Is the sum of null space still a null space for the same matrix?

Question

Is the sum of null space still a null space for the same matrix?I mean if $\vec v_1$ and $\vec v_2$ are both null space vector for matrix $\mathbf H$,now if $\vec w=\alpha \vec v_1 +\beta \vec v_2$,now matter what the $\alpha$ and $\beta$ are,is $\vec w$ still a null space vector of $\mathbf H$?

Answer 1

Yes, because $H(\alpha \vec {v_1}+\beta \vec {v_2})=\alpha H(\vec {v_1})+\beta H(\vec {v_2})=\alpha 0+ \beta 0=0$.

Answer 2

Yes, this is an implication (or part of the proof) that the null space is a linear subspace.