For basis vector in the context of crystals, see Crystal structure. For a basis and dimension examples pdf general concept in physics, see Frame of reference.

For other uses, see Basis. In more general terms, a basis is a linearly independent spanning set. Given a basis of a vector space V, every element of V can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. This picture illustrates the standard basis in R2.

A basis B of a vector space V over a field F is a linearly independent subset of V that spans V. The numbers ai are called the coordinates of the vector x with respect to the basis B, and by the first property they are uniquely determined. A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets.