linearitas

Linearitas, or linearity, is a fundamental concept describing when a relation or system preserves the basic operations of addition and scalar multiplication. In mathematics, a function f between vector spaces is linear if for all vectors x and y and all scalars α and β, f(αx + βy) = αf(x) + βf(y). Equivalently, f(0) = 0 and f(x + y) = f(x) + f(y). Linear maps can be represented by matrices, and the composition of linear maps is linear.

A related idea is an affine transformation, which has the form f(x) = Ax + b. Affine maps are not linear unless the offset b is zero, because they do not necessarily preserve the origin.

In systems theory and physics, linearity implies the superposition principle: the response to a sum of inputs equals the sum of the responses to each input. Linear time-invariant systems are completely characterized by their impulse responses, and their behavior is described by convolution.

In statistics and econometrics, linearity often means the model is linear in the parameters: y = Xβ + ε is linear in β, even if the regressors involve nonlinear transformations. Models that are nonlinear in the parameters or variables may require different estimation techniques or transformations to approximate linearity.

Practical linearity is often an approximation valid within a limited operating range. Many real-world systems exhibit nonlinear behavior when inputs become large or interact in complex ways, necessitating nonlinear models or linearization around a point using methods such as Taylor expansion.

See also: linear transformation, superposition principle, vector space, linear regression.