In mathematics, particularly linear algebra, an orthogonal basis for an inner product space
{\displaystyle V}
is a basis for
{\displaystyle V}
whose vectors are mutually orthogonal.
If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis.
Any orthogonal basis can be used to define a system of orthogonal coordinates
Orthogonal (not necessarily orthonormal) bases are important due to their appearance from curvilinear orthogonal coordinates in Euclidean spaces, as well as in Riemannian and pseudo-Riemannian manifolds.
In functional analysis, an orthogonal basis is any basis obtained from an orthonormal basis (or Hilbert basis) using multiplication by nonzero scalars.
The concept of an orthogonal basis is applicable to a vector space
(over any field) equipped with a symmetric bilinear form
, where orthogonality of two vectors
means
For an orthogonal basis
{\displaystyle \langle e_{j},e_{k}\rangle ={\begin{cases}q(e_{k})&j=k\\0&j\neq k,\end{cases}}}
is a quadratic form associated with
(in an inner product space,
Hence for an orthogonal basis
are components of
in the basis.
The concept of orthogonality may be extended to a vector space over any field of characteristic not 2 equipped with a quadratic form
Starting from the observation that, when the characteristic of the underlying field is not 2, the associated symmetric bilinear form
allows vectors
to be defined as being orthogonal with respect to