Suppose both $A$ and $B$ are matrices with dimensions $n \times m$ and $n \times p$ respectively. Assume $m + p < n $ and $A \neq 0$ and $B \neq 0$. Further, assume Im($B$) $\nsubseteq$ Im($A$). Then, it means that columns of $B$ are independent of columns of $A$. Hence, I can claim the following: \begin{equation} rank([A\ \ B]) = rank(A) + rank(B) \end{equation}

I believe this should be correct (if not correct me please). However, I was hoping to find a reference for it. It would be nice if someone could can suggest me a good book that contains many useful identities related to rank of matrices (like the one I just showed). Alternatively, I would like to get a formal proof. Thank you in advance!