Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space

V

{\displaystyle V}

with finite dimension is a basis for

V

{\displaystyle V}

whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space

R

n

{\displaystyle \mathbb {R} ^{n}}

is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for

R

n

{\displaystyle \mathbb {R} ^{n}}

arises in this fashion.

An orthonormal basis can be derived from an orthogonal basis via normalization.

The choice of an origin and an orthonormal basis forms a coordinate frame known as an orthonormal frame.

For a general inner product space

V

,

{\displaystyle V,}

an orthonormal basis can be used to define normalized orthogonal coordinates on

V

.

{\displaystyle V.}

Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of

R

n

{\displaystyle \mathbb {R} ^{n}}

under the dot product. Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram–Schmidt process.

In functional analysis, the concept of an orthonormal basis can be generalized to arbitrary (infinite-dimensional) inner product spaces.