**141**

votes

**20**answers

19k 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 ...

**59**

votes

**8**answers

10k 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 ...

**57**

votes

**19**answers

19k 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. ...

**54**

votes

**3**answers

10k 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 ...

**53**

votes

**9**answers

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 ...

**52**

votes

**6**answers

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, ...

**52**

votes

**3**answers

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 ...

**50**

votes

**11**answers

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

**5**answers

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 ...

**45**

votes

**2**answers

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 ...

**45**

votes

**0**answers

2k views

### A naive question about the double 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 ...

**44**

votes

**7**answers

3k views

### How to prove this determinant is positive?

Given the matrices $ A_i=
\biggl(\begin{matrix}
0 & B_i \\
B_i^T & 0
\end{matrix} \biggr)
$, where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove that $\det(I + ...

**43**

votes

**1**answer

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 ...

**41**

votes

**1**answer

1k 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 ...

**38**

votes

**4**answers

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 ...

**36**

votes

**19**answers

7k 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 ...

**36**

votes

**4**answers

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 ...

**35**

votes

**16**answers

30k 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 ...

**34**

votes

**6**answers

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 ...

**34**

votes

**1**answer

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 $A\otimes_k B$
...

**30**

votes

**21**answers

9k 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 ...

**30**

votes

**3**answers

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

**2**answers

2k 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:
...

**28**

votes

**3**answers

2k views

### A curious determinantal inequality

In my study, I come across the following curious inequality, which I do not know a proof yet (so I am asking it here).
Let $A, B$ be $n\times n$ (Hermitian) positive definite matrices. It is very ...

**28**

votes

**2**answers

2k 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

**16**answers

7k 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, ...

**27**

votes

**4**answers

914 views

### Why there is a relation among the second-order minors of a symmetric $4\times 4$ matrix?

A $4\times 4$ symmetric matrix
$$
\left(
\begin{array}{cccc}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{12} & a_{22} & a_{23} & a_{24} \\
a_{13} & a_{23} & a_{33} & ...

**27**

votes

**7**answers

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 ...

**27**

votes

**2**answers

967 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

**3**answers

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

**4**answers

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

**2**answers

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 ...

**26**

votes

**2**answers

959 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

**1**answer

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

**10**answers

5k 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

**13**answers

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 ...

**25**

votes

**9**answers

7k 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 ...

**25**

votes

**6**answers

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 ...

**25**

votes

**4**answers

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 ...

**25**

votes

**7**answers

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 ...

**23**

votes

**6**answers

4k 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 ...

**23**

votes

**6**answers

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 ...

**22**

votes

**6**answers

14k views

### Determinant of sum of positive definite matrices

Say $A$ and $B$ are symmetric, positive definite matrices.
I've proved that $\det(A+B) \ge \det(A) + \det(B)$ in the case that $A$ and $B$ are two dimensional.
Is this true in general for ...

**22**

votes

**12**answers

7k 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 ...

**22**

votes

**16**answers

3k 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

**5**answers

6k 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

**4**answers

1k views

### Dividing by two in the category of vector spaces

Does every invertible linear map $M$ between $V \oplus V$ and $W \oplus W$ naturally yield an invertible linear map $L$ between $V$ and $W$?
Here "naturally" means "in an $GL(V) \times ...

**22**

votes

**1**answer

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 ...

**22**

votes

**3**answers

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

**3**answers

1k views

### Example for column rank $\neq$ row rank

The proof that column rank = row rank for matrices over a field relies on the fact that the elements of a field commute. I'm looking for an easy example of a matrix over a ring for which column rank ...