finite vector space

[[concept]]

finite vector space

$V$ is finite if every linearly independent set is finite.

ie, if for every set $E \subseteq V$ such that $\forall v_{1}, \dots, v_{N} \in E$ $\sum_{i = 1}^{N} a_{i} v_{i} = 0 ⟹ a_{1} = \dots = a_{N} = 0$ and $E$ has finite cardinality, then $V$ is finite-dimensional. Otherwise, $V$ is infinite-dimensional.

Example

$C ([0, 1])$ is infinite dimensional because $E = {f_{n} (x) = x^{n} ∣ n \in N \cup {0}}$ is a linearly independent set with non-finite cardinality.

Mentions

Mentions

TABLE file.mday as "Last Modified"
FROM [[]]
 
FLATTEN choice(contains(artist, this.file.link), 1, "") + choice(contains(author, this.file.link), 1, "") + choice(contains(director, this.file.link), 1, "") + choice(contains(source, this.file.link), 1, "") as direct_source
 
WHERE !direct_source
SORT file.mday ASC
SORT file.name ASC

const { dateTime } = await cJS()
 
return function View() {
	const file = dc.useCurrentFile();
	return <p class="dv-modified">Created {dateTime.getCreated(file)}     ֍     Last Modified {dateTime.getLastMod(file)}</p>
}