vector space union

let $\text{[math]}$ be a vector space over $\text{[math]}$ , and let $\text{[math]}$ be subvectors
the broken link: blk:def-union of 2 vector spaces is given by
$\text{[math]}$

the union operation on vector spaces doesnt necessarily yield another vector space.

let $\text{[math]}$ and $\text{[math]}$ , we have $\text{[math]}$ is a collection of vectors of 2 elements that have a coordinate of 0, this set isnt enclosed under addition:
$\text{[math]}$

$\text{[math]}$ is a vector subspace of $\text{[math]}$ iff $\text{[math]}$ or $\text{[math]}$ .