# Does a linear time system imply that for a zero input, a zero output is given?

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.

• Linear functions must map $0$ to $0$ because $T(0)=T(0+0)=T(0)+T(0)\implies T(0)=0$. – John Douma Oct 31 '18 at 21:15
• @JohnDouma Thank you! That was my intuition, but I didn't manage to express it. If that's worth the effort to write an actual answer, I'd be happy to accept! – Felix Oct 31 '18 at 21:17
• Thank you but it wouldn't be much of an answer because it is a standard trick most folks who have taken Linear Algebra were taught. I am glad I was able to help. – John Douma Oct 31 '18 at 21:19
• @JohnDouma Yeah, you'd think that would occur to someone studying the damn thing. Oh well, not the first lapse of brain function, and certainly not the last. – Felix Oct 31 '18 at 21:22