Which statement best characterizes linear systems with respect to input-output relationships?

Which statement best characterizes linear systems with respect to input-output relationships?

• A. A linear combination of inputs must produce the same linear combination of outputs.
• B. The system output is always proportional to input.
• C. The system can only handle one input.
• D. The output is equal to the integral of the input.

Answer: (A)

Linear systems follow the superposition principle: the response to a weighted sum of inputs is the same weighted sum of responses. In other words, if you combine inputs linearly, the outputs combine linearly in the same way. This means a linear combination of inputs yields the same linear combination of outputs, which is exactly what the statement says. Why this is the best description: it captures the fundamental behavior of linear systems across any set of inputs, not just a single input or a special case. The other ideas describe specific situations rather than the general property. Saying the output is always proportional to the input describes only a simple gain for one input and ignores what happens when multiple inputs are present. Saying the system can handle only one input limits generality, which isn’t a defining feature of linear systems. Saying the output is the integral of the input describes a particular linear operator, not the general input–output relationship of all linear systems.

Linear systems follow the superposition principle: the response to a weighted sum of inputs is the same weighted sum of responses. In other words, if you combine inputs linearly, the outputs combine linearly in the same way. This means a linear combination of inputs yields the same linear combination of outputs, which is exactly what the statement says.

Why this is the best description: it captures the fundamental behavior of linear systems across any set of inputs, not just a single input or a special case. The other ideas describe specific situations rather than the general property. Saying the output is always proportional to the input describes only a simple gain for one input and ignores what happens when multiple inputs are present. Saying the system can handle only one input limits generality, which isn’t a defining feature of linear systems. Saying the output is the integral of the input describes a particular linear operator, not the general input–output relationship of all linear systems.