Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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122
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19answers
16k views

Geometric Interpretation of Trace

This afternoon I was speaking with some graduate students in the department and we came to the following quandry; Is there a geometric interpretation of the trace of a matrix? This question ...
55
votes
19answers
17k views

Why were matrix determinants once such a big deal?

I have been told that the study of matrix determinants once comprised the bulk of linear algebra. Today, few textbooks spend more than a few pages to define it and use it to compute a matrix inverse. ...
53
votes
9answers
6k views

Geometric proof of the Vandermonde determinant?

The Vandermonde matrix is the $n\times n$ matrix whose $(i,j)$-th component is $x_j^{i-1}$, where the $x_j$ are indeterminates. It is well known that the determinant of this matrix is $$\prod_{1\leq ...
53
votes
6answers
5k views

Existence of a zero-sum subset

Some time ago I heard this question and tried playing around with it. I've never succeeded to making actual progress. Here it goes: Given a finite (nonempty) set of real numbers, $S=\{a_1,a_2,\dots, ...
53
votes
3answers
2k views

Does linearization of categories reflect isomorphism?

Given a category $C$ and a commutative ring $R$, denote by $RC$ the $R$-linearization: this is the category enriched over $R$-modules which has the same objects as $C$, but the morphism module between ...
49
votes
8answers
8k views

Should the formula for the inverse of a 2x2 matrix be obvious?

As every MO user knows, and can easily prove, the inverse of the matrix $\begin{pmatrix} a & b \\\ c & d \end{pmatrix}$ is $\frac{1}{ad - bc} \begin{pmatrix} d & -b \\\ -c & a ...
47
votes
11answers
5k views

Why are matrices ubiquitous but hypermatrices rare?

I am puzzled by the amazing utility and therefore ubiquity of two-dimensional matrices in comparison to the relative paucity of multidimensional arrays of numbers, hypermatrices. Of course ...
45
votes
2answers
2k views

vector balancing problem

I believe the solution posted to the arXiv on June 17 by Marcus, Spielman, and Srivastava is correct. This problem may be hard, so I don't expect an off-the-cuff solution. But can anyone suggest ...
44
votes
5answers
3k views

Does this formula have a rigorous meaning, or is it merely formal.

I hope this problem is not considered too "elementary" for MO. It concerns a formula that I have always found fascinating. For, at first glance, it appears completely "obvious", while on closer ...
43
votes
3answers
8k views

Slick proof?: A vector space has the same dimension as its dual if and only if it is finite dimensional

A very important theorem in linear algebra that is rarely taught is: A vector space has the same dimension as its dual if and only if it is finite dimensional. I have seen a total of one proof ...
42
votes
1answer
3k views

Geometric interpretation of characteristic polynomial

The coefficients of lowest and next-highest degree of a linear operator's characteristic polynomial are its determinant and trace. These have well-known geometric interpretations. But what about its ...
38
votes
1answer
764 views

Determinant of a determinant

Consider an $mn \times mn$ matrix over a commutative ring $A$, divided into $n \times n$ blocks that commute pairwise. One can pretend that each of the $m^2$ blocks is a number and apply the $m ...
36
votes
4answers
4k views

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 ...
35
votes
4answers
4k views

Does the fact that this vector space is not isomorphic to its double-dual require choice?

Let $V$ denote the vector space of sequences of real numbers that are eventually 0, and let $W$ denote the vector space of sequences of real numbers. Given $w \in W$ and $v \in V$, we can take their ...
34
votes
1answer
1k views

Is this lemma in elementary linear algebra new?

Is anyone familiar with the following, or anything close to it? Lemma. Suppose $A$, $B$ are nonzero finite-dimensional vector spaces over an infinite field $k$, and $V$ a subspace of $A\otimes_k B$ ...
32
votes
19answers
6k views

Wonderful applications of the Vandermonde determinant

This semester I am assisting my mentor teaching a first-year undergraduate course on linear algebra in Peking University, China. And now we have come to the famous Vandermonde determinant, which has ...
31
votes
6answers
2k views

Is the non-triviality of the algebraic dual of an infinite-dimensional vector space equivalent to the axiom of choice?

If $V$ is given to be a vector space that is not finite-dimensional, it doesn't seem to be possible to exhibit an explicit non-zero linear functional on $V$ without further information about $V$. The ...
29
votes
21answers
8k views

Why linear algebra is fun!(or ?)

Edit: the original poster is Menny, but the question is CW; the first-person pronoun refers to Menny, not to the most recent editor. I'm doing an introductory talk on linear algebra with the ...
29
votes
3answers
2k views

How much linear algebra can be done with graphs?

Let G be a finite directed acyclic graph, with sources $A=\{a_1,\ldots,a_n\}$ and sinks $B=\{b_1,\ldots,b_n\}$, with edge weights $w_{ij}$. The weight of a directed path P is the product of weights of ...
29
votes
0answers
959 views

a naive question about the second dual of a vector space

Let $K$ be a field. Are there non-scalar endomorphisms of the endofunctor $$ V\mapsto V^{**}/V $$ of the category of $K$-vector spaces? I asked a related question on Mathematics Stackexchange, but ...
28
votes
2answers
1k views

Symmetric powers and duals of vector bundles in char p

Suppose that $X$ is a smooth projective variety (eg $P^n$) and $E$ is a vector bundle (eg the tangent bundle). If the characteristic is zero, then taking symmetric powers "commutes" with taking duals: ...
27
votes
2answers
888 views

Continuous functions $f$ with $f(A)$ linearly independent when $A$ is independent

Is there any characterization of continuous functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ such that for any linearly independent set $A$ (over the rationals) $f(A)$ is also linearly independent ?
27
votes
2answers
1k views

The probability for a symmetric matrix to be positive definite

Let me give a reasonable model for the question in the title. In ${\rm Sym}_n({\mathbb R})$, the positive definite matrices form a convex cone $S_n^+$. The probability I have in mind is the ratio ...
27
votes
3answers
2k views

Perron-Frobenius “inverse eigenvalue problem”

The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only ...
27
votes
4answers
1k views

Probability of zero in a random matrix

Let $M(n,k)$ be the set of $n\times n$ matrices of nonnegative integers such that every row and every column sums to $k$. Let $P(n,k)$ be the fraction of such matrices which have no zero entries, ...
26
votes
16answers
24k views

Linear Algebra Texts?

Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in very negative ways to ...
26
votes
2answers
880 views

What are the algebras for the double dualization monad?

Let $k$ be a field, and let $\mathbf{Vect}$ denote the category of vector spaces (possibly infinite-dimensional) over $k$. Taking duals gives a functor $(\ )^*\colon \mathbf{Vect}^{\mathrm{op}} \to ...
26
votes
1answer
1k views

solving linear equations made difficult

(Note: This is a what's-in-the-literature question, not a what's-mathematically-true question, but I believe both are considered valid kinds of MathOverflow question.) I saw this amusing derivation ...
25
votes
10answers
3k views

real symmetric matrix has real eigenvalues - elementary proof

Every real symmetric matrix has at least one real eigenvalue. Does anyone know how to prove this elementary, that is without the notion of complex numbers?
25
votes
6answers
1k views

Does seeing beyond the course you teach matter? The case of linear algebra and matrices

This question is indeed very important for me. Thus I hope you bear with my subjective explanations for a few minutes. I am an "excellent" lecturer, at least according to course evaluation forms ...
24
votes
9answers
6k views

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 ...
23
votes
15answers
6k views

Linear Algebra Proofs in Combinatorics?

Simple linear algebra methods are a surprisingly powerful tool to prove combinatorial results. Some examples of combinatorial theorems with linear algebra proofs are the (weak) perfect graph theorem, ...
23
votes
8answers
5k views

“A gentleman never chooses a basis.”

Around these parts, the aphorism "A gentleman never chooses a basis," has become popular. Is there a gentlemanly way to prove that the natural map from V to V** is surjective if V is finite ...
23
votes
2answers
1k views

When exactly and why matrix multiplication became a part of undergraduate curriculum?

The story about Heisenberg inventing matrices and matrix multiplication in 1925 is very well known and well documented. Few weeks later Born and Jordan picked this and recognized the matrix ...
22
votes
13answers
2k views

Elementary applications of linear algebra over finite fields

I'm teaching axiomatic linear algebra again this semester. Although the textbooks I'm using do everything over the real or complex numbers, for various reasons I prefer to work over an arbitrary ...
22
votes
16answers
2k views

functions satisfying “one-one iff onto”

Hello Everybody. I need some more examples for the following really interesting phenomenon: A function from the class ... is one-one iff it is onto. Some ...
22
votes
5answers
5k views

Linearity of the inner product using the parallelogram law

A norm on a vector space comes from an inner product if and only if it satisfies the parallelogram law. Given such a norm, one can reconstruct the inner product via the formula: 2<u,v> = |u + ...
22
votes
7answers
2k views

When is a monic integer polynomial the characteristic polynomial of a non-negative integer matrix?

Suppose $P(x)$ is a monic integer polynomial with roots $r_1, ... r_n$ such that $p_k = r_1^k + ... + r_n^k$ is a non-negative integer for all positive integers $k$. Is $P(x)$ necessarily the ...
22
votes
1answer
1k views

If $V$ is a vector space with a basis. $W\subseteq V$ has to have a basis too?

Suppose $V$ is a vector space, we say that $\mathcal B$ is a basis for $V$ if: Every $v\in V$ can be written as a linear combination of elements of $\mathcal B$; If $\sum\alpha_i b_i = 0$, where ...
21
votes
6answers
3k views

A slick proof of the Bruhat Decomposition for GL_n(k)?

On one of my exams last year, we were given a problem (we chose five or six out of eight problems) on an exam, the goal of which was to prove the Bruhat decomposition for $GL_n(k)$. I was one of the ...
21
votes
12answers
6k views

Why are tensors a generalization of scalars, vectors, and matrices?

Take two vector spaces $V$ and $W$ over a field $F$. One may form the tensor product $V\otimes W$ and it fulfills an universal property. Elements of $V\otimes W$ are called tensors and they are linear ...
21
votes
6answers
3k views

Is there a preferable convention for defining the wedge product?

There are different conventions for defininig the wedge product $\wedge$. In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$, in Spivak, we find ...
21
votes
8answers
9k views

Fast Matrix Multiplication

Suppose we have two $n$ by $n$ matrices over particular ring. We want to multiply them as fast as possible. According to wikipedia there is an algorithm of Coppersmith and Winograd that can do it in ...
21
votes
4answers
2k views

Using linear algebra to classify vector bundles over P^1

There is a theorem of Grothendieck stating that a vector bundle of rank $r$ over the projective line $\mathbb{P}^1$ can be decomposed into $r$ line bundles uniquely up to isomorphism. If we let ...
21
votes
3answers
2k views

Understanding zeta function regularization

I attended a talk this morning on Ray-Singer torsion, in which Rafael Siejakowski introduced zeta function regularization in a compelling way. The goal is to define the determinant of a positive ...
21
votes
2answers
635 views

Factorization of a real matrix into Hermitian x Hermitian. Is it stable ?

It is known (see Theorem 4.1.7 in R. Horn & C. Johnson) that every matrix $A\in M_n(\mathbb R)$ (real entries) can be written as the product $HK$ of two Hermitian matrices (complex entries). Of ...
21
votes
3answers
2k views

Splitting the determinant polynomial into linear factors - a Dedekind problem

Here's the question in a nutshell. For some $n\in\mathbb N$, we consider the polynomial $\det\left(\left(X_{i,j}\right) _ {1\leq i\leq n,\ 1\leq j\leq n}\right)\in\mathbb Z\left[X_{i,j}\mid 1\leq ...
21
votes
0answers
991 views

Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices

In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...
20
votes
7answers
2k views

Modern developments in finite-dimensional linear algebra

Are there any major fundamental results in finite-dimensional linear algebra discovered after early XX century? Fundamental in the sense of non-numerical (numerical results, of course, are still ...
20
votes
4answers
1k views

The sum of same powers of all matrices modulo p

The following is a problem from our department algebra competition for students: Non-question. An experimental-math geek was trying to raise all matrices $17\times17$ over the field with 17 ...