Questions tagged [vector-spaces]

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11
votes
1answer
331 views

Examples of vector spaces with bases of different cardinalities

In this question Sizes of bases of vector spaces without the axiom of choice it is said that "It is consistent [with ZF] that there are vector spaces that have two bases with completely different ...
0
votes
0answers
163 views

Eigenvalues without the axiom of choice

Without the Axiom of Choice (AC), we can find models of ZF set theory in which some vector spaces have no base, and also models in which some vector spaces have bases of different cardinalities. The ...
0
votes
0answers
129 views

Eigenbases without the Axiom of Choice

I understand that in ZF set theory without the Axiom of Choice (AC), it is consistent to have models in which there exist vector spaces over some (unspecified) field $k$ without a basis. So in ...
0
votes
0answers
98 views

Axiom of Choice and bases of $k$-vector spaces, $k$ fixed

I know that from ZF + the Axiom of Choice (AC) follows that every vector space has a basis. And, conversely, Blass proved that in ZF set theory, the assumption that every vector space has a basis ...
4
votes
0answers
72 views

Hamel basis with all coordinate functionals discontinuous

If $X$ is an infinite dimensional (separable) Banach space, can one find a Hamel basis $(x_\alpha)_{\alpha\in\Lambda}$ such that all coordinate functionals $x_\alpha^*(x_\beta)=\delta_{\alpha,\beta}$ ...
0
votes
0answers
88 views

Large subgroups of infinite-dimensional vector spaces

Let $V$ be an infinite-dimensional vector space over $\mathbb{Q}$. Consider a proper subgroup $W$ of $V, +$ with the following property: each vector line $L$ (which we see as a subgroup of $V, +$) has ...
17
votes
3answers
717 views

Existence of translation-invariant basis on $C_c(\mathbb R)$

Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...
-2
votes
1answer
66 views

A generalized norm function in $\mathbb{R}^n$ [closed]

We defined a new norm. The norm of $x \in \mathbb{R}^n$ is defined as $$ N_P(x) = \min \{t \geq 0 : x \in t\cdot P\} \enspace,$$ where $P$ is a centrally symmetric and convex body centered at the ...
2
votes
1answer
76 views

Low-Hamming weight vectors in low-dimensional subspaces of $\mathbb{F}_p^n$

What is the maximum number vectors of Hamming weight at most $w$ in a $d$-dimensional subspace of $\mathbb{F}_p^n$, where $w,d,p$ are constant and $p$ is odd. (The Hamming weight is the number of ...
1
vote
1answer
97 views

Is trace of a slice of an elementary function of a matrix also elementary?

Let we have an elementary function $f(W)$, applicable to a matrix. Now consider the function $g(x)=\operatorname{tr} f(W+x),$ where $x$ is scalar. Is $g(x)$ necessarily an elementary function? Simple ...
2
votes
1answer
105 views

A matrix identity

Suppose $A=(a_{jk})_{j,k=1}^n$ is a symmetric complex valued matrix, that is to say, $a_{jk}=a_{kj}$ for all $j,k=1,\dotsc,n$. Suppose that given any two linearly independent vectors $\alpha=(\alpha^j)...
0
votes
0answers
56 views

Decomposition an $A$-module to irreducible ones

Let $V$ be a complex vector space (i.e, over the filed of complex numbers) and $A$ be a complex algebra. Suppose that $V$ is an $A$-module. Under what proper condition(s) there are irreducible ...
0
votes
0answers
38 views

The relation between unit quaternions and the virtual rotation of an angular velocity vector?

