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### Lebesgue measure of set of $y\in\mathbb{R}^n$ such that $x,y,Ay$ are linearly dependent

I've asked this question here on math.stackexchange, but I have been unable to solve this yet, so I'm hoping I can get some advice here.
Consider a vector $x\in \mathbb{R}^n$ and a real $n\times n$ ...

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**1**answer

47 views

### Find matrices which generate a given set of vectors

I've got a finite set of vectors in $Z^3$, which I know to be of the form $A^{i_1}B^{i_2}A^{i_3}B^{i_4}\dots v$ for some unknown vector $v\in Z^3$ and two unknown matrices $A$ and $B$ with integer ...

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### Analogous to a PDE but where independent variable is a function

Consider, as an example for my question, a density function $u(\boldsymbol{x},t)$ on a vector field $\boldsymbol{x}$ at some time $t$. The flow velocity vector of the density is given by ...

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72 views

### Help with notation for the state of a dynamical system defined by a PDE

Before my question let me briefly describe a simplified version of the dynamical system I'm working with. Suppose that I have a density function $m(\boldsymbol{x},t)$, that describes the abundance of ...

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### The Euler characteristic of Hilbert series

The Hilbert series of a graded vector space $V=\bigoplus_{n\mathbb Z}V_n$ is the (ordinary) generating function of the dimensions of its homogeneous components, $h_V(t)=\sum_{n\in\mathbb Z}t^n\dim ...

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73 views

### When a homogeneous map between vector spaces is also additive?

Suppose to have two real vector spaces $V$ and $W$ and an injective map $T:V\rightarrow W$ such that $T(\alpha v)=\alpha T(v)$ for all $v\in V$ and $\alpha \in\mathbb{R}$. Do there exist some ...

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767 views

### Does the linear automorphism group determine the vector space?

I was recently thinking about what it means to put structure on a set. It seems to me that, in my area (representation theory), the two main ways of imposing structure on a set $X$ are:
...

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148 views

### Linearly independent family of sequences of rationals with a cardinal equal to the continuum

I'm coming back to this question. Is it possible to have "an explicit" linearly independent family of sequences of rationals with a cardinal equal to the continuum?
PS: sorry for the duplicate on the ...

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57 views

### The space of sequences of rationals and its dimension [duplicate]

In the following page, I give an example of a vector space not isomorphic to its double dual.
I use the space $E$ of sequences of reals. Its dimension (over the field of the reals) is the one of the ...

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### Results true in a dimension and false for higher dimensions

Some theorems are true in vector spaces for a given dimension $n$ but become false in higher dimensions.
Here are two examples:
A positive polynomial not reaching its minimum. Impossible in ...

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62 views

### angle inequality in n-dimensional vector space [closed]

Does anyone has answer for the following doc product problem?
Let A,B,C be three vectors of magnitude of 1.
Let A*B = Cos(x) ( * means dot product)
B*C = Cos(y)
...

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**1**answer

145 views

### Linear map with two “incompatible” representations

Let $K$ be a field and let $V$ be the set of sequences $\{v_1,v_2,\dots\}$ of elements of $K$. If $A=\{a_1,a_2,\dots\}$ is also a sequence of elements of $K$, then it defines an endomorphism of $V$ ...

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**1**answer

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### Space of matrices B for which there is a solution to Bx=c for a given c

Let $F$ be a field, $k$ and $m$ natural numbers with $k \leq m$, and $c \in F^m$.
Is there some name for the set $\mathcal{B}_c = \{ B \in F^{m \times k}\, | \,\, \exists x \in F^k $s.t. $ Bx = c\}$ ...

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39 views

### Speed of convergence of vector expansions in non orthogonal basis

Suppose we have a finite-norm vector $X$ in a Hilbert space, and we want to construct its expansion in a certain (infinite) basis $V_k$, $X=\sum_k a_k V_k$. If the basis is orthonormal, then we know ...

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118 views

### Is an integral against a probability measure in the convex hull of the range?

This may be really obvious but I don't see it. Let $f:\Omega \to \mathbb R^n$ be integrable with respect to a probability measure $\mu$. Does it follow that $\int_\Omega f \, d\mu$ is in the convex ...

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### Sets of cardinalities of bases without choice

For a vector space $V$, let $BS(V)$ be the set of cardinalities (not necessarily $\aleph$s) of bases of $V$. Of course, in ZFC each $BS(V)$ is a singleton, but supposing the axiom of choice fails, it ...

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### Properties of vector spaces without AC

With AC, it is easy to see that any vector space is injective, and free, therefore alse flat and projective.
Without AC, vector spaces can be not free. Are they must be projective modules? Flat ...

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276 views

### Subset of vectors whose sum has a large norm

In Rudin - Real & Complex Analysis we have the following
Lemma 6.3. If $z_1, \ldots, z_n \in \mathbb{C}$ then there is a subset $S \subseteq \{1,\ldots,n\}$ for which
$$\left|\sum_{k \in S} ...

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### Finite generation of vector identities

This question is partially motivated by Looking for a comprehensive referece for vector identities, although that question may not be appropriate for MO.
Consider the set $\mathcal{E}$ of all valid ...

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121 views

### Grassmannian frames in the Grassmannian

I am new to the Grassmannian. I have read about Grassmannian frames in $\mathbb R^n$. My question is can we define Grassmannian frames in a Grassmannian space $Gr(k,n)$ just like in $\mathbb R^n$? ...

