Enhetsbasering

Enhetsbasering, in linear algebra, refers to a basis for a vector space in which each basis vector has unit length (norm 1). A unit basis does not by itself require the vectors to be mutually orthogonal, but when the vectors are also pairwise orthogonal, the basis is called orthonormal. In Euclidean spaces, an orthonormal basis simplifies many computations, while a non-orthogonal unit basis can still serve as a valid coordinate system.

In a vector space of dimension n, any basis consisting of n linearly independent vectors with norm 1 constitutes an enhetsbasering. For example, in R3, the standard basis e1, e2, e3 is an orthonormal unit basis. A non-orthogonal unit basis could be formed by three unit vectors that are not mutually perpendicular, which remains a valid basis as long as they are linearly independent.

A key practical distinction is how coordinates are computed. If the basis is orthonormal, the coordinates of a vector x relative to the basis can be obtained directly via inner products: the i-th coordinate is x · ei. If the basis is not orthogonal, the coordinates must be found by solving the linear system Bc = x, where B is the matrix whose columns are the basis vectors, or by using the Gram matrix.

Enhetsbasering is closely related to concepts such as basis, orthogonality, and change of basis. Techniques like the Gram–Schmidt process can transform any linearly independent set into an orthonormal basis, illustrating the relationship between unit-length vectors and efficient representations.