Bilineaarinen

Bilineaarinen is a Finnish term used in mathematics and related fields to describe a map or relationship that is linear in each of two arguments. Formally, a bilinear map B: V × W → F, where V and W are vector spaces over a field F, satisfies linearity in the first argument when the second is fixed and linearity in the second argument when the first is fixed: B(a x1 + b x2, y) = a B(x1, y) + b B(x2, y) and B(x, a y1 + b y2) = a B(x, y1) + b B(x, y2) for all x, x1, x2 ∈ V, y1, y2 ∈ W, and a, b ∈ F.

If V = W and the codomain is F, B is called a bilinear form. A standard example is the dot product in Euclidean space, B(x, y) = x · y. Every bilinear form can be represented by a matrix A such that B(x, y) = x^T A y. When B is symmetric, i.e., B(x, y) = B(y, x), it is associated with a quadratic form Q(x) = B(x, x).

Applications of bilinear forms and bilineaarinen concepts appear across disciplines, including physics and engineering (inner products, moment-like quantities), computer graphics (transformation pipelines and shading calculations), and statistics (covariance-like quantities). In digital image processing, bilinear interpolation is a specific technique that uses the four nearest pixels to estimate intermediate values during resampling.

In summary, bilineaarinen describes structures that are linear in each of two arguments, with wide use in linear algebra, geometry, and applied sciences.