mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, particularly

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...

, an orthogonal basis for an

inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...

V

is a

basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...

for

V

whose vectors are mutually

orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

. If the vectors of an orthogonal basis are normalized, the resulting basis is an

orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...

As coordinates

Any orthogonal basis can be used to define a system of

orthogonal coordinates In mathematics, orthogonal coordinates are defined as a set of ''d'' coordinates q = (''q''1, ''q''2, ..., ''q'd'') in which the coordinate hypersurfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate su ...

V.

Orthogonal (not necessarily orthonormal) bases are important due to their appearance from

curvilinear In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is invertible, l ...

orthogonal coordinates in

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

s, as well as in Riemannian and

pseudo-Riemannian In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which t ...

manifolds.

In functional analysis

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...

, an orthogonal basis is any basis obtained from an orthonormal basis (or Hilbert basis) using multiplication by nonzero scalars.

Extensions

Symmetric bilinear form

The concept of an orthogonal basis is applicable to a

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

V

(over any

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

) equipped with a

symmetric bilinear form In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinea ...

\langle \cdot, \cdot \rangle,

where '' orthogonality'' of two vectors

v

and

w

means

\langle v, w \rangