathbb{R}^d$ such ...
2
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0answers
60 views

Solution of equation on vector field

I have a vector field function $\vec{J}: {\bf R}^3\to {\bf R}^3$ looking like: $$ \vec{J}(\vec{r}) = (\vec{B} \times \vec{v}(\vec{r}))\rho(\vec{r}) $$ with a (very well behaved) real, positive, ...
3
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1answer
84 views

The “semi-symmetric” algebra of a vector space

If $V$ is a vector space over a field $K$, then the symmetric algebra $S(V)$ is defined as the tensor algebra $T(V)$ factorized by the two-sided ideal generated by $x\otimes y-y\otimes x$, with $x,y\...
0
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1answer
56 views

How to calculate the camera 3D position if I know 4 points in the picture of the camera [closed]

I've got a picture of - let's say a - table and I measured 4 Points on this table. I define my table as "ground level", so my Points got the coordinates (px, py, 0). Is it possible to calculate the ...
5
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0answers
376 views

Spectral theory without topology

How much of spectral theory can be developed just working with vector spaces (finite or infinite dimensional) without referring to a choice of topology ? Something along these lines, for example: ...
1
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1answer
93 views

A vector Jacobi identity [closed]

Let $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ be three solenoidal vector fields i.e. $\nabla\cdot\mathbf{A}=\nabla\cdot\mathbf{B}=\nabla\cdot\mathbf{C}=0$. Can we prove that $\mathbf{A}\times\nabla\...
0
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1answer
75 views

Reconstructing Euclidian space from distance matrix

The setup. Let's say that we have a set of objects $O_i$ for which we have a dissimilarity measure $M(O_1,O_2)$. With this we can build a distance matrix $D_{ij}$. Let's also assume that we have NO ...
1
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0answers
71 views

Shortest vectors in tensor product and maximal lattices in tensor product

$\mathcal L$ and $\mathcal L'$ be full rank lattices in $\mathbb R^n$ with shortest vectors $v_1,\dots,v_n$ and $v_1',\dots,v_n'$ respectively where $$\|v_1\|_2\leq\dots\leq\|v_n\|_2$$ $$\|v_1'\|_2\...
4
votes
1answer
127 views

Is the free modular lattice linear?

Dedekind proved that the free modular lattice on 3 generators is realisable by the intersections and sums of 4-dimensional subspaces in 8-space. Birkhoff showed that the free lattice is infinite if it ...
1
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0answers
47 views

Existence of subspace which is totally non-invariant under unitary transformation

Given a complex Hilbert space $\mathcal{H}$ of dimension $d$ - interpreted as vectorspace over $\mathbb{R}$ with dimension $2d$. And the space $L(\mathcal{H},\mathbb{C}^4)$ of all linear operators ...
1
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0answers
125 views

Does the 2 category of Groupoids Admit the Vector Space Monad?

We can see here in Jacob's 2013 paper, that he seems to state that a particular kind of multiset monad is actually a vector space monad. 3.2. Vector spaces. For a semiring S one can define the ...
1
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1answer
126 views

Equivalence of matrices over a vector space

I'm trying to characterize equivalence classes of matrices over a vector space. Specifically, let $V$ be a vector space over a field $K$, let $M \in M_n(V)$ be an $n \times n$ matrix with entries in $...
6
votes
1answer
210 views

Jordan form on an invariant vector subspace

Let $\mathbb{F}$ be a field and $V$ an $\mathbb{F}$-vector space. Let $\operatorname{T}\in\mathrm{End}(V)$ be an $\mathbb{F}$-linear operator. It is well known that if $\dim V<\infty$ then $\...
0
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1answer
87 views

Why Expected squared length of a projected vector on reduced dimensionality coordinates is k/d?

For the proof of Johnson-Lindenstrauss algorithm by Dasgupta and Gupta, they comment in their paper that for a vector $Z \in R^k $, the expected squared length, $E[L]$ (where $L = \|Z\|^2$) of the ...
12
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2answers
558 views

The geometry of $\mathbb{R}^n$

Let $X,Y$ be finite-dimensional real normed spaces. Consider the set of linear operators $L(X,Y)$ between the two spaces. Then we define the set of equivalence classes $$G(X,Y):=\left\{[T]; T,S \in ...
-1
votes
1answer
72 views

transformation of two measures on different space

Let $\{e_1,e_2,...,e_n\}=E $ be the standard bases of $\mathbb{R}^n$, and $U\subset\mathbb{R}^n$ be a linear space generated by $\{e_1,e_2,...,e_n\}$. Let $\Sigma_U$ be the smallest $\sigma-$ field ...
6
votes
1answer
371 views

Action of upper triangular matrices

Let $M,N$ be two $n\times m$ matrices with $n\leq m$ and coefficients in an algebraically closed field of characteristic zero $K$, both of full rank $n$. Do there exist two upper triangular matrices ...
3
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0answers
105 views

Behavior of vector field that is the arg min of a convex function

Say $f(x,y)$ is a real valued function that is strongly convex and twice differentiable in both $x$ and $y$. Here $x$ and $y$ are real vectors of different dimensions. We define $y^\star(x)= \arg \...
9
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1answer
566 views

Axiom(s) of choice and bases of vector spaces

I'm not sure this question is more suitable for MO or for MSE, so feel free to move it to MSE if necessary. I work here in ZF theory. Consider the following statements: $(C)$ Axiom of choice: for ...
6
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0answers
168 views

Growth in a vector space

I am interested in "growth" in finite vector spaces. I have been wondering about the veracity of the following statement: Statement: For every $\varepsilon>0$ there exists $b\in\mathbb{Z}^+$ ...
2
votes
1answer
237 views

