Questions about the properties of vector spaces and linear transformations, including linear systems in general.
Questions tagged [linear-algebra]
5673 questions
67
votes
4 answers
explicit big linearly independent sets
In the following, I use the word "explicit" in the following sense: No choices of bases (of vector spaces or field extensions), non-principal ultrafilters or alike which exist only by Zorn's Lemma (or AC) are needed. Feel free to use similar…
Martin Brandenburg
- 61,443
62
votes
9 answers
Can a vector space over an infinite field be a finite union of proper subspaces?
Can a (possibly infinite-dimensional) vector space ever be a finite union of proper subspaces?
If the ground field is finite, then any finite-dimensional vector space is finite as a set, so there are a finite number of 1-dimensional subspaces, and…
Anton Geraschenko
- 23,718
43
votes
6 answers
"A gentleman never chooses a basis."
Around these parts, the aphorism "A gentleman never chooses a basis," has become popular.
Question. Is there a gentlemanly way to prove that the natural map from $V$ to $V^{**}$ is surjective if $V$ is finite-dimensional?
As in life, the exact…
Richard Dore
- 5,205
36
votes
5 answers
Is there a version of inclusion/exclusion for vector spaces?
I am asking for a way to compute the rank of the 'join' of a bunch of subspaces whose pairwise intersections might be non-zero. So in the case n=2 this is just $\dim(A_1+A_2) = \dim(A_1) + \dim(A_2) - \dim(A_1\cap A_2)$. For general $n$, I don't…
mingming
- 509
33
votes
3 answers
Singular values of matrix sums
This is a follow-up question to this one about eigenvalues of matrix sums. Suppose you have matrices $A$ and $B$, and know their singular values. What can you say about the singular values of $A+B$?
For Hermitian matrices and eigenvalues, this…
Peter Shor
- 6,272
31
votes
1 answer
Can you partition the sphere into orthonormal bases?
I've been writing some linear algebra problems with colleagues, and the following question occurred to us:
Let $S^2$ denote the unit sphere in $\mathbb{R}^3$. Does there exist a
partition $S^2=\bigsqcup_\alpha B_\alpha$ such that each $B_\alpha$…
Dustin G. Mixon
- 7,563
25
votes
3 answers
Largest number of vectors with pairwise negative dot product
What is the largest $m$ such that there exist $v_1,\dots,v_m \in \mathbb{R}^n$ such that for all $i$ and $j$, $1\leq i< j\leq m$, we have $v_i \cdot v_j < 0$.
Also, the preview screen is not displaying set braces for LaTeX. Is that just the…
user5810
23
votes
5 answers
Smallest non-zero eigenvalue of a (0,1) matrix
What's the smallest absolute value possible of a non-zero eigenvalue of an $n$ by $n$ square matrix whose entries are either $0$ or $1$ (all operations are over $\mathbb{R}$)? I would be interested in estimates or bounds as I imagine an exact…
Simd
- 3,195
20
votes
4 answers
Rings over which every module is free
We know that modules over skewfields are free. Is the converse true? In other words, is it true that a nontrivial ring over which every module is free is a skewfield?
If the ring A is commutative, then writing that for any proper ideal I of A, the…
Benoit Jubin
- 1,049
20
votes
2 answers
Spectral radius on 0-1 vectors.
Let $A$ be an $n\times n$ symmetric substochastic matrix (i.e. all entries are non-negative and each row adds up to $1$ or less).
Call a vector $v \in \mathbb{R}^n$ an indicator if $v \neq 0$ and each coordinate of $v$ is either $0$ or $1$. Define…
Pablo Lessa
- 4,194
19
votes
4 answers
Problems concerning subspaces of $M_n(\mathbb{C})$
Let $M_n(\mathbb{C})$ denote the n times n matrices over the complex number field. N be a subspace of $M_n(\mathbb{C})$.
If all the matrices in N are non-invertible , what is the maximum the dimension of N can be?
If all the matrices in N commute…
zhaoliang
- 353
18
votes
3 answers
Torsion in GL_n(Z)
Fix some $n \geq 3$. It's hopeless to classify the torsion elements in $\text{GL}_n(\mathbb{Z})$, but I have a couple of less ambitious questions. It's well-known that for any odd prime $p$, the map $\phi_p : \text{GL}_n(\mathbb{Z}) \rightarrow…
Andy Putman
- 43,430
14
votes
2 answers
Involutions in GL_n(Z)
Is there a classification of involutions in $\text{GL}_n(\mathbb{Z})$?
Here's some more details about what I mean. Consider $f \in \text{GL}_n(\mathbb{Z})$ such that $f^2=1$. Regard $f$ as an automorphism of $\mathbb{Z}^n$. Extend $f$ to an…
New to this
- 245
14
votes
3 answers
angle between subspaces
Let $E$ be a finite dimensional real inner product space. I want to define the angle between two subspaces $E_1$ and $E_2$. This has a fairly obvious meaning if $E_1$ is 1-diemsnional: Take the angle between any non-zero vector in $E_1$ and its…
John Hubbard
- 411
13
votes
3 answers
Relationship between determinants.
Given an orthogonal matrix $O$ with dimensions $4n \times 4n$ and $\det O = -1$, how to prove that
$\det[O_{11} - O_{22} + i (O_{12} + O_{21})] = 0$?
Here $O$ is a block matrix $[[O_{11}, O_{12}], [O_{21}, O_{22}]]$, and all blocks have equal…
Anton Akhmerov
- 605