All Questions
Tagged with linear-algebra fields
23 questions
43
votes
18
answers
5k
views
Results in linear algebra that depend on the choice of field
Linear algebra as we learn it as undergraduates usually holds for any field (even though we usually learn it for the complex, or real, numbers).
I am looking for a list of concepts, and results, in ...
13
votes
3
answers
2k
views
Which polynomials are determinants of a symmetric matrix with linear entries?
Let $k$ be a field. Can each degree $n$ polynomial $P(t) \in k[t]$ be written as the determinant of the matrix $A + tB$, where $A$ and $B$ are two symmetric $(n \times n)$-matrices with entries in $k$?...
- 5,163
12
votes
5
answers
1k
views
Does k(X) have a k-basis for every set X, without AC?
This question is inspired by Pace Nielsen's recent question Does a left basis imply a right basis, without AC?.
For any field $k$, the field $k(x)$ of rational functions in one variable has an ...
- 35.2k
11
votes
1
answer
2k
views
Sum of commuting semisimple operators
Let $V$ be a finite dimensional vector space over a field $K$. An operator $T:V\to V$ is called semi-simple if every $T$-invariant subspace of $V$ has a $T$-invariant complement(for algebraically ...
- 2,233
10
votes
2
answers
2k
views
When is a bilinear form equivalent to a trace form?
Associated to a finite, separable field extension $L/K$, there is a natural nondegenerate bilinear form, the trace form, defined by $$\langle x,y \rangle := \mathrm{Tr}_{L/K}(xy)$$
Now, given a ...
- 1,142
8
votes
4
answers
3k
views
Finite dimensional vector spaces over a complete but not-necessarily-valued field
I'm essentially reopening this old question of Ricky Demer which was never fully answered.
Essentially the original question: Suppose we have a topological field $F$ which is complete, Hausdorff, and ...
- 2,585
8
votes
2
answers
746
views
Field extension containing the eigenvectors of a Hermitian matrix
Let H be a (finite-dimensional) Hermitian matrix with algebraic numbers for its entries, all of which lie in some minimal field extension of the rational numbers; call this field ℚ(H) for short. ...
- 2,649
7
votes
3
answers
2k
views
Is there a field which is the union of finitely many proper subfields?
Is there a field which is the union of finitely many proper subfields?
- 79
6
votes
2
answers
462
views
Splitting subspaces and finite fields
Hellow. I'm sure that the following is truth, but I can't prove it.
Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a chain of finite fields and
$A = \{\theta\...
5
votes
1
answer
204
views
Is it true that every subspace contain a primitive element?
Let $R = GF(q), q = p^r$, be a field with identity $e$, where $p$ is a prime number. Let $S=GF(q^n)$ be an extension of $R, n\geq 2$ and $K = GF(q^{mn})$ be an extension of $S$, where $m$ is prime. ...
4
votes
1
answer
2k
views
Fields whose embeddings into the complex numbers are invariant under complex conjugation
Is there a general notion/description of fields $K$ such that the image of any embedding $K \hookrightarrow \mathbb{C}$ is invariant under complex conjugation, thus inducing an involution on $K$ which ...
- 41
4
votes
1
answer
354
views
Does the weak approximation theorem hold for general topological fields?
The weak approximation theorem states that given a field $F$ and nontrivial inequivalent absolute values $|\cdot|_1,\ldots,|\cdot|_n,$ and letting $F_i$ denote $F$ with the topology from $|\cdot|_i$, ...
- 2,585
2
votes
2
answers
1k
views
What is it called if a vector space doesn't have an additive inverse?
so, you have, for any two members of the algebraic structure A and B and any nonnegative real values a, b:
two operations: * and +, such that
a*A + b*A = (a+b)*A is in the structure
A + B = B + A ...
- 598
2
votes
1
answer
987
views
Surjectivity of bilinear forms.
It is not uncommon to describe interesting classes of field extensions by declaring that an extension $L|K$ belongs to that class if some type of problem with $K$-coefficiens has a property over $L$ ...
- 4,015
2
votes
0
answers
59
views
Tensor product of two transcendental flat algebras is not a field?
I'm considering the correctness of the following assertion, which is related to linear disjointness (I'm trying to generalize it to subalgebras), What does "linearly disjoint" mean for ...
- 173
2
votes
0
answers
92
views
System of linear equations in positive characteristic
Let $K$ be a field of positive characteristic $p$. Consider the system of $\mathbb F_p$-linear equations
$$\left\{\begin{array}{ccl}
a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n&=&b_1\\
a_{11}x^p_1+a_{...
- 3,998
2
votes
0
answers
214
views
Is there a symmetric basis for $\mathbf{Q}(x,y)$?
Consider $\mathbf{Q}(x,y)$, the rational functions in $x$ and $y$, as a vector space over $\mathbf{Q}$.
Let $\sigma$ be the map interchanging $x$ and $y$. Is there a basis for $\mathbf{Q}(x,y)$ ...
user44143
1
vote
0
answers
246
views
Frobenius twist of a field
Let $k$ be a field of characteristic $p>0$ (not necessarily perfect). Consider the Frobenius endomorphism $F : k \to k$, $x \mapsto x^p$. I am curious about what happens when we take $k$ as a $k$-...
1
vote
0
answers
268
views
Determinant and restriction of scalar
Let $E/F$ be a finite separable extension of fields, and $V$ a finite dimensional vector space over $E$. Let $T\in\operatorname{End}_EV$ be a linear operator on $V$, and let $\det(T)$ be its ...
- 833
1
vote
0
answers
107
views
Complementation in an extension field
If $E$ is an extension field of $F$, is $F$ necessarily (without assuming the axiom of choice) complemented as a vector subspace of $E$? (Of course the answer is easily yes if the extension is finite....
- 11.4k
0
votes
1
answer
164
views
On separable field extensions [closed]
Let $F\subseteq K$ be a finite separable field extension with $a_1,..., a_n$ an $F$-basis for $K$. Is it true that the matrix $A := [\mbox{tr}(a_ia_j)]$ is non-singular ?
- 447
0
votes
0
answers
108
views
Generalization of SVD algorithm
Let $K$ be a field, $A\in K^{n\times m}$ and $\lVert \cdot \rVert$ the Euclidean norm. Consider the problem: Find a $v\in K^m$ such that
\begin{align}
\lVert Av\rVert=\min_{\lVert x\rVert=1}\lVert Ax\...
- 157
-1
votes
1
answer
247
views
Maximal commutative subrings of the endomorphism ring of a vector space
Let $\mathbb{F}$ be a field, and $\mathbf{V}$ a possibly uncountably generated $\mathbb{F}$-vector space. Let $\mbox{End}_\mathbb{F}(\mathbf{V})$ be the endomorphism ring of $\mathbf{V}$. That the ...
- 1,959