klinear

In mathematics, a map f: V -> W between vector spaces V and W over a field k is called k-linear (or a k-linear transformation) if for all vectors v, w in V and all scalars a, b in k, f(a v + b w) = a f(v) + b f(w). Equivalently, f preserves addition and scalar multiplication. Consequently, f(0) = 0 and f preserves all linear combinations.

Examples include the identity map id_V, the zero map, and any map of the form f(v) = c v where c is a fixed scalar in k. More generally, any linear transformation between vector spaces is k-linear. If bases are chosen for V and W, a k-linear map is represented by a matrix A over k such that f(v) = A v relative to those bases.

Matrices and representations: When V = k^n and W = k^m with standard bases, k-linear maps correspond to m by n matrices with entries in k, acting by matrix multiplication. The set of all k-linear maps from V to W is denoted Hom_k(V, W). If V and W are finite-dimensional, the dimension of Hom_k(V, W) is m n.

Extensions and context: The concept extends to bilinear and multilinear maps, which are linear in each argument. k-linear maps are the morphisms in the category of vector spaces over k. If k is a familiar field such as R or C, the terminology R-linear or C-linear is often used. In more general settings, k-linear maps refer to linear maps of modules over a ring or division ring.