# Questions tagged [vector-spaces]

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $x_\alpha^*(x_\beta)=\delta_{\alpha,\beta}$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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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 $x,y\... 1answer 66 views ### How to calculate the camera 3D position if I know 4 points in the picture of the camera [closed] I've got a picture of - let's say a - table and I measured 4 Points on this table. I define my table as "ground level", so my Points got the coordinates (px, py, 0). Is it possible to calculate the ... 0answers 394 views ### Spectral theory without topology How much of spectral theory can be developed just working with vector spaces (finite or infinite dimensional) without referring to a choice of topology ? Something along these lines, for example: ... 1answer 96 views ### A vector Jacobi identity [closed] Let$\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\...
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 ...