A linear system is a model based on some kind of linear operator. Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case. By definition, they satisfy the properties of superposition and scaling:

Given two inputs

as well as their respective outputs

then a linear system must satisfy

for any and .

The mathematical properties of linear systems are also better (more transparent, at least) than in the general case. For example, typical differential equations of linear time-invariant systems are well adapted to analysis using the Laplace transform in the continuous case, and the Z-transform in the discrete case (especially in computer implementations).

Another perspective is that solutions to linear systems comprise a system of functions which act like vectors in the geometric sense. The behavior of the resulting system can be described as a sum of the linear parts. In nonlinear systems, there is no such relation.

See also

• Linear system of divisors in algebraic geometry.
• System analysis