Let $A$, $B$ and $C$ be three matrices where $A$ and $B$ have full column rank. Let $x$, $y$ and $z$ be three vectors where $Ax$, $By$ and $Cz$ are well defined.
When $Ax \neq By + Cz$ for any $x \neq 0$, $y \neq 0$ and $z \neq 0$, does this imply $Ax \neq By$ or $Ax \neq Cz$?? Or, under what conditions, does $Ax \neq By + Cz$ imply $Ax \neq By$ or $Ax \neq Cz$??