Questions tagged [vector-spaces]
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172
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Correspondence of forms of the same solution of a system of linear equations [migrated]
I am currently reading Mathematics for Machine Learning book (p.27-28, freely available). And I am really confused with how two different forms of a solution of a system of linear equations correspond ...
- 11
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0
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92
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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:
$$\langle x,y \...
- 60.4k
1
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0
answers
158
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Reconstructing an object from its shadow
I'm looking into the section "Reconstructing an object from its shadow" in the book Introduction to the Mathematics of Medical Imaging by Charles L. Epstein.
I have two questions
The ...
- 5,632
2
votes
1
answer
77
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Difference of probabilities of two random vectors lying in the same set
Suppose I have to random vectors:
$$\mathbf{z} = (z_1, \dots, z_n)^T, \quad \mathbf{v} = (v_1, \dots, v_n)^T$$
and set $A \subset \mathbb{R}^n$.
I want to find an upper bound $B$ for the following ...
- 118k
1
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0
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72
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Dedekind complete quotients of Riesz spaces
Suppose that $E$ is a Dedekind complete Riesz space and let $J$ be an order ideal of $E$. Then one can form a quotient Riesz space $E/J$. I am interested which properties $J$ needs to possess so that ...
- 3,964
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1
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82
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Find efficiently greatest difference between $2$ vectors from set of vectors [closed]
Let us have a list of vectors in a $3$D space.
Is there a more efficient way to find the greatest difference between any two of them than combining each, computing the size of their difference, and ...
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8
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3
answers
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How many non-orthogonal vectors fit into a complex vector space?
I am sitting on a problem, where I have a complex vector space of dimension $D$ and a set of normalized vectors $\{v_k\}$, $k\in\{1,2,\dots,N\}$ that are supposed to satisfy
$$\lvert\langle v_j\vert ...
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0
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85
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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 ...
- 33.2k
6
votes
1
answer
157
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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\...
- 60.4k
1
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1
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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 ...
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1
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126
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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 ...
- 61.5k
1
vote
0
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154
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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 $2$-...
- 11.4k
32
votes
4
answers
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Definition of inner product for vector spaces over arbitrary fields
Is there a canonical definition of the concept of inner products for vector spaces over arbitrary fields, i.e. other fields than $\mathbb R$ or $\mathbb C$?
- 33.9k
1
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0
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153
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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.
...
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7
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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 ...
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2
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Exponentiation of vector spaces?
This question occurred to me while thinking on another one here, Name for an operation on matrices?
Can one define in an invariant way a binary operation on finite-dimensional vector spaces - let us ...
- 11.4k
-4
votes
1
answer
118
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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 ...
- 5,571
0
votes
1
answer
118
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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}\...
- 178k
0
votes
1
answer
128
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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 ...
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0
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87
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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
4
votes
1
answer
387
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, ...
- 42.2k
1
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1
answer
398
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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 ...
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148
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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
91
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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\...
- 11.4k
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votes
0
answers
293
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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 ...
- 60.4k
4
votes
1
answer
313
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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{...
- 2,341
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2
answers
204
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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 ...
- 24.2k
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votes
1
answer
281
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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 ...
- 178k
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0
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83
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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 ...
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85
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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 $$A=\{(x,y) \in \mathbb Q^{\geq ...
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1
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Which lattices have non-trivial linear representations?
Suppose we have a a bounded lattice $L$. We might ask: does there exist a non-trivial linear representation of $L$, i.e. a lattice homomorphism $\rho: L \to \text{Sp}(V)$, where $V$ is a non-zero ...
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2
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Reducing $9\times9$ determinant to $3\times3$ determinant
Consider the $9\times 9$ matrix
$$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$, ...
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0
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50
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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 ...
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0
answers
83
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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 ...
- 60.4k
2
votes
0
answers
109
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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 ...
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vote
2
answers
100
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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 ...
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3
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1
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424
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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 $...
- 11.6k
3
votes
0
answers
65
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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 ...
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1
vote
0
answers
617
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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 ...
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0
answers
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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 ...
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1
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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-...
- 97
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votes
1
answer
260
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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:
\...
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3
votes
1
answer
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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 ...
- 60.4k
1
vote
1
answer
121
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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 ...
- 356
1
vote
0
answers
237
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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 ...
- 1,838
3
votes
1
answer
174
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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 ...
- 36.3k
2
votes
1
answer
2k
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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 ...
- 4,618
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vote
0
answers
61
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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, $...
- 133
16
votes
1
answer
819
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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 ...
- 38.2k
0
votes
0
answers
175
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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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