Null space of Sum of matrces

Question

I want to find the null space of sum of matrices. I believe the following statement is correct:

N(A+B+C+...) ⊇ N(A) ∩ N(B) ∩ N(C) ∩ ...

However, I cannot find a proof. I would appreciate a theoretical proof or any textbook/publication reference that I can cite. In my case, the matrices are typically square, full-rank and symmetric.

Answer 1

If $x \in \mathcal N(A_j)$ for $j = 1, 2, \ldots$, then $0 = A_1 x = A_2x = \ldots$, so $A_1x + A_2 x + A_3x = (\sum_j A_j)x = 0$. Therefore, $x \in \mathcal N(\sum_j A_j)$.

Note: no additional matrix structure assumptions were required.