The vector-spaces tag has no wiki summary.

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**24**answers

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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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**0**answers

12 views

### 3D Vector projection on a Plane [migrated]

I want to Project a Vector on to a Plane. Assume, you have a Central Point (1,1,1) and you want to move (0,0,3) in z-direction. How can I project the end of this movement (point) on a plane with ...

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**1**answer

56 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)
...

**1**

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**1**answer

138 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 $V$ ...

**2**

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**1**answer

84 views

### 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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**0**answers

33 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 ...

**2**

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**1**answer

82 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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**1**answer

214 views

### 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 ...

**4**

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**2**answers

363 views

### 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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**2**answers

257 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} ...

**4**

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**1**answer

137 views

### 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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**0**answers

108 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 $\mathbb R^n$? ...

**1**

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**3**answers

190 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 ...

**3**

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**0**answers

77 views

### 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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**1**answer

193 views

### 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 ...

**3**

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**1**answer

309 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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**0**answers

315 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 ...

**0**

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**0**answers

212 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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**0**answers

121 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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**1**answer

338 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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**1**answer

215 views

### 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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**1**answer

146 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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**1**answer

99 views

### 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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**1**answer

184 views

### 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} : ...

**2**

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**0**answers

272 views

### 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 ...

**3**

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**1**answer

290 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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**2**answers

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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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**1**answer

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### 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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**1**answer

440 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

707 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.

**2**

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**1**answer

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### 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 ...

**3**

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**1**answer

683 views

### 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 ...