Linear system: definition, properties, representations and applications

Core properties

Two basic conditions characterize linearity:

• Additivity — output(x1 + x2) = output(x1) + output(x2).
• Homogeneity (scaling) — output(a·x) = a·output(x) for scalar a.

When both hold, the system is linear. Many mathematical descriptions — matrices, linear operators and differential operators — formalize these properties, often referred to as linear mappings.

Common representations

Linear systems appear in several mathematical forms depending on context:

• Algebraic: systems of linear equations and matrix transformations.
• Differential or difference equations: linear ordinary differential equations for continuous time, linear recurrences for discrete time.
• State-space form: a set of first-order linear equations describing internal state evolution and outputs.
• Frequency-domain: transfer functions and convolution kernels for linear time-invariant cases.

History and development

Concepts of linearity emerged with linear algebra and classical analysis; tools such as transforms (Fourier, Laplace) and eigenvalue methods became central for solving linear problems. In the 20th century, formal systems engineering and control theory adopted and extended these ideas, making linear models foundational in signal processing and automatic control.

Uses and examples

Linear models are widely used because they are tractable and often good approximations near operating points: electrical circuit analysis (small-signal models), mechanical vibrations, basic economic models, filter design in signal processing, and controller design. Even when real behavior is nonlinear, engineers often linearize around equilibria to obtain useful insight and design controllers.

Distinctions and notable facts

Most natural systems are inherently nonlinear, so exact linearity is uncommon; such systems are said to be non-linear. A particularly important subclass is linear time-invariant (LTI) systems, where tools like convolution, impulse response and transfer functions apply and frequency-domain methods yield powerful results. Stability, controllability and observability are key analysis concepts that are simpler to assess for linear models than for nonlinear ones.