Questions tagged [vector-spaces]
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195 questions
15
votes
1
answer
872
views
(AC) and existence of basis of $\{0,1\}^X$ for any set $X$
For any set $X$, let $\{0,1\}^X$ be the collection of all functions $f:X\to\{0,1\}$. We make it into a vector space over the field $\mathbb{F}_2$ by endowing it with pointwise addition modulo $2$ and ...
- 54.4k
2
votes
0
answers
120
views
Is there a 'determinant' of a two-variable function when treated as a linear map?
A two variable real function $F(y,x)$, defined on $[c,d] \times [a,b]$, can be thought of as a linear map between functions of different domains by:
$$
g(y) = \int^a_bF(y,x)f(x)dx
$$
This has very ...
- 121
5
votes
0
answers
104
views
What does the automorphism group of a norm cone look like in general?
Let $(V, \lVert \cdot \rVert)$ be a finite-dimensional real normed space, and let $C \subseteq \mathbb R \oplus V$ be the norm cone of $V$; that is, $C$ consists of all $(t, v)$ for which $\lvert t \...
- 181
-1
votes
2
answers
118
views
Constructing an orthonormal set with given projections in a direct sum decomposition
Let $V$ be an $n$-dimensional real inner product space. Suppose we have $k\leq n/2$ orthonormal vectors $u_1, u_2, \dots, u_k \in V$.
Assume that there exist pairwise orthogonal subspaces $A,B,C \...
- 11
42
votes
14
answers
4k
views
Alternative proofs that two bases of a vector space have the same size
Let's stick to finite-dimensional vector spaces. I'm teaching linear algebra right now, and soon I'll have to prove that any two bases of a finite-dimensional vector space have the same size. I've ...
1
vote
0
answers
67
views
Vector matroids and projections of vector spaces (reference and literature request)
I am new in matroid theory, and currently I am working with so-called matroids of a vector space.
Definition. Let $V \subseteq \mathbb F^n$ be a vector space. A matroid of subspace $\mathbf M(V)$ is a ...
- 123
1
vote
0
answers
173
views
In which cases Hahn-Banach theorem holds for pseudotopological Hausdorff locally convex linear spaces?
In which cases Hahn-Banach theorem holds for pseudotopological Hausdorff locally convex linear spaces? I would be grateful for references.
Some definitions for context.
Pseudotopological space is a ...
0
votes
0
answers
76
views
vector membership and tensor product
Given a $m$-dimensional nonzero complex vector $\bar{x}$ and a set $S=\{\bar{y}_1,\cdots,\bar{y_n}\}$,for any $k$,
$$
\bar{x}^{\otimes k}\in span\{\bar{y}_1^{\otimes k},\cdots,\bar{y_n}^{\otimes k}\}
$...
- 1,533
5
votes
0
answers
182
views
Abelian category containing both vector spaces and profinite vector spaces
Let $k$ be a field. Consider the following categories:
$A$: The category of finite-dimensional $k$-vector spaces.
$B$: The category of all $k$-vector spaces.
$C$: The category of profinite $k$-vector ...
- 399
0
votes
0
answers
101
views
Which positive convex cones in $\mathbb{R}^{n}$ are closed under componentwise meets?
The vector space $\mathbb{R}^{n}$ has a natural lattice structure: for $\mathbf{a} = (a_1, \dots, a_n)$ and $\mathbf{b} = (b_1, \dots, b_n)$
$\mathbf{a} \wedge \mathbf{b} = (\min(a_1,b_1), \dots, \min(...
- 265
3
votes
1
answer
216
views
Bounded operators with closed complemented ranges
Let $B$ be a Banach space and $A:B\rightarrow B$ a bounded operator such that $A\left( B\right) $ is closed and there is some closed subspace $E\subset B$ such that $B=A\left( B\right) \oplus E$. Is ...
- 153
3
votes
0
answers
235
views
How are ideals that generate isomorphic algebras related?
Given the tensor algebra $T(V)$ for a finite-dimensional vector space $V$, and ideals $I_1,I_2\in T(V)$ (I only care about quadratic ones, if that matters), how are $I_1$ and $I_2$ related if I have ...
