Lines in $\mathbb{F}_3^n$

Question

Considering the vector space $\mathbb{F}_3^n$, I need to show that, given three vectors $u,v,w$, the set $\{u,v,w\}$ is an affine subspace if and only if $\forall i\in\{1,...,n\}:u_i+v_i+w_i=0$.

Answer 1

(I will refer to them as $v_0, v_1, v_2$ instead of $u, v, w$ for convenience). If they are an affine subspace, we may write $$ v_i = a + i w $$ for some $a, w \in \mathbb{F}_3^n$. In more detail, your space consists of at most three points, and thus must be $0$- or $1$-dimensional. So it fits on a line or $w=0$; either way the above holds. Then $$ v_0 + v_1 + v_2 = 3a + (0+1+2)w = 3(a + w) = 0. $$ Conversely, suppose $v_0 + v_1 + v_2 = 0$. Let $a = v_0$ and $w = v_1 - v_0$. Then certainly $v_i = a + iw$ for $i = 0, 1$. Also, $$ v_2 = -v_0 - v_1 = -2a -w = a+2w $$ as desired.