Reference (or proof) for the following identity in Linear Algebra

Question

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!

Answer 1

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.

Answer 2

Try $A=\begin{bmatrix}0 & 1\\\\ 1 & 0\\\\ 0 & 0\\\\ 0 & 0\\\\ 0 & 0\end{bmatrix}$ and $B=\begin{bmatrix}0 & 1\\\\ 0 & 0\\\\ 1 & 1\\\\ 0 & 0\\\\ 0 & 0\end{bmatrix}$. Then $rank([A,B])=3$ but $rank(A)=rank(B)=2$.

What is true however, is $rank([A, B])=rank(A)+rank(E_{A}B)$, where $E_{A}$ is the projection defined by $E_{A}=I-AA^{\dagger}$. See p.4 in this paper (the results are much older, going at least back to Marsaglia-Styan in the 70s, but Tian's paper gives a good recap and is easily accesible).