A system is linear if it is both homogeneous and additive. Equivalently, in a single property:
If and , then
for any constants and any inputs .
Setting recovers additivity; setting recovers homogeneity. So linearity packages the two pieces into one statement.
What linearity buys you
Linearity gives the superposition principle: if you can decompose an input as a sum of pieces and know how the system responds to each piece, the response to the whole is the sum of the individual responses. This is the key fact that makes everything in this course possible. The convolution formula, the Fourier series, the Laplace transform — all rest on being able to add up the responses to simple basis signals.
Linear vs nonlinear
Most real-world systems are not strictly linear. A transistor amplifier saturates for large inputs. A diode is nonlinear in its I–V relationship. A spring is approximately linear only for small displacements. The course strategy when faced with a nonlinear system is to linearize: approximate it by a linear system valid for small excitations around an operating point, then apply linear theory.
This is the basis of small-signal analysis in electronics. The full nonlinear circuit is replaced by a linear “small-signal model” near a DC bias point, and we use linear tools to compute the response to perturbations. It’s an approximation, but a remarkably useful one.
What’s exactly linear
Systems built out of resistors, capacitors, inductors, and ideal integrators are exactly linear. Op-amps in their linear operating region (negative feedback, output not saturated) are also linear; op-amp circuits with nonlinear feedback (Schmitt triggers, log amplifiers, comparators) are not. Diodes, BJTs, MOSFETs, and any other element with a nonlinear I–V curve are nonlinear in general. The “ideal-component” qualifier matters: real components have parasitic nonlinearities (saturation, distortion) that limit linearity to a finite range. This is why so much of electrical engineering reduces to linear analysis — the basic passive components and the linear-region behavior of active devices are linear.
For everything in this course from Chapter 4 onward, the systems we study will be linear. Combined with Time-invariance, we get LTI — the class of systems characterized by an impulse response and analyzable by convolution.