Linear system

From Academic Kids

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
<math>x_1(t) \,<math>
<math>x_2(t) \,<math>
as well as their respective outputs
<math>y_1(t) = H \left( x_1(t) \right)<math>
<math>y_2(t) = H \left( x_2(t) \right)<math>
then a linear system must satisfy
<math>\alpha y_1(t) + \beta y_2(t) = H \left( \alpha x_1(t) + \beta x_2(t) \right)<math>
for any <math>\alpha \,<math> and <math>\beta \,<math>.

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

Template:Math-stub

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