# Questions tagged [vector-spaces]

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104
questions

**-2**

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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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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**

votes

**1**answer

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**

vote

**2**answers

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 ...

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**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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**

vote

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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

**1**answer

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**

votes

**0**answers

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**

vote

**0**answers

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

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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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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**

vote

**1**answer

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**

votes

**1**answer

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**

vote

**0**answers

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

**1**answer

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 ...

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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 ...

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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 ...

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vote

**1**answer

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

**1**answer

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**

votes

**1**answer

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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**2**answers

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

**1**answer

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**

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**1**answer

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 ...

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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**

votes

**1**answer

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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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

**1**answer

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

**1**answer

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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**2**answers

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}...

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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

**1**answer

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**

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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}...

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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 $(m\times n)$...

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**2**answers

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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**0**answers

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}, \...

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**0**answers

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 ...

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vote

**1**answer

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 ...

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votes

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**2**answers

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 ...