# Questions tagged [vector-spaces]

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142
questions

5
votes

0
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180
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### 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

76
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

91
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

88
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

368
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

57
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### 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

187
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

65
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### 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

222
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

150
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

434
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

86
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

156
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

163
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

923
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### 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

59
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, $...

15
votes

1
answer

544
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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 ...

0
votes

0
answers

173
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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 ...

0
votes

0
answers

152
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

115
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

99
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

91
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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 ...

17
votes

3
answers

765
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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 ...

-2
votes

1
answer

81
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

101
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

100
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

108
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

70
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 ...

1
vote

0
answers

38
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

0
answers

172
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### $2$-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

0
answers

117
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### 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 ...

41
votes

2
answers

2k
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### 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

1
answer

477
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
votes

0
answers

90
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

0
answers

87
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

1
answer

128
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

1
answer

173
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

1
answer

180
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### 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

0
answers

40
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### 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

68
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

356
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

0
answers

161
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

439
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

512
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

570
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

0
answers

19
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

315
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

182
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

61
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

135
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 ...