$W_1\cap W_2=\{0\}$ is equivalent to $V_1\cap V_2=\{0\}$, where $W_i$ ($V_i$) is the space of row (column) vectors.

How to show?

Precisely, if $A_1,A_2$ are two $m\times n$ matrices. Let $W_1,W_2$ be the space of row vectors of $A_1$ and $A_2$ respectively; $V_1,V_2$ be the space of column vectors of $A_1$ and $A_2$ respectively. Show that $W_1\cap W_2=\{0\}$ is equivalent to $V_1\cap V_2=\{0\}$.

It sounds like $rank\left(A\atop B\right)=rank(A)+rank(B)$ if and only if $W_1\cap W_2=\{0\}$. But how to proceed?