Ortanormaalina

Ortanormaalina is the property of a collection of vectors in an inner product space to be orthogonal to each other and of unit length. A set {v1, v2, ..., vk} is orthonormal if for all i and j, the inner product satisfies <vi, vj> = 0 when i ≠ j and <vi, vi> = 1. In real spaces this means the vectors are pairwise perpendicular and of unit length; in complex spaces the inner product is conjugate symmetric and unit length is defined with respect to the standard inner product.

Orthonormal sets simplify many linear-algebra operations. The projection of a vector x onto the span of an orthonormal set {v1, ..., vk} is given by the sum ∑i <x, vi> vi, and the coefficients are simply the inner products <x, vi>. The squared norm of x’s projection equals ∑i |<x, vi>|^2.

An orthonormal basis is an orthonormal set that spans the entire space. In finite-dimensional inner product spaces, any basis can be converted to an orthonormal basis via the Gram–Schmidt process. The matrix whose columns are orthonormal vectors is orthogonal in the real case and unitary in the complex case, preserving inner products and norms: U*U = I.

Applications appear across mathematics, physics, and engineering. Orthonormal bases underpin Fourier-like expansions, signal processing, quantum mechanics, and numerical linear algebra, where coordinates of vectors relative to the basis are simply the inner products with the basis vectors.

Related concepts include orthogonality (vectors with zero inner product) and normalization (adjusting a vector to unit length).