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Results tagged with vector-spaces
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user 22989
46
votes
Accepted
Do vector spaces without choice satisfy Cantor-Schroeder-Bernstein?
Without the axiom of choice, it is possible that there is a vector space $U\neq 0$ over a field $k$ with no nonzero linear functionals.
Let $V$ be the direct sum of countably many copies of $U$, and $ …
- 35.2k
11
votes
Properties of vector spaces without AC
I think vector spaces must still be flat (tensor product is exact). I don't think any of the steps in the following proof use choice, although it's quite possible I'm mistaken:
Finite dimensional vec …
- 35.2k
5
votes
Accepted
Linearly independent family of sequences of rationals with a cardinal equal to the continuum
Choose a bijection $\alpha:\mathbb{N}\to\mathbb{Q}$, and for each $x\in\mathbb{R}$ let
$$a(x)_i=\begin{cases}0&\mbox{ if $\alpha(i)<x$}\\1&\mbox{ if $\alpha(i)\geq x$}\end{cases}.$$
Then the set of s …
- 35.2k
4
votes
Accepted
Maximal commutative subrings of the endomorphism ring of a vector space
Even for a $2$-dimensional vector space (so we're looking at subrings of the ring $M_2(\mathbb{F})$ of $2\times2$ matrices over $\mathbb{F}$) there are nonisomorphic maximal commutative subrings.
Bo …
- 35.2k