The null space vector of matrix means that it is orthogonal to the the matrix?

Question

If now there is a matrix $\mathbf H$,and assume it has null space vector,that is $\vec n \mathbf H=\vec 0$.

Now if i want to find a vector $\vec v$ which is orthogonal to $\mathbf H$.So is $\vec v =\vec n$?

Answer 1

No. $nH=0$ and $vH=0$ does not imply $v=n$. For example, $nH=0$ and $(2n)H=0$ but $n\neq 2n$ (assuming $n\neq0$). If only the zero vector $n=0$ gives $nH=0$, then $nH=0$ and $vH=0$ implies $n=v=0$.