I want to have a result ${\rm rank}(X) = {\rm rank}(X_1) + {\rm rank}(X_2)$ where $X=[X_1, X_2]$, $X_1,X_2 \neq 0$ and $X_1$ and $X_2$ are $n \times k_1$ and $n \times k_2$ with $n>k_1,k_2$.
I'm wondering under what condition this equality holds. If $X_1 a \neq X_2 b$ holds for any vectors $a \neq 0$ and $b \neq 0$, does the above equality hold? If so, how can I prove it ?
Any comments would be appreciated!