Polynomials and matrices in $\Bbb F_q$

Given a polynomial $p(x,y)\in\Bbb F_q[x,y]$ of $(x,y)$ degree $(n_x,n_y)$ ($n_x,n_y\geq0$ and $n_x,n_y\in\Bbb Z$) where $q=p^\alpha$ with $p$ a prime and $\alpha\in\Bbb N$ how many different matrices $...
5
votes
1answer
551 views

Generalizing Big O notation to arbitrary vector spaces

I'm constructing a Coq library for Big-O notation. Naturally, I'd like it to be as general as possible. The Wikipedia page on Big-O notation says The generalization to functions taking values in ...
16
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2answers
369 views

exponential functors on finite dimensional complex vector spaces

Denote by $Vect^{fin}_{\mathbb{C}}$ the category of finite dimensional complex vector spaces. We will call a pair $(F,\tau)$ that consists of a functor $F \colon Vect^{fin}_{\mathbb{C}} \to Vect^{fin}...
1
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0answers
97 views

On dimension of Segre embedding of lattice translations

Consider three lattices $L_1$, $L_2$ in $\Bbb Z^{n+1}$ and $L$ in $\Bbb Z^{2n+1}$. Let $L_1+v_1$, $L_2+v_2$ and $L+v$ be their respective translationsfor some $v_1,v_2\in\Bbb Z^{n+1}\backslash\{(0,\...
2
votes
1answer
519 views

maximum of orthogonal vectors

$$v_1=(x_1,x_2,x_3\cdot\cdot\cdot,x_n)$$is such a vector. By changing its signs and positions of each component $x_i$, we can get different vectors. When n is odd, it's impossible for any of ...
3
votes
0answers
112 views

Probability of orthogonal vectors?

Denote $\mathcal I_m=\{-m,-m+1,\dots,0,\dots,m-1,m\}$ where $m\geq0$ is an integer. Pick a uniformly random vectors $$\hat a=(a_n,a_{n-1},\dots,a_1)\in(\mathcal I_{m_1}+\sqrt{-1}\cdot\mathcal I_{m_2}...
2
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0answers
180 views

Finite-dimensional vector spaces vs. matrices over a semiring

The category $\mathbf{Vect}_k$ of vector spaces over some field $k$ is weakly equivalent to the category $\mathbf{V}_k$ whose objects are finite sets $n$, and whose morphisms $m\to n$ are $(m\times n)$...
8
votes
2answers
346 views

Relation between well-orderings of $\mathbb{R}$, and bases over $\mathbb{Q}$

The following question arose from a discussion about the definability of bases of $\mathbb{R}$ as a $\mathbb{Q}$-vector space. (ZF without AC) something we can note is that the existence of a (...
2
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0answers
87 views

Stochastic approximation in two dimensions

I have a question about a stochastic approximation method presented in J. R. Blum, Ann. Math. Stat. 25(4): 737 (1954). Given a $k$-dimensional vector $\bf x$ and $k$ random variables $Y^1_{\bf x}, \...
1
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0answers
388 views

Orthonormal basis of matrices

I am asking if somebody knows how to do or is aware of the following construction: Let $n \in \mathbb{N}$ be given, then take an arbitrary matrix $A \in \mathbb{C}^{n \times n}$. Then, there are maps ...
1
vote
1answer
195 views

The definition of the face of a convex set by a nonnegative affine linear polynomial

My question comes from the paper: https://arxiv.org/abs/0911.2750 (p.2~p.3) For $n\in \mathbf{N}$ Let $X = (X_1,\ldots,X_n)$ be an $n$-tuple of variables. Let $\mathbf{R}[X]$ denote the ...
-2
votes
1answer
138 views

English Translation of French Verseurs [closed]

Trying to read Lamaitres 1948 paper on Quaternions, in reply to Klein's Verlangen program, but can not find a translation of term Verseurs, which is even a section heading: "Un quaternion dont la ...
1
vote
1answer
302 views

Vector bundles and equivariant vector spaces

It seems commonly accepted that most of the results of equivariant geometry for vector spaces yield analog result for vector bundles. In so far as I understand it, the reason for that is the ...
-2
votes
1answer
44 views

Rotating a known vector over two axis-es to result to another known vector [closed]

Lets assume i have a known vector, for example x = [1,0,0] After 2 rotations, one over the y axis and one over the z axis, i result in a vector which in this example is x' = [0.5774, 0.5774, 0.5774] ...
-1
votes
1answer
172 views

Maximal commutative subrings of the endomorphism ring of a vector space

Let $\mathbb{F}$ be a field, and $\mathbf{V}$ a possibly uncountably generated $\mathbb{F}$-vector space. Let $\mbox{End}_\mathbb{F}(\mathbf{V})$ be the endomorphism ring of $\mathbf{V}$. That the ...
11
votes
1answer
670 views

Pfaffian equals complex determinant?

Let $V$ be a Euclidean vector space and let $V^{\mathbb{C}} = V \oplus V$ be its complexification, with complex structure $$J = \begin{pmatrix} 0 & -\mathrm{id}\\ \mathrm{id} & 0 \end{pmatrix}....
3
votes
2answers
615 views

Linear space with (Hamel) basis and the axiom of choice

It is true that the axiom of choice is equivalent to the statement that every linear space has a Hamel basis. There are some linear spaces which definitely don't need axiom of choice to possess (...
0
votes
2answers
713 views

Different inner products for vector spaces of random variables

The inner product that appears in most books on probability is the covariance $\langle X,Y \rangle = E[XY]$ (considering that $X$ and $Y$ are zero mean real random variables). Are there other inner ...