Ortogonaalset

Ortogonaalset is a term used in Estonian mathematics to denote a set of vectors that are mutually orthogonal in an inner product space. A finite set {v1, v2, ..., vn} is orthogonal if for all i ≠ j, the inner product ⟨vi, vj⟩ equals zero. In real or complex spaces with the standard inner product, this means their dot product is zero. If the vectors are nonzero, an orthogonal set is automatically linearly independent, since any linear combination that yields zero must have all coefficients zero.

An orthogonal set is orthonormal if, in addition, every vector has unit length, i.e., ⟨vi, vi⟩ = 1 for all i. Orthogonal (and orthonormal) sets are useful because they simplify many computations; for example, projections onto the subspace spanned by an orthogonal set are easy to compute, and coordinates of a vector relative to an orthogonal basis are simply inner products with the basis vectors.

Gram-Schmidt is a standard procedure to convert a linearly independent set into an orthogonal (and then orthonormal) set. In finite dimensions, any linearly independent set can be orthogonalized to form an orthogonal basis for its span, and an orthonormal basis can be obtained by normalizing the orthogonal vectors.

For a finite orthogonal set {v1,...,vn}, the Gram matrix G with entries Gij = ⟨vi, vj⟩ is diagonal with diagonal entries ||vi||^2, so det(G) equals the product of the squared norms. In infinite dimensions, an orthonormal set may be complete (forming an orthonormal basis) or incomplete, with the former enabling series expansions of vectors in a Hilbert space.

See also: orthonormal basis, inner product, Gram-Schmidt, projection, Hilbert space.