Basis

Basis is a fundamental concept in linear algebra. A basis of a vector space V is a set B of vectors such that every element of V can be written uniquely as a finite linear combination of vectors in B. Equivalently, B is linearly independent and spans V. The number of vectors in B is called the dimension of V. For finite-dimensional spaces, all bases have the same size.

For example, in the real vector space R^n the standard basis consists of the unit vectors e1, …, en. With a chosen basis, every vector has a coordinate tuple relative to that basis, and the vector is reconstructed by the corresponding linear combination. Changing the basis amounts to applying an invertible matrix: the coordinates transform by multiplication by the change-of-basis matrix.

In inner product spaces one can seek an orthonormal basis, where vectors are mutually orthogonal and of unit length. The Gram–Schmidt process constructs such a basis from any linearly independent set. The dual basis, defined in the dual space, consists of linear functionals that pick out the corresponding coordinates: f^i(v_j) = δ^i_j for the basis {v_j}.

In infinite-dimensional spaces, the term basis can refer to different concepts. A Hamel (algebraic) basis spans the space using finite linear combinations; a Schauder basis allows convergent infinite series. Not every infinite-dimensional space has a countable basis, and the choice of basis affects the form of representations and convergence properties.