TITLE: Show $\operatorname{rank}(A) + \operatorname{rank}(B) \ge \operatorname{rank}(A+B)$
QUESTION [23 upvotes]: Show $\operatorname{rank}(A) + \operatorname{rank}(B) \ge \operatorname{rank}(A+B)$, where $A,B \in M_{m\times n}(\mathbb{F})$.

I'm trying to think in terms of linear transformations.
We can define $T_a, T_b:\mathbb{F}^n\rightarrow \mathbb{F}^m$.
I know that $\dim_{\mathbb F}\operatorname{Im} T_a, \dim_{\mathbb F}\operatorname{Im} T_b \le m$.
What should I do next?

REPLY [6 votes]: How about this: Let $rk A: =a$ , $rk B:=b$. Now we can do Gaussian elimination on both, to end with two matrices $A'$, $B'$ with , respectively, $a$ and $b$ non-zero rows. But the sum $A'+B'$ will have at most max{a,b} nonzero rows. Then $a+b \geq$ Max{$a,b$}