a result from rank-nullity theorem

Question

Suppose that we have $m$ vectors in $\mathbb{R}^n,$ $v_1,v_2,...,v_m$, such that

$X^Tv_i=0,~\forall i.$

Prove that if $\dim(\langle v_1,...,v_m\rangle)=l$ then the answers of above equation ($X$) is $n−l$ dimension.

Can anyone prove this with Rank-nullity theorem?

Answer 1

You are looking for the $\dim (\text{nul}(V))$ with $V = (v_1,\ldots,v_m)$