standardbasisen

Standardbasisen is a term used to refer to the standard basis in a finite-dimensional coordinate space, typically encountered in linear algebra. In English contexts, it is usually called the standard basis. The concept provides a canonical, simple coordinate system for representing vectors.

In the conventional setting of R^n, the standardbasisen consists of n unit vectors e1, e2, ..., en where ei has a 1 in the i-th position and 0 in all other positions. Any vector x in R^n can be written uniquely as a linear combination x = x1 e1 + x2 e2 + ... + xn en, where the coordinates (x1, x2, ..., xn) are the components of x with respect to the standardbasisen.

Generalizing beyond R^n, the standardbasisen applies to any finite-dimensional vector space V over a field F that is identified with F^n via a chosen coordinate isomorphism. Under that identification, the standardbasis vectors correspond to the images of the unit coordinate vectors, giving a coordinate-free reference frame for expressing vectors and linear maps.

Properties of the standardbasisen include linear independence, spanning the space, and providing a straightforward matrix representation for linear transformations: the matrix of a linear map with respect to the standardbasisen is simply its action on the basis vectors. The identity transformation has the identity matrix in this basis.

Applications include simplifying vector coordinates, performing matrix multiplication, and serving as a reference for changes of basis and coordinate calculations. See also basis, coordinate vector, and change of basis. In German-language contexts, references may use Standardbasis or Standardbasen.