Lineaariset

Lineaariset is Finnish an adjective form used to describe phenomena, models, or relations that exhibit linearity. The term derives from lineaarinen, and in Finnish it is commonly applied to plural or collective contexts (for example lineaariset suhteet or lineaariset yhtälöt). The word reflects the idea of proportional change and additive structure across disciplines.

In mathematics lineaarisuus refers to properties preserved under addition and scalar multiplication. A lineaarinen muunnos (linear transformation) satisfies f(x + y) = f(x) + f(y) and f(c x) = c f(x). Lineaariset yhtälöt are equations in which the unknowns occur to the first power and are not multiplied together, such as ax + by = c. Lineaariset järjestelmät, or systems of linear equations, can be analyzed using matrix methods and lineaarinen algebra. Lineaaritermit also describe linear functions, lineaariset mallit, and other structures that obey the principle of superposition.

In applied contexts lineaariset appear in statistics, economics, physics, and engineering. Lineaarinen malli and lineaarinen regressio refer to models assuming a linear relationship between variables. Lineaarinen optimointi (linear programming) concerns optimization where the objective function and constraints are linear. In physics and engineering, lineaarisuus underpins the superposition principle and linear response theory, providing tractable approximations for complex systems.

See also: lineaarinen algebra, lineaarinen ohjelmointi, lineaariset yhtälöt, lineaarinen regressio.