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Questions tagged [linear-algebra]

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

9
votes
5answers
800 views

Non-conjugate words with the same trace

Let n>=2, p a large prime, G = SL_n(Z/pZ). If n=2, there are words that, while not conjugate in the free group, do have identical trace in G. For example, tr(g h^2 g^2 h)= tr(g^2 h^2 g h) for all g, ...
19
votes
4answers
3k views

Linear transformation that preserves the determinant

It seems "common knowledge" that the following holds: Let $T$ be a linear transformation on nxn matrices with complex coefficients that preserves the determinant. Then there exists matrices U and V ...
18
votes
4answers
3k views

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, ...
2
votes
1answer
765 views

Theta Functions and Cousins

So I am (barely) familiar with the construction of the theta function of an integral lattice $L$. The theta function, as I understand it, is defined as the function which takes a variable $z$ and ...
1
vote
1answer
259 views

Request for info on the space of commuting matrices preserving a flag.

Fix a flag of subspaces V1 in V2 in V3, etc. all in Cn. Consider the space of pairs of commuting linear transformations A and B such that: A preserves the flag (i.e. A(Vi) is in Vi), and B strictly ...
40
votes
6answers
7k views

“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 $V$...
5
votes
2answers
1k views

What's the correct notion of determinant of a bilinear pairing?

By a pairing on a vector space $V$, I mean a linear map $A : V \otimes V \to R$. If $V$ is $n$-dimensional ($n < \infty$), then I can define the determinant of $A$ by considering the canonical ...
17
votes
2answers
2k views

Zeta-function regularization of determinants and traces

The short answer to my question may be a pointer to the right text. I will give all the background I know, and then ask my questions in list form. Let A be an operator (on an infinite-dimensional ...
25
votes
5answers
9k 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\langle u,v\rangle ...
7
votes
6answers
5k views

How to approximate a solution to a matrix equation? [closed]

Suppose a matrix equation Ax = b has no solution (b is not in the column space of A) How ...
-1
votes
2answers
3k views

When does a matrix equation have a solution? [closed]

Given a matrix equation $Ax=b$ where $A$ is a matrix and $b$ is a column vector, what is a condition that would ensure that there is a column vector $x$ that satisfies the equation? Assume the ...
48
votes
9answers
13k 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 ...