ortonormaalsed

Ortonormaalsed, also known as "orthonormal" in English, refers to a set of vectors in a vector space that are both orthogonal and normalized. In mathematical terms, a set of vectors is considered orthonormal if each pair of distinct vectors is perpendicular (orthogonal) and each vector has a unit length (normalized). This concept is fundamental in linear algebra, particularly in the context of vector spaces equipped with an inner product.

In an orthonormal set, the inner product of any two different vectors is zero, indicating perpendicularity. The inner product of a vector with itself equals one, indicating normalization. These properties simplify many operations, such as projections and coordinate transformations, making orthonormal systems especially useful in applications like signal processing, quantum mechanics, and computer graphics.

An orthonormal basis is a special case where the set of orthonormal vectors not only is orthogonal and normalized but also spans the entire vector space. This basis provides a convenient framework for representing vectors uniquely and for simplifying calculations related to linear transformations.

Constructing an orthonormal set often employs the Gram-Schmidt process, which transforms a linearly independent set of vectors into an orthonormal set. The use of orthonormal systems facilitates efficient numerical computations and theoretical analyses across many scientific and engineering disciplines.

Overall, orthonormality ensures that vectors are both directionally independent and scaled uniformly, enabling clear and efficient mathematical and computational operations within the vector space.