lineárnylinear

Lineárnylinear is a term encountered occasionally in mathematical literature to describe a function that is linear in each of its two arguments. The phrase is a tautological emphasis of double linearity and is not standard across all textbooks, but it appears in some Slovak-language discussions of bilinear and multilinear structures.

Definition and meaning. Let V and W be vector spaces over a field F, and let f: V × W → F (or into another vector space) be a map. The term lineárnylinear denotes that f is linear in the first argument when the second is fixed, and linear in the second argument when the first is fixed. Equivalently, for all x1, x2 ∈ V, y1, y2 ∈ W, and a, b ∈ F, one has f(a x1 + b x2, y) = a f(x1, y) + b f(x2, y) and f(x, a y1 + b y2) = a f(x, y1) + b f(x, y2). This is the standard property of bilinear maps.

Properties and representation. A lineárnylinear map is bilinear, meaning it is linear separately in each argument. In coordinates, such maps often take the form f(x, y) = x^T A y for some matrix A, linking them to bilinear forms and tensor products. The concept underpins many constructions in linear and multilinear algebra, including dual spaces and rank-one decompositions.

Examples and applications. Common examples include f(x, y) = x^T M y on real vector spaces, or simple bilinear forms like the dot product when adapted to appropriate spaces. Bilinear forms appear in geometry, physics, and optimization, where they encode energy, projections, and interaction terms.

See also. Bilinear form; Tensor product; Multilinear algebra.