1
vote
0
answers
108
views
Reference request - Fourier multiplier of vector valued function
I would like to understand the concept of multiplier for vector valued functions and find appropriate references for the multiplier theorems out there.
For instance say that we would like to express $\...
- 133
1
vote
0
answers
118
views
Is there a name for "applying linear operations to vector sequences from the right"?
Let $v_1,...,v_n\in\Bbb R^d$ be a sequence of vectors. When we say that we "linearly transform" this sequence, we mean that we apply a linear transformation $T\in\Bbb R^{d\times d}$ to each ...
- 14.5k
4
votes
1
answer
278
views
Problem in Probability Theory and Functional Analysis
Let's consider the vector space V of bounded scalar functions, which includes the constant function 1. We assume that any uniform limit of a bounded monotonic sequence of functions from V also ...
- 51
2
votes
0
answers
131
views
Representation of Dirac-delta distribution in subspace of functions
Suppose I have a subspace $V\subset L^2(\Omega)$ where $\Omega\subset \mathbb{R}^d$ is a bounded and closed set. $V$ is defined by
\begin{align}
V=\text{span}(\{\varphi_i(x): i=1,2,\dots,n\})
\end{...
- 103
0
votes
1
answer
120
views
If the matroids associated to two finite subsets of the same vector space are isomorphic, are these two finite subsets linearly equivalent?
Let $E$ be a finite subset of ${\mathbb{F}_2}^n$, the $n$-dimensional vector space over the finite field $\mathbb{F}_2$ of $2$ elements. Let $M_E$ denote the associated matroid on $E$ where the ...
- 49
2
votes
1
answer
388
views
Question on a vector inequality
Is it true that
$$
\min\left( \begin{aligned}
&\|\mathbf{u}\| + \|\mathbf{v}\| - \|\mathbf{u} + \mathbf{v}\|, \\
&\|\mathbf{u}\| + \|\mathbf{w}\| - \|\mathbf{u} + \mathbf{w}\|, \\
&\|\...
- 171
1
vote
1
answer
123
views
When is a $1$-varifold $V$ the associated varifold of the reduced boundary of some Caccioppoli set?
Let $v_1$, $v_2$, $\cdots$, $v_l\in\mathbb{R}^n$ be unit vectors, $\mathbb{R}_v^+:=\{\lambda v:\lambda>0\}\subset\mathbb{R}^n$ be the ray in $v$'s direction; $n_1$, $n_2$, $\cdots$, $n_l>0$ be ...
- 111
0
votes
1
answer
196
views
Least squares cross product equations
I've tried, unsuccessfully, to either solve or find a solution to something along lines of:
find $\bar{a}$, $\bar{b}$ nearby to some initial guess that satisfies $\bar{c} = \bar{a} \times \bar{b}$.
...
- 101
-1
votes
2
answers
818
views
$p$-norm of random variables and weighted $L^p$ space resemblance
I noticed a very similar relationship between weighted $L^p$ space (denoted $L_w^p$) and normed vector space of random variables. I want to unify these two spaces but there always seems to be a ...
- 63
7
votes
0
answers
864
views
Dimension inequality for subspaces in field extensions
Let $K\subset L$ be a field extension and $A, B\subset L$ be $K$-subspaces of $L$ of finite positive dimensions. Assume further that for every $a, b \in L$ and every nontrivial proper finite ...
- 379
8
votes
0
answers
310
views
For which norms does closest projection never increase norm?
Let $X$ be a finite-dimensional normed space, and $V\subset X$ a vector subspace.
For a point $x\in X$ we define a closest projection of $x$ onto $V$ to be a minimizer $v^\star$ of $\|v-x\|$ amongst $...
- 137
1
vote
0
answers
107
views
Are these kinds of bases for $\mathbb{F}_2^q$ seen as a vector space studied?
In the context of my research, I have to work with sets of vectors $\left\{y_i\right\}_{i\in[n+1]}$ of $\mathbb{F}_2^n$ such that the following property is true:
$$\forall i\in[n+1], \left\{y_i\oplus ...
