A set S of vectors is a spanning set if any vector can be written as a linear combination of elements in S. If, in addition, the vectors in S are linearly independent, then S is a basis, and any element of S may be referred to as a basis vector. Any vector can be written uniquely as a linear combination of the basis vectors. In 3-dimensional space, any set of three non-coplanar vectors is a basis. In 2-dimensional space, any set of 2 non-parallel vectors is a basis.

More generally, a vector space V over a field F is finite-dimensional if it has a finite basis; all bases then contain the same number of vectors—the dimension of V. The introduction of a basis identifies V with F^dimV and uniquely assigns coordinates to each vector. Any linearly independent set can be extended to a basis, and any spanning set contains a basis. Assuming the axiom of choice, any vector can be shown to have a basis. Compare complete orthonormal basis, dual basis, Hamel basis, orthonormal basis, Schauder basis.