ortonormala

Ortonormala, often written as orthonormal in English, refers to a property of a set of vectors in an inner product space where each vector has unit length and is orthogonal to every other vector. Formally, a set {v1, ..., vk} is orthonormal if the inner product satisfies <vi, vj> = 0 for i ≠ j and <vi, vi> = 1 for all i. In many contexts, the term ortonormală is used in Romanian-language texts.

An orthonormal basis is an orthonormal set that also spans a subspace. In a finite-dimensional inner product space, any orthonormal basis provides a convenient coordinate system: the vectors in the basis are pairwise orthogonal unit vectors, and every vector in the subspace can be expressed uniquely as a linear combination of them.

Key properties include that the coefficients in the expansion of a vector x onto an orthonormal basis are simply the inner products with the basis vectors: x = sum_i <x, vi> vi, and the squared norm satisfies ||x||^2 = sum_i |<x, vi>|^2. The matrix whose columns are an orthonormal set has the property that its transpose is its inverse (U^T U = I), which implies that projections and rotations preserve lengths and angles.

Gram–Schmidt is a common method to obtain an orthonormal set from a linearly independent set: starting with w1, ..., wk, one constructs orthonormal vectors u1, ..., uk spanning the same subspace. Orthonormal bases are central in applications such as Fourier analysis, principal component analysis, quantum mechanics, and computer graphics, because they simplify projection, reconstruction, and spectral decompositions.