Ortonormalien

Ortonormalien, also known as orthonormal bases, are a fundamental concept in linear algebra and vector spaces. An orthonormal set is a collection of vectors that are mutually orthogonal and each have a unit length. In other words, for any two distinct vectors u and v in the set, their dot product is zero (u · v = 0), and the magnitude of each vector is one (||u|| = 1).

Orthonormal sets are particularly useful in various mathematical and scientific disciplines, including physics, engineering, and computer science. They provide a convenient way to represent and manipulate vectors, as the orthogonality condition simplifies many calculations. For instance, in quantum mechanics, orthonormal bases are used to describe the state of a quantum system.

In finite-dimensional vector spaces, an orthonormal set with the same number of vectors as the dimension of the space is called an orthonormal basis. Any vector in the space can be uniquely expressed as a linear combination of the basis vectors. This property makes orthonormal bases essential tools in linear algebra and related fields.

Orthonormal sets can be constructed using the Gram-Schmidt process, which takes a set of linearly independent vectors and produces an orthonormal set with the same span. This process is widely used in numerical linear algebra and other applications where orthonormal bases are required.