Let $M$ be a matrix of size $n$x$m$ over the reals. The following statement is clearly right:
$(\exists x,y \in R^{n} ,x \neq y \; \land M*x =M*y) \rightarrow$ $M$ is non unitarary matrix
Does the opposite direction hold as well? i.e. does:
$M$ is non unitarary matrix $\rightarrow (\exists x,y \in R^{n} ,x \neq y \; \land M*x =M*y)$
?