Consider full column rank (injective) matrices $A \in \mathbb{R}^{n\times q}$ and $B\in \mathbb{R}^{n\times d}$ for some $n\geq d,q\geq 1$. Furthermore, assume that $A^T B$ is of full column rank entailing that $q\geq d$.
Let $R(A)$ and $R(B)$ be the range of the matrices and let $P_A= A(A^TA)^{-1}A^T$ be the orthogonal projection matrix onto $R(A)$. It holds that $P_A B\not = 0$.
Suppose that I know that there exists an element $x\not = 0$ with $$ x\in R(P_AB)^\perp \cap R(B)^\perp. $$
Can one show that this implies $R(B)\subseteq R(A)$, if not, why?
Let $n\ge 3$ and $d=q=1$, let $A$ and $B$ be single column vectors, non orthogonal and non parallel.
Then $P_AB$ is a nonzero vector parallel to $A$, so $R(P_AB)=R(A)={\rm span}(A)$, and any vector orthogonal to both $A$ and $B$ will serve as $x$.