These vectors are a basis in the sense that any other vector can be expressed uniquely as a linear combination of these. For example, every vector v in three-dimensional space can be written uniquely as
where ei denotes the vector with a 1 in the ith coordinate and 0's elsewhere.
However, an ordered orthonormal basis is not necessarily a standard basis. For instance the two vectors representing a 30° rotation of the 2D standard basis described above, i.e.
are also orthogonal unit vectors, but they are not aligned with the axes of the Cartesian coordinate system, so the basis with these vectors does not meet the definition of standard basis.
from I into a ring R, which are zero except for a finite number of indices, if we interpret 1 as 1R, the unit in R.
The existence of other 'standard' bases has become a topic of interest in algebraic geometry, beginning with work of Hodge from 1943 on Grassmannians. It is now a part of representation theory called standard monomial theory. The idea of standard basis in the universal enveloping algebra of a Lie algebra is established by the Poincaré–Birkhoff–Witt theorem.