lineariteter

Lineariteter, or linearity, is a property used across mathematics, physics and engineering to describe how a system responds to inputs. A linear system or function behaves in a predictable, proportionate way: doubling the input doubles the output, and combining inputs produces outputs that combine in the same way.

Mathematically, a function f: V → W between vector spaces is linear if for all u, v in V and all scalars α, β we have f(αu + βv) = αf(u) + βf(v). Equivalently, linear maps preserve vector addition and scalar multiplication, and in particular f(0) = 0. When the domain is the real numbers, a linear function takes the form f(x) = ax, with a constant a, provided the function must pass through the origin; a nonzero intercept would make it affine rather than linear.

Many objects are linear: linear transformations represented by matrices, and differential operators that satisfy superposition. The superposition principle states that if f and g are solutions or responses, then αf + βg is also a solution or response for any scalars α and β. This underpins linear systems theory, linear differential equations, and linear models in statistics.

Linearity is contrasted with nonlinear behavior, where additivity or homogeneity fails. Examples of nonlinearities include f(x) = x^2, sin(x), or products of variables, which do not preserve scaling or sums.

In practice, linearity enables simple analysis and predictable behavior. When a problem is nonlinear, local linearization around a point—using derivatives or Jacobians—often provides useful approximations.