Let $\mathbf{x}\neq \mathbf{0}$ and $\mathbf{y}\neq \mathbf{0}$ be $n \times 1$ vectors, $\mathbf{A}\neq \mathbf{0}$ and $\mathbf{B}\neq \mathbf{0}$ be $m \times n$ matrices with $m>n$, and, for some $u < m$, let
$$ \mathbf{C} \equiv [\mathbf{I}_{u} , \mathbf{0}]_{u\times m}, $$
with $\mathbf{I}_{u}$ denoting the $u \times u$ identity matrix.
Are there general conditions on matrices $\mathbf{A}$ and $\mathbf{B}$ so that for given $\mathbf{x}$ and $\mathbf{y}$ $$ \mathbf{C A x} = \mathbf{C B y} \implies \mathbf{A x} = \mathbf{B y} ? $$
I believe there should be conditions on the rank of these matrices or perhaps a relationship between them that should make it work but I haven't been able to work them out properly.
Matrix $C$ just cut last $m-u$ rows of matrices $A$ and $B$. Let
$$ A= \begin{bmatrix}A_1\\A_2\end{bmatrix}\qquad B= \begin{bmatrix}B_1\\B_2\end{bmatrix} $$
where $A_1$ and $B_1$ are $u \times u$ matrices. Then
$$ CAx = CBy \Leftrightarrow A_1x = B_1y $$
for any $A_2$, $B_2$. So, $$ C A x = C B y \implies A x = B y $$ will have place (for every $x$, $y$, in two sides) if and only if $$ A_2x = B_2y $$ (for every $x$, $y$).