I am working with a calculus employed in multi rigid body dynamics problems introduced by professor E. Haug (c.f. this book). Let's define $\boldsymbol{\mathrm{e}} \in \mathcal{R}^4$ as a set of unit ...
-2
votes
1answer
133 views

$f$ is the complexification of a map if $f$ commutes with almost complex structure and standard conjugation. What if we had anti-commutation instead? [closed]

I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying backwards as much as possible, but I have been stuck on the concepts of almost complex structures ...
1
vote
0answers
36 views

Characterization of dimension over $\mathbb{Q}$ of infinite sums of rational functions

Let $P(n)=(n+r_1)(n+r_2)...(n+r_k)$ be a polynomial with simple, rational, negative roots (i.e. $r_i>0$) and degree $k\geq 2$ (I stick with negative roots as I don't have to worry about dividing by ...
3
votes
0answers
161 views

$2$-vector spaces and algebraic $2$-stacks

I am thinking about higher Artin stacks in the sense of Simpson, concretely I would like to calculate the dimension and compare these two cases: $\mathfrak{X}_{1}=$ Higher linear stack classifying (...
2
votes
0answers
109 views

Are there any known algebras or vector spaces, where absolute value, modulus or norm is connected to the factors of $\pi$ or $e^{-\gamma}$?

I am currently working on an algebra of divergent integrals and series, and all the elements of that space consist of a regular part (which is a real or complex number) and irregular part (which is ...
40
votes
2answers
2k views

Do vector spaces without choice satisfy Cantor-Schroeder-Bernstein?

If $V \hookrightarrow W$ and $W \hookrightarrow V$ are injective linear maps, then is there an isomorphism $V \cong W$? If we assume the axiom of choice, the answer is yes: use the fact that every ...
4
votes
1answer
286 views

Real eigenvectors of complex matrices

Let $A$ be a nonsingular complex $(3 \times 3)$-matrix (that is, an element of $\mathrm{GL}_3(\mathbb{C})$). Then what are some of the best-known criteria which guarantee $A$ to have real eigenvectors ...
0
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0answers
51 views

Minimize a vector from a matrix operation

I want to minimize a certain vector that results from a matrix operation with some constraints and i don't exactly know how to tackle this problem. Lets say we have $$ (L+A)*s = v $$ L is the ...
2
votes
0answers
84 views

Topology of the set of polynomials with bounded real algebraic varieties (inside the v. s. of polynomials in $n$ variables and up to degree $d$)

Set $x=(x_{1}, \dots, x_{n}).$ Consider the set $\mathbb{R}[x]_{d}$ of polynomials with coef. in $\mathbb{R}$ in $n$ variables up to degree $d.$ This set can be seen as a finite-dimensional vector ...
4
votes
1answer
122 views

Looking for an algorithm that finds lowest cost linear subspace containing the vector

Let $v_1, v_2, \dots, v_N, u$ are vectors in $\mathbb{R}^n$, each defined by $n$ integer numbers. Typically, $n<N$, and each vector has only few (one to four) non-zero coefficients. Additionally, ...
3
votes
1answer
172 views

Can one turn finite-dimensional vector subspaces into a cancellative semigroup?

Let $V$ be a vector space over some field and let ${\rm Fin}\,V$ be the family of all finite-dimensional subspaces of $V$. Is it possible to turn ${\rm Fin}\,V$ into an commutative cancellative ...
0
votes
1answer
144 views

Reverse the construction of a basis for a tensor product of vector spaces [closed]

If $V,W$ are infinite-dimensional vector spaces with basis {${v_i}$} and {${w_j}$} respectively it holds that $V\otimes W$ has as basis {${v_i⊗w_j}$}. What about the reciprocal? That is: if {${v_i}$} ...
0
votes
0answers
33 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
votes
1answer
54 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
vote
2answers
297 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
votes
0answers
143 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
votes
1answer
407 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
votes
1answer
465 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
votes
1answer
396 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
vote
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
231 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
votes
0answers
142 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
vote
0answers
55 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
129 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
votes
0answers
66 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
votes
1answer
98 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
votes
1answer
66 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
votes
0answers
394 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
vote
1answer
96 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
votes
1answer
112 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 ...
2
votes
0answers
80 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
135 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
vote
0answers
48 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
vote
0answers
134 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
vote
1answer
173 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
373 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
votes
1answer
115 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
votes
2answers
597 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 ...