standardbas

Standardbas is not a widely used term in formal mathematics. When encountered, it typically refers to what is more commonly called the standard basis (also known as the canonical basis) of a vector space. In a finite-dimensional vector space over a field, the standard basis consists of n vectors e1, e2, ..., en, where each e_i has a 1 in the i-th coordinate and 0 in all other coordinates. In R^n, these are the coordinate unit vectors (1,0,...,0), (0,1,0,...,0), ..., (0,...,0,1).

The standard basis is linearly independent and spans the space, so any vector v can be uniquely expressed as a linear combination v = v1 e1 + v2 e2 + ... + vn e_n. The coordinates of a vector with respect to this basis form its coordinate vector (v1, v2, ..., vn). Matrix representations are closely tied to the standard basis: the identity matrix corresponds to the standard basis, and the columns of a matrix are the images of the basis vectors under a linear transformation. Changing from the standard basis to another basis B is accomplished by a change-of-basis matrix whose columns are the coordinates of B’s basis vectors in the standard basis.

Context and variants: The standard basis is sometimes called the canonical or coordinate basis. While “standardbas” may appear as a nonstandard spelling, a proper mathematical use generally prefers standard basis terminology. Outside mathematics, the term could occasionally be encountered as a proper noun for a project, dataset, or organization.

See also: standard basis; canonical basis; basis (linear algebra).