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($B$) $\cap$ Im($A$) = {0}. 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! 



Reference (or proof) for the following identity in Linear Algebra
I conjecture that you didn't ask what you meant to ask. The reason for my conjecture is that you wrote "assume Im$(B)\not\subseteq$ Im$(A)$ ... it means that columns of $B$ are independent of columns of $A$." In fact, it doesn't mean that at all. The assumption Im$(B)\not\subseteq$ Im$(A)$ allows some (though not all) of the columns of $B$ to be linear combinations of columns of $A$ (or, indeed, to be columns of $A$). If you really intended the hypothesis to be Im$(B)\not\subseteq$ Im$(A)$, then there are lots of counterexamples, and Felix has given one of them. But if you intended the hypothesis to be that the columns of $B$ are independent of those of $A$, in the sense that no non-zero linear combination of the columns of $B$ equals a linear combination of the columns of $A$ (in other words, Im$(B)\cap$ Im$(A)=\{0\}$), then your proposed equation about ranks is OK, simply because all columns of $[A\ \ B]$ are linearly independent.


