Linearë

Linearë is a mathematical term used in Albanian-language texts to denote objects and relations that exhibit linearity. In practice, linearë concepts include linear equations, linear maps, and linear spaces, all characterized by a simple, proportional interaction between their components.

A map f between vector spaces V and W over a field F is called linearë if it preserves addition and scalar multiplication: f(u + v) = f(u) + f(v) for all u, v in V, and f(a u) = a f(u) for every scalar a in F. A structure that satisfies these properties is called a linearë space or, in some contexts, a linearë transformation. Linearë spaces are the setting for many results in linear algebra, including basis, dimension, and coordinate representation.

Linearë equations are equations whose unknowns appear linearly: for example, a1 x1 + a2 x2 + … + an xn = b, where the coefficients ai and the constant b are fixed. Systems of linearë equations can be analyzed using matrices and vector algebra, with solution methods such as Gaussian elimination or matrix factorizations. The principle of superposition underpins linearë models, allowing complex responses to be decomposed into sums of simpler parts.

Applications of linearë concepts span science and engineering, economics, computer graphics, and data analysis, wherever relationships can be approximated or described as linear. Related terms include linearity, linear transformation, and linear system. In Albanian mathematics, the adjective linearë is used to describe these standard linear structures and operations.