- 167
0
votes
0
answers
165
views
Is there an inner product on $\mathbb{F}_p\left[S_n\right]$ for which $\langle x, x \rangle \ne 0$ for all $x$?
Let $\mathbb{F}_p\left[S_n\right]$ be the group algebra of the symmetric group $S_n$ over the finite field $\mathbb{F}_p$.
One can define an "inner product" in the usual way:
$$\langle x,y \...
- 601
1
vote
0
answers
218
views
Reconstructing an object from its shadow
I'm looking into the section "Reconstructing an object from its shadow" in the book Introduction to the Mathematics of Medical Imaging by Charles L. Epstein.
I have two questions
The ...
- 11
2
votes
1
answer
134
views
Difference of probabilities of two random vectors lying in the same set
Suppose I have to random vectors:
$$\mathbf{z} = (z_1, \dots, z_n)^T, \quad \mathbf{v} = (v_1, \dots, v_n)^T$$
and set $A \subset \mathbb{R}^n$.
I want to find an upper bound $B$ for the following ...
- 33
0
votes
1
answer
119
views
Find efficiently greatest difference between $2$ vectors from set of vectors [closed]
Let us have a list of vectors in a $3$D space.
Is there a more efficient way to find the greatest difference between any two of them than combining each, computing the size of their difference, and ...
- 109
0
votes
0
answers
189
views
Totally isotropic space for bilinear pairing over ring
A duplicate of this:
Consider the following well-known inequality: Let $b$
be a non-degenerate symmetric bilinear pairing over a (finite-dimensional) $\mathbb F$-vector space $V$ and $W$
a totally ...
- 327
7
votes
1
answer
190
views
Constructing countable threelds of finite dimension
A threeld is a generalization of a field, with three operations, such that the $F$ is a field with respect to the first (outer) and second (middle) operations (call it the outer field), and $F\...
- 2,871
8
votes
3
answers
1k
views
How many non-orthogonal vectors fit into a complex vector space?
I am sitting on a problem, where I have a complex vector space of dimension $D$ and a set of normalized vectors $\{v_k\}$, $k\in\{1,2,\dots,N\}$ that are supposed to satisfy
$$\lvert\langle v_j\vert ...
1
vote
1
answer
108
views
Is this notion of being "fully" convex closed under set addition?
While reading through "Linear Operators: General theory" by "Jacob T. Schwartz", reading the corollary to II.10.1 which states that for a compact convex subset $C$ of some ...
- 216
0
votes
1
answer
212
views
Name for a monoid on the basis of a vector space?
Is there a name for the structure of a vector space with a monoid defined on its basis?
Given a vector space V over a field F, we can choose a basis and define a monoid on it. Now we can use each ...
- 481
1
vote
0
answers
204
views
Centraliser of a finite group
Let $G=\operatorname{Sp}(8,K)$ be a symplectic algebraic group over an algebraically closed field of characteristic not $2$.
We have a vector space decomposition $V_8=V_2\otimes V_4$ where the $2$-...
- 491
1
vote
0
answers
328
views
General linear group in infinite dimensions
Let $V$ be a vector space over the field $k$. Upon assuming the Axiom of Choice, we know that $V$ has a well-defined dimension $N$, and hence a well-defined basis $B$. Suppose that $N$ is not finite.
...
- 4,837
21
votes
7
answers
6k
views
Why do infinite-dimensional vector spaces usually have additional structure?
On Mathematics Stack Exchange, I asked the following question: Why are infinite-dimensional vector spaces usually equipped with additional structure? Although it received one good answer, I feel that ...
- 1,538
-4
votes
1
answer
226
views
Coordinate free computation of the second derivative of a functional [closed]
Let $F(g(f))$
be a functional that sends functions of a vector variable (from $n$-dimensional vector space) to $\mathbb R$.
$g$
is some function of scalar valued functions $f$.
I'm interested in a ...
- 1
0
votes
1
answer
167
views
Can you help me prove this vector identity?
It could be that the preprint where I found this identity has a typo or that it is simply wrong, but I have been trying to see if this is true:
\begin{equation}
\int \left(\nabla\times F_{\bf B}\...
