Question

2.17 Prove that a system is linear if and only if 1. It is homogeneous, i.e., for all input signals x(t) and all real numbers α, we have 2. It is additive, i.e., for all input signals xi (t) and x2(t), we have In other words, show that the two definitions of linear systems given by Equations (2.1.39) and (2.1.40) are equivalent. s.PNG Edit & Create Add to a creation Sh Linearand NonlinearSystems. Linear systems are systems for which the super- position property is satisfied, i.e., the system's response to a linear combination of the inputs is the linear combination of the responses to the corresponding inputs. A system Tis linear if and only if, for any two input signals xi(t) and x2(t) and for any two scalars a and B, we have A system that does not satisfy this relationship is called nonlinear. Linearity can also be defined in terms of the following two properties: J [ax(t)]

Answer 1

Proof is fairly straightforward. Just have to do a few clever substitutions.

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