Közbázis

Közbázis is a Hungarian term that translates to "co-basis" in English. It is a concept used in linear algebra and functional analysis, specifically in the context of vector spaces and their dual spaces. A co-basis is a set of linear functionals that form a basis for the dual space of a vector space. If V is a vector space with basis B = {v1, v2, ..., vn}, then its dual space V* consists of all linear functionals from V to the underlying field. A co-basis for V* is a set of linear functionals {f1, f2, ..., fn} such that fi(vj) = delta(ij), where delta(ij) is the Kronecker delta (1 if i=j, and 0 if i!=j). This means that each functional in the co-basis picks out a specific component of a vector in the original basis. This relationship between a basis and its co-basis is fundamental for many operations in linear algebra, including expressing vectors in terms of their components and calculating the action of linear transformations on vectors. The concept is particularly useful when working with inner product spaces and in the study of differential geometry.