- 327
0
votes
0
answers
132
views
Construct a vector space whose elements are sets
I would like to construct a vector space whose elements are convex and closed subsets of $\mathbb{R}^n$.
A natural idea is as follows.
For any two sets $S_1, S_2 \subseteq \mathbb{R}^n$, define the ...
- 159
0
votes
1
answer
183
views
Seeking closed-form solution for vector equation
I'm working on a problem that involves vectors and scalar values, and I'm looking for a closed-form solution. I hope someone can help me with this or provide insights into how to approach it. Here's ...
- 111
4
votes
1
answer
644
views
Boolean algebra of the lattice of subspaces of a vector space?
Recall that a Boolean algebra is a complemented distributive lattice. The set of subspaces of a vector space comes very close to being a boolean algebra. It satisfies all the required properties, ...
- 1,203
0
votes
0
answers
226
views
Given optimality of L1 norm, prove that absolute value of sum of a vector with proper sign is less than 1?
Problem:
Given a domain $\mathcal{D}\subset\mathbb{R}^{l}$, we can find $l$
points $\boldsymbol{v}_{i}\in\mathcal{D}$, $i=1,\cdots,l$. Each
point is a column vector with dimension $l\times1$. They ...
- 1
1
vote
1
answer
573
views
Dimension of a kernel of a linear map
Let $\mathbb{F}$ be a field of characteristic $2$, $n$ be a positive integer and $f_n:\bigoplus\limits_{i=1}^n\mathbb{F}\sigma_{i}\mapsto \bigoplus\limits_{i,j=1,i<j}^n\mathbb{F}\sigma_{i,j}$ be a ...
- 1,029
1
vote
1
answer
135
views
Counting the number of summands in a vector space over characteristic $2$ to get a direct sum
Let $\mathbb{F}$ be a field of characteristic $2$ and define $S$ to be the set of all triples $(i,j,k)\in\lbrace 1,\dotsc,n\rbrace^3$ with $\left|i-j\right|=1$, $\left|i-k\right|>1$, and $\left|j-k\...
- 1,029
5
votes
1
answer
482
views
Automorphisms of vector spaces and the complex numbers without choice
In Zermelo-Fraenkel set theory without the Axiom of Choice (AC), it is consistent to say that there are models in which:
there are vector spaces without a basis;
the field of complex numbers $\mathbb{...
- 4,837
3
votes
1
answer
352
views
Are all Helmholtz decompositions related?
Suppose $V\subset \mathbb{R}^3$ be non-empty and at least twice differentiable (Smooth) and let $S$ be the surface that encloses $V$ (for example a sphere). Let $\textbf{F}\in \mathbb{R}^3$ be a ...
user353746
0
votes
2
answers
297
views
Does surface integral preserve the curl operation?
Suppose $V\subset \mathbb{R}^3$ be non-empty and at least twice differentiable (Smooth) and let $S$ be the surface that encloses $V$ (for example a sphere). Let $\textbf{F}\in \mathbb{R}^3$ be a ...
user353746
0
votes
0
answers
106
views
Arithmetic triangles and unimodality of its rows
Let's consider the sequence of coefficients of $\prod_{i}\frac {1-x^{d_i}} {1-x}$, where $d_i$ is a monotonically increasing nonnegative integer sequence.
How to prove that the coefficients form an ...
2
votes
0
answers
138
views
To show $\{(x,y) \in \mathbb Q^{\geq 0} \times \mathbb Q^{\geq 0}~:~ mn+1 \mid m^x+n^y \}$ is subset of the lattice $\{\vec u+i \vec v+j \vec w\}$?
I am writing two definitions, the $1$st one is a cover in some sense while the $2$nd one is a lattice:
Definition 1: If $m,n$ are integers bigger than $1$, then the set $$A=\{(x,y) \in \mathbb Q^{\geq ...
- 928
7
votes
1
answer
270
views
Which lattices have non-trivial linear representations?
Suppose we have a a bounded lattice $L$. We might ask: does there exist a non-trivial linear representation of $L$, i.e. a lattice homomorphism $\rho: L \to \text{Sp}(V)$, where $V$ is a non-zero ...
- 191