My question is exactly that. The definition of a linear (discrete) time system is, as with most definitions of linearity, that a linear combination of signals given as input must correspond to the same combination of individual outputs. I.e:
$$T(\alpha_1\overline{x_1} + \alpha_2\overline{x_2}) = \alpha_1T(\overline{x_1}) + \alpha_2T(\overline{x_2})$$
Does this imply that given a zero signal, the output must also be zero?
As background, I was given some signals and responses of a system and a task to determine the linearity. If alphas are set to zero, the signals may be whatever, and naturally multiplying the responses by zero yields a zero response. But by combining vectors such that the input is zero, alphas are most likely non-zero. So additionally I'm wondering, is there a way to say something meaningful about the system based on those few signal-response pairs.
Proof:
Let $T:U\to V$
$T(\alpha \cdot 0_U)=T(0 \cdot u)=0\cdot T(u)=0_V.$