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?

Hint:For the homogeneous system $Ax=0$, there exists $n-m$ linearly independent solutions, where $n$ is the number of variables and $m$ equals to Rank($A$).
