# Questions tagged [vector-spaces]

The vector-spaces tag has no usage guidance.

164
questions

7
votes

3
answers

988
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

65
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 ...

0
votes

1
answer

107
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 ...

1
vote

0
answers

151
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 $2$-...

1
vote

0
answers

106
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.
...

17
votes

7
answers

3k
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 ...

-4
votes

1
answer

102
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 ...

0
votes

1
answer

115
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}\...

0
votes

0
answers

83
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 ...

0
votes

1
answer

115
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 ...

4
votes

1
answer

325
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, ...

0
votes

0
answers

133
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
vote

1
answer

371
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
vote

1
answer

81
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\...

4
votes

1
answer

276
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{...

3
votes

1
answer

280
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 ...

0
votes

2
answers

191
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 ...

0
votes

0
answers

81
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

84
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 ...

7
votes

1
answer

184
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 ...

3
votes

2
answers

416
views

### Reducing $9\times9$ determinant to $3\times3$ determinant

Consider the $9\times 9$ matrix
$$M = \begin{pmatrix} i e_3 \times{} & i & 0 \\
-i & 0 & -a \times{} \\
0 & a \times{} & 0 \end{pmatrix}$$
for some vector $a \in \mathbb R^3$, ...

0
votes

0
answers

48
views

### A query regarding complex vector decomposition

Given a complex vector $V$ of length $n^2$. Each complex entry in the vector is of size (number of digits or bits required to express the complex number) $c$ for some constant $c$.
Is it always ...

7
votes

0
answers

291
views

### A minimal semigroup generating subset of the additive reals

I asked this on MSE, but I was told to ask it here because it is a difficult question. Consider the additive magma of the real numbers, $(\mathbb{R};+)$. Does there exist a subset $S$ of the reals ...

2
votes

0
answers

83
views

### Name for closure property: set of maps closed under taking $(f,g)\mapsto (f-g)/2$

Suppose that $F$ is a collection of functions mapping some set $\Omega$ to $\mathbb{R}$, with the following closure property: whenever $f,g\in F$, we also have $(f-g)/2\in F$. Is there a name for this ...

2
votes

0
answers

106
views

### Right unitor in star-autonomous categories

1.Context
Let $(C, \otimes, I, a, l,r)$ be a monoidal category. Here $r$ denotes the right unitor.
Suppose $S: C^{op} \xrightarrow{\sim} C$ is an equivalence of categories with inverse $S’$. Assume ...

1
vote

2
answers

98
views

### Name for the weight function defined as the integer sum of coordinate entries from ${\mathbf F}_p$

In ${\mathbb F}_p^n$, $p$ prime one may define a weight function on vectors in various ways such as Hamming, or Lee weight. (These two weights correspond nicely to the respective distances from $\bar ...

3
votes

1
answer

404
views

### Image of a quadratic form is a closed cone

Let $Q : E \to F$ be a quadratic form induced by a symmetric bilinear form $B : E \times E \to F$ defined in a finite dimensional real normed vector space $E$, with values in the normed vector space $...

3
votes

0
answers

65
views

### Field automorphisms of projective spaces without the axiom of choice

Suppose P is a projective space over the field $k$. If P has finite dimension $n$, we can fix a base. Relative to this base, the full automorphism group of P can be described by the action on the ...

1
vote

0
answers

525
views

### Notation for the space of eventually-zero sequences

An eventually-zero sequence is a real-valued sequence $(x_n)_{n=1}^\infty$ for which there exists an $N\in\mathbb{N}$ such that $x_n=0$ for each $n\geq N$. The space of eventually-zero sequences ...

2
votes

0
answers

96
views

### Proving the non-existence of canonical isomorphisms

From time to time, during my undergraduate lectures on linear algebra appears the following question from the most smart students in the class. I asked to my algebra colleagues but I have not received ...

-1
votes

1
answer

233
views

### Injection vs. bijection between $V^\star \otimes V^\star$ and $(V \otimes V)^\star$ [closed]

It is a well-known fact that, if $V$ is a vector space over a field $k$, then $V^\star \otimes V^\star$ embeds into $(V \otimes V)^\star$.
It turns out to be an isomorphism when $V$ is a finite-...

0
votes

1
answer

242
views

### Application of the Frechet derivative [closed]

$f\colon U\subset \mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$ is differentiable at $x_{0}$ if there exist a linear transformation $T\colon \mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$, such that:
\...

3
votes

1
answer

533
views

### Sum of $q$-binomial coefficients

Denote by $ \binom{n}{k}_q = \prod_{i=0}^{k-1} \frac{ q^{n-i} - 1 }{ q^{k-i} - 1 } $, $ k = 0, 1, \ldots, n $, the $ q $-binomial (Gaussian) coefficients. These numbers are symmetric, in the sense ...

1
vote

1
answer

113
views

### Nilpotent matrices with (Motzkin-Taussky) property L

One of the consequences of the well-known Motzkin-Taussky theorem (https://www.jstor.org/stable/1990825) is the following : if two complex matrices $A, B$ generate a vector space of diagonalisable ...

0
votes

0
answers

214
views

### Vector convolution?

I am working on a research problem which leads to the following optimization problem:
\begin{equation}
\hat{M} = \operatorname*{arg\,max}_M \Bigl\lVert\sum_{k=0}^{M-1} {\mathbf y}_k \exp\left(-j 2\pi ...

3
votes

1
answer

169
views

### Looking for a paper on axiomatic orthogonality in a vector space

I am looking for a paper "Linear spaces with disjoint elements and their conversion into vector lattices" by A. I. Veksler.
It was published in 1967 in Research Notes of Leningrad State ...

2
votes

1
answer

2k
views

### Under which conditions: dim(W1 + W2 + W3) = dim(W1) + dim(W2) + dim(W3) − dim(W1 ∩ W2) − dim(W2 ∩ W3) − dim(W3 ∩ W1) + dim(W1 ∩ W2 ∩ W3) [closed]

Let $V$ be a finite dimensional vector space over a field $K$, and let $W_1$, $W_2$ and $W_3$ be subspaces of $V$. By analogy with the inclusion-exclusion principle for sets, and taking into account ...

1
vote

0
answers

61
views

### Derivative of a function of ordered variables

Can I differentiate
$$(\pmb{y}^o - (\pmb{x}\cdot\pmb{a})^o)^\top(\pmb{y}^o - (\pmb{x}\cdot\pmb{a})^o)$$ with respect to $\pmb{a}$? (I want to minimize the expression with respect to $\pmb{a}$.)
Here, $...

16
votes

1
answer

754
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

0
answers

174
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

0
answers

171
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

0
answers

130
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

0
answers

117
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

0
answers

93
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

3
answers

840
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

1
answer

122
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

1
answer

154
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

1
answer

109
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

1
answer

109
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

0
answers

95
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 ...