coefënten

Coefënten is a term used in linear algebra to denote the set of scalar coefficients in a linear combination that represents a vector relative to a chosen basis. The word is formed to indicate a formal distinction between coefficients as numerical components and their collective arrangement in a coordinate representation. In most texts the standard term "coordinates" is used, but coefënten appears in didactic contexts to emphasize component values.

Definition: Let V be a vector space over a field F and B = {b_i} a basis. For any v in V there exist unique c_i in F such that v = sum_i c_i b_i. The ordered family (c_i) is the coefënten of v with respect to B; each c_i is a coefënt. This representation is not unique across different bases; changing the basis changes the coefënten accordingly.

Example: In R^3 with the standard basis e1, e2, e3, the coefënten of the vector (4, -1, 7) relative to this basis is (4, -1, 7). If a new basis is formed by rotating e1, e2, e3, the coefënten change according to the corresponding transformation matrix.

Properties and usage: The coefënten depend on the chosen basis; they provide a coordinate vector that allows algebraic operations to be performed in a fixed representation. Linear maps act on coefënten via matrices: if T: V → W is a linear map and [v]_B denotes the coefënten of v, then [T(v)]_{B'} = M [v]_B where M encodes T in the pair of bases B and B'. The concept extends to modules and coordinate systems in higher mathematics and computer science.

See also: coordinate vector, basis, coefficient, coordinate transformation. Notes: The term coefënten is uncommon in standard texts, which usually refer to coordinates or coefficients; it may appear in specialized or historical writings.