Apologies for the brief nature of this question, but it is something that I don't think was clarified in a previous post on this topic – Finding a spanning set for a null space.
When we say that the null space of the matrix $A$ is equal to the span of some set of vectors:
$N(A) = span(\vec v_1, \vec v_2,$ $… \vec v_n)$
Are we effectively saying that:
$A(\vec v_1) + A(\vec v_2) +$ $… A(\vec v_n) = \vec 0$
Best Answer
To be precise, $$N(A)=\text{Span}\{v_1,...,v_n\},$$ indeed implies that $$Av_1+\cdot +Av_n=0,$$ or even that for all $\alpha _1,...,\alpha _n\in\mathbb R$ (if you work in a $\mathbb R-$vector space) that $$\alpha _1Av_1+...+\alpha _nAv_n=0.$$
But the converse doesn't hold. If $N(A)$ has $n$ dimension, that $v_1,...,v_n\in N(A)$ and that there are free, then indeed $N(A)=\text{Span}\{v_1,...,v_n\}$. But $Av_1+...+Av_n=0$ just allow you to conclude that $v_1+...+v_n\in N(A)$ and unfortunately nothing more.