spanning set

consider the vector space $\text{[math]}$ , let $\text{[math]}$ , then $\text{[math]}$ is a spanning set of $\text{[math]}$ and $\text{[math]}$ is a finitely spanned set

$\text{[math]}$
$\text{[math]}$ because the third vector is the sum of the first two which makes it a linear combination of them

which means the first set is a spanning set but the second isnt

$\text{[math]}$ is spanned by the set $\text{[math]}$ :
$\text{[math]}$ $\text{[math]}$ is spanned by the following $\text{[math]}$ vectors:
$\text{[math]}$

$\text{[math]}$ is the space of polynomials over the field $\text{[math]}$ with a single polynomial $\text{[math]}$
$\text{[math]}$ isnt finitely spanned, to prove this, we assume in contradiction that the set $\text{[math]}$ spans $\text{[math]}$
a linear combination of polynomials cant give us a degree that is higher than the degree of the polynomial with the highest degree in said linear combination, we write:
$\text{[math]}$ let $\text{[math]}$ be the polynomial with the highest degree, $\text{[math]}$ , we know $\text{[math]}$ by necessity because $\text{[math]}$ contains all the polynomials over the field $\text{[math]}$
$\text{[math]}$

let $\text{[math]}$ be a vector space over the field $\text{[math]}$ , and let $\text{[math]}$ be sets of vectors, then $\text{[math]}$

if $\text{[math]}$ is finitely spanned, if $\text{[math]}$ then $\text{[math]}$ is finitely spanned too