basiscomponent

Basiscomponent is a term in linear algebra referring to the scalar coefficient in the linear combination of basis vectors that expresses a vector in a given basis. Given a vector space V over a field F and a basis {e1, ..., en}, any v in V can be written uniquely as v = sum_{i=1}^n a_i e_i. The numbers a_i are the basiscomponents of v with respect to that basis. If {f_i} is the dual basis satisfying f_i(e_j) = δ_{ij}, then a_i = f_i(v). When the basis is orthonormal in an inner product space, a_i equals the inner product ⟨v, e_i⟩.

Basiscomponents depend on the chosen basis; changing the basis changes the components via the corresponding change-of-basis matrix. They provide coordinates of v in that basis and are used to reconstruct v by the linear combination of basis vectors. Computation in practice often uses coordinate vectors, where the basiscomponent vector is [a_1, ..., a_n]^T.

Examples: In R^3 with the standard basis, the vector v = (3, -2, 5) has basiscomponents 3, -2, and 5 with respect to that basis. In a nonstandard basis {b1, b2, b3}, the same v would have different basiscomponents a_i.

Applications include solving linear systems, transforming vectors under linear maps, and performing coordinate geometry in different bases. The concept underpins the idea that a single vector can be viewed simultaneously in many coordinate systems.

See also: basis, dual basis, coordinate vector, change of basis, linear transformation.