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Results tagged with linear-algebra
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user 164965
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
1
vote
Accepted
If a vector space has a basis then its dual vector space has a basis
Here are two statements you might find interesting:
With base field $k\subseteq\mathbb C$ your axiom implies that any free $\mathbb Z$-action has a choice of representatives.
For each prime $p,$ ther …
- 1,916
3
votes
Accepted
Example of an inverse system which suddenly "jumps" in size in a specific "controlled" way?
Here is my argument, which assumes $|2^\omega|<|2^{\omega_1}|$ and uses an infinite base field. See Tim Campion's answer https://mathoverflow.net/a/376790/164965 for the general case.
Take the base fi …
- 1,916
11
votes
Accepted
Existence of translation-invariant basis on $C_c(\mathbb R)$
No, there is no shift-invariant basis of $V=C_c(\mathbb R).$ I'll use the formulation in YCor's answer, so we need to show that $V$ is not a free $B$-module where $B=\mathbb C[T^r:r\in \mathbb R],$ wi …
- 1,916