Is $n−m$ always the largest possible number of linearly independent vectors in this vector space? Fix linear independent vectors $a_1, . . . , a_m ∈ \mathbb{R}^n$, and let $S$ be the vector space of such that 
$S:=$ {$x∈ \mathbb{R}^n:a_i⋅x=0∀1≤i≤m$} .
The vector space $S$ always has at least $n − m$ linearly independent vectors (solutions to $a_i⋅x=0$).
Is $n−m$ always the largest possible number of linearly independent solutions or is it possible for me to find a counterexample?

Yes, it is always true. Let $b_1,\dots,b_r$ be a basis for $S$. You can show that the set $\{a_1,\dots,a_m,b_1,\dots,b_r\}$ is linearly independent (check this), which implies that $m+r\le n$, so $r\le n-m$.
