basisformules

Basisformules are compact formulas used to define and manipulate bases in vector spaces and related mathematical structures. In Dutch-language mathematics, the term is often used to describe the key relations that govern how a basis is constructed, represented, and transformed across different contexts, such as linear algebra, functional analysis, and numerical methods.

In a vector space V, a basis E = {e1, ..., en} is a set that is linearly independent and spans V. Every vector v in V can be written uniquely as v = sum_i c_i e_i, where the coefficients c_i are the coordinates of v relative to E, denoted [v]_E. The change of basis between two bases E and B is described by a matrix P whose columns are the B-coordinates of E; coordinates relate by [v]_E = P [v]_B and [v]_B = P^{-1} [v]_E. The dimension of V equals n, the number of basis vectors.

Constructing a basis can be done by row reducing a set of vectors to identify pivot columns, by Gram–Schmidt to obtain an orthonormal basis, or by selecting standard basis functions in a function space (for example monomials or trigonometric functions). For function spaces, basisformules include expansion v = sum_i c_i φ_i with φ_i as basis functions.

Dual basis and orthogonality are also part of basisformules: for a basis E = {e_i} there is a dual basis E* = {e^i} in the dual space such that e^i(e_j) = δ^i_j. In inner product spaces with an orthonormal basis, coefficients are c_i = ⟨v, e_i⟩.

Applications include solving linear systems, changing representations between bases, and tasks in signal processing, approximation theory, and finite element methods.