orthonormé

Orthonormé is a term used in linear algebra to describe a set of vectors within a vector space. A set of vectors is considered orthonormal if it meets two specific conditions. First, each vector in the set must be a unit vector, meaning its magnitude or norm is equal to one. This is the "normal" part of the term. Second, every pair of distinct vectors within the set must be orthogonal, meaning their dot product is zero. This signifies that the vectors are perpendicular to each other.

The concept of orthonormal sets is fundamental in many areas of mathematics and physics. For instance, in Euclidean space, the standard basis vectors, such as (1, 0, 0), (0, 1, 0), and (0, 0, 1) in three dimensions, form an orthonormal set. These vectors are mutually perpendicular and each has a length of one. The Gram-Schmidt process is a common algorithm used to convert any linearly independent set of vectors into an orthonormal set.

Orthonormal bases simplify many calculations and theoretical developments. For example, when representing a vector in an orthonormal basis, the coordinates are simply the projections of the vector onto each basis vector, which are easily calculated using dot products. This property makes Fourier series and spectral analysis particularly elegant. In functional analysis, orthonormal sets of functions play a crucial role in decomposition and approximation theorems. The invariance of dot products under a change of basis to an orthonormal basis simplifies many geometric considerations.