Let $v=a\_0+a\_1x+...+a\_nx^n$, if $a\_0+a\_1x+...+a\_nx^n=0$, then all coefficient $a\_i\in F$ are $0$ as variables are all with different degrees so it is linearly independent (Here I am no sure how to correctly prove it). In addition, although the span looks apparent, but I am still not sure how to show that. Could someone help?