Lineaarseteks

Lineaarseteks is a term used in Estonian mathematics to denote a linear set, typically understood as a linear subspace of a vector space. In a vector space V over a field F, a subset S is called a lineaarseteks if there exists a generating set B ⊆ V such that S = spanF(B). Equivalently, S is nonempty and closed under vector addition and scalar multiplication.

In finite-dimensional settings, lineaarseteks have a finite basis. The dimension of a lineaarseteks S is the size of its basis, and S is naturally isomorphic to F^k where k = dim(S). Well-known examples include the zero subspace {0}, the entire space V, and any span of a finite collection of vectors from V.

Key properties include closure under common subspace operations: the intersection of lineaarseteks is a lineaarseteks, the sum of two lineaarseteks is a lineaarseteks, and quotients by subspaces yield new vector spaces. To construct a lineaarseteks, one selects a generating set and computes its span, often determining a basis via methods such as row reduction.

Applications of lineaarseteks are widespread in linear algebra: they underpin solving systems of linear equations, studying linear transformations, and geometric descriptions of directions, planes, and higher-dimensional subspaces through the origin. They also provide the framework for dimension counting, projections, and the construction of quotient spaces.

Etymology and usage note: the term reflects the Estonian word for linear, with a suffix that forms a substantive noun. In more specialized texts, “lineaarseteks” may be used together with discussions of linear sets in finite geometry, where the concept is extended in specific ways to projective spaces.