Standard basis

In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as $\mathbb {R} ^{n}$ or $\mathbb {C} ^{n}$ ) is the set of vectors, each of whose components are all zero, except one that equals 1.^[1] For example, in the case of the Euclidean plane $\mathbb {R} ^{2}$ formed by the pairs $(x, y)$ of real numbers, the standard basis is formed by the vectors

For broader coverage of this topic, see Canonical basis.

Here the vector e_x points in the x direction, the vector e_y points in the y direction, and the vector e_z points in the z direction. There are several common notations for standard-basis vectors, including {e_x, e_y, e_z}, {e₁, e₂, e₃}, {i, j, k}, and {x, y, z}. These vectors are sometimes written with a hat to emphasize their status as unit vectors (standard unit vectors).

These vectors are a basis in the sense that any other vector can be expressed uniquely as a linear combination of these.^[2] For example, every vector v in three-dimensional space can be written uniquely as

In the $n$ -dimensional Euclidean space $\mathbb {R} ^{n}$ , the standard basis consists of n distinct vectors

Standard bases can be defined for other vector spaces, whose definition involves coefficients, such as polynomials and matrices. In both cases, the standard basis consists of the elements of the space such that all coefficients but one are 0 and the non-zero one is 1. For polynomials, the standard basis thus consists of the monomials and is commonly called monomial basis. For matrices ${\mathcal {M}}_{m\times n}$ , the standard basis consists of the m×n-matrices with exactly one non-zero entry, which is 1. For example, the standard basis for 2×2 matrices is formed by the 4 matrices

Other usages[edit]

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.

Gröbner bases are also sometimes called standard bases.

In physics, the standard basis vectors for a given Euclidean space are sometimes referred to as the versors of the axes of the corresponding Cartesian coordinate system.

Standard basis

Other usages[edit]

Canonical units

Examples of vector spaces § Generalized coordinate space