lineaarsele

Lineaarsele is a hypothetical concept in abstract linear algebra used to describe a class of linear, structure-preserving maps on a finite-dimensional vector space V equipped with a nondegenerate bilinear form B. In this framework, a lineaarsele is a linear operator T: V → V that preserves B in the sense that B(Tx, Ty) = B(x, y) for all x, y ∈ V. The term echoes the Finnish word lineaarinen (linear) and is used in thought experiments to contrast with general linear operators that do not preserve B.

Examples: If B is the standard dot product on R^n, lineaarsele maps are exactly the isometries of Euclidean space, such as rotations and reflections. For a symplectic form ω, the lineaarsele maps form the symplectic group Sp(2n, R). In general, the group of all lineaarsele operators is called the isometry group of (V, B).

Properties: The set of lineaarsele operators on (V, B) forms a subgroup of the general linear group GL(V). It is closed under composition and inverses, hence a group; it is a Lie group when V is finite dimensional and B is nondegenerate. The dimension of this group depends on B.

Variations: Different choices of B yield different lineaarsele groups; for degenerate B the structure is more intricate. There are also covariant and contravariant versions depending on how one transforms dual spaces.

Applications: Lineaarsele concepts appear in geometry, physics (preservation of metrics and conserved quantities), computer graphics (rigid motions), and in teaching as a unifying view of isometries and structure-preserving maps. The term is mainly used in hypothetical or pedagogical contexts rather than as a standard technical term.