A linear system is a mathematical model of a system based on the use of a linear operator.
Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case.
As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, and telecommunications. For example, the propagation medium for wireless communication systems can often be
modeled by linear systems.A general deterministic system (mathematics)|deterministic system can be described by operator, H, that maps an input, x(t), as a function of t to an output, y(t), a type of Black box (systems)|black box description. Linear systems satisfy the properties of Superposition principle|superposition and Scaling (geometry)|scaling or Homogeneous_function|homogeneity. Given two valid inputs:x_1(t) \,
:x_2(t) \,
as well as their respective outputs
:y_1(t) = H \left \{ x_1(t) \right \}:y_2(t) = H \left \{ x_2(t) \right \}then a linear system must satisfy
:\alpha y_1(t) + \beta y_2(t) = H \left \{ \alpha x_1(t) + \beta x_2(t) \right \}for any scalar (mathematics)|scalar values \alpha \, and \beta \,.
The system is then defined by the equation H(x(t)) = y(t), where y(t) is some arbitrary function of time, and x(t) is the system state. Given y(t) and H, x(t) can be solved for. For example, a simple harmonic oscillator obeys the differential equation:
:m \frac{d^2(x)}{dt^2} = -kx
If H(x(t)) = m \frac{d^2(x(t))}{dt^2} + kx(t), then H is a linear operator. Letting y(t) = 0, we can rewrite the differential equation as H(x(t)) = y(t), which shows that a simple harmonic oscillator is a linear system.
The behavior of the resulting system subjected to a complex input can be described as a sum of responses to simpler inputs. In nonlinear systems, there is no such relation.This mathematical property makes the solution of modelling equations simpler than many nonlinear systems.
For time-invariant system|time-invariant systems this is the basis of the impulse response or the frequency response methods (see LTI system theory), which describe a general input function x(t) in terms of unit Dirac's delta function|impulses or frequency components.Typical differential equations of linear time-invariant system|time-invariant systems are well adapted to analysis using the Laplace transform in the continuous function|continuous case, and the Z-transform in the discrete mathematics|discrete case (especially in computer implementations).
Another perspective is that solutions to linear systems comprise a system of function (mathematics)|functions which act like vector (geometric)|vectors in the geometric sense.
A common use of linear models is to describe a nonlinear system by linearization. This is usually done for mathematical convenience.

A linear system is a mathematical model of a system based on the use of a linear operator.
Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case.
As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, and telecommunications. For example, the propagation medium for wireless communication systems can often be
modeled by linear systems.A general deterministic system (mathematics)|deterministic system can be described by operator, H, that maps an input, x(t), as a function of t to an output, y(t), a type of Black box (systems)|black box description. Linear systems satisfy the properties of Superposition principle|superposition and Scaling (geometry)|scaling or Homogeneous_function|homogeneity. Given two valid inputs:x_1(t) \,
:x_2(t) \,
as well as their respective outputs
:y_1(t) = H \left \{ x_1(t) \right \}:y_2(t) = H \left \{ x_2(t) \right \}then...