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**3**answers

201 views

### Formalism for moving from a metric space into a vector space for mathematical/statistical modeling given a data

I have a metric space $(X,d)$. I have a physical situation (data) where each physical entity corresponds to an $x \in X$.
I want to do some mathematical/statistical modeling of this data, but the ...

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### Is there a known generalization of the Schmidt decomposition based on a maximal set of “locally orthogonal” vectors?

I came across the following unusual generalization of the Schmidt decomposition in my work, which I describe below. I would like to know if this structure has been studied before so I can read more ...

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### Is Taylor expansion related to Helmholtz decomposition?

The Taylor expansion of a vector field $f(x)$ to the order of one is
$$f(x)=f(x_0)+Jf(x_0)\cdot\Delta x+o(\Delta x)$$
where $Jf$ is Jacobian of the vector field and $\Delta x=x-x_0$.
Suppose we ...

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**1**answer

424 views

### What is the significance of matrix ordered algebras?

I am trying to grok matrix ordered operator algebras, but I am having a hard time understanding their significance from the definition. Here is the definition (or at least, one way of stating it):
...

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328 views

### A conjecture about vector space (repost from math.SE)

This post is copied from math.SE in the following link:
http://math.stackexchange.com/questions/456398/a-conjecture-about-vector-space
I have posted the question two days ago, but receive no answer ...

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217 views

### Finding the effective maximum number of subspaces in a finite dimensional vector space

Hi mathoverflow community, may be some one may give me a hint on the following problem before I spend much time on brute force search.
For $q$ a prime number and $n=6$, let $\mathbb {F}_{q}^{n}$ be ...

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140 views

### Lattice basis reductions and finding minimal values

While reading several articles about lattice basis reduction I am left with a few questions.
For one, I came across this piece of text
Let $\alpha$ and $\beta \in \mathbb{R}$. Also let $X>0$ and ...

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353 views

### Visual pictures of rotation and torsion

In vector analysis / differential geometry we have rotation and torsion. The formalisms are certainly well known. But how could I best explain the geometric pictures and their difference (as a coach ...

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### Historical precursor for Peano's axioms of a linear space?

Peano is typically credited with giving the first abstract definition of a vector space (1888):
http://www-history.mcs.st-and.ac.uk/HistTopics/Abstract_linear_spaces.html
Apparently, Peano credits ...

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156 views

### Is there a wedge which operates on multiple vector spaces?

Let's say I have two vector spaces $V,W$ , and we have the graded algebras $\Lambda(V),\Lambda(W)$, each with an operation $\wedge$. I'd like to know if there are "many" $\wedge$ operators, or if ...

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### Minimal generating sets of monoids acting on finite vector spaces.

Let $V$ be a finite dimensional vector space over $\mathbb{Z}_2$ with a linear map $f_i : V \to V$ for each $i$ in some finite index set $I$.
Then one can always find some subset $G \subseteq V$ of ...

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### Endomomorphisms of Chain Complexes of vector spaces and determinants

Let $C_{\ast} : \cdots \to A_{2} \to A_{1} \to A_{0} \to 0$ be a chain complex of finite dimensional vector spaces over a field $K$.
And let $f_{\ast} : C_{\ast} \to C_{\ast}$ and $g_{\ast} : ...

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### Morphisms of Spectral Sequences and alternating products

Let $E_{a,b}^{r}, F_{a,b}^{r}$ be two (co)homologica first quadrant spectral sequences of vector spaces over a field $K$, and $f : E \to F$ be a morphism of spectral sequences.
Assume that morphisms ...

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348 views

### Intersection of vector spaces

Let $\{v_{1,j},\ldots, v_{n,j}\}$ be a basis of the $n$-dimensional vector space $V$ for $j=1\ldots k$ (and assume $2k\leq n$). Let $V_i$ be the subspace spanned by $v_{i,1},\ldots, v_{i,k}$ for ...

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### Ultrafilters over vector spaces

Perhaps my question is naive, but let me try.
Take a (real or complex) vector space $V$ and consider an ideal $\mathcal{I}$ of subsets of $V$ with the following property (call it (*)): for each ...

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703 views

### Who first proved that the dimension of a vector space is unique?

every vector space is known to have a basis (assuming the axiom of choice). This is attributed to Georg Hamel (http://de.wikipedia.org/wiki/Georg_Hamel). Moreover, any two bases have the same ...

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459 views

### Fuzzy vector similarity

Hi all,
I have two multi-dimensional vectors representing documents $\vec{a}$ and $\vec{b}$.
Considering cases where there is no overlap between $a$ and $b$ ($a \cap b = \emptyset $), traditional ...

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**2**answers

777 views

### How to prove the Hahn-Banach constructively

I am just wondering, how to prove the Hahn-Banach theorem constructively for a finite dimensional normed vector space.
Thanks in advance for any helpful answers.

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**1**answer

766 views

### When do 0-preserving isometries have to be linear?

Let $\langle \mathbf{V},+,\cdot,||.|| \rangle$ be a normed vector space over $\mathbb{R}$.
Let $f : \mathbf{V} \to \mathbf{V}$ be an isometry that satisfies $f(\mathbf{0}) = \mathbf{0}$ .
What ...

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### Inequalities and bounds for relating p-norms (Reference request)

Hello all, I'm trying to find a good resource for a discussion on the relation between say, the p-norm of a vector (from a finite dimensional vector space) and its Euclidean norm. In my search on the ...