vector subspace

let $\text{[math]}$ be a vector space, if $\text{[math]}$ we say $\text{[math]}$ is a subspace of $\text{[math]}$ and we write $\text{[math]}$ only if the following is true:

$\text{[math]}$ ,
for $\text{[math]}$ its also true that $\text{[math]}$ (addition closure),
for $\text{[math]}$ (scalar multiplication closure).

if $\text{[math]}$ then $\text{[math]}$ is a vector space

let $\text{[math]}$ then $\text{[math]}$ , let $\text{[math]}$ then $\text{[math]}$
to be continued..

strictly speaking, a subspace is a vector space included in another larger vector space. therefore, all properties of a vector space, such as being closed under addition and scalar multiplication still hold true when applied to the subspace.

let the set of vectors $\text{[math]}$ , consider the following set:
$\text{[math]}$ by the definition of this set, a vector $\text{[math]}$ is in the set $\text{[math]}$ only if $\text{[math]}$
for $\text{[math]}$ to be a subspace of $\text{[math]}$ , it has to meet 3 conditions:
condition 1: $\text{[math]}$ , in this case $\text{[math]}$
$\text{[math]}$ condition 2: addition closure, meaning if $\text{[math]}$ then $\text{[math]}$
$\text{[math]}$ condition 3: multiplication closure, meaning if $\text{[math]}$ then $\text{[math]}$
$\text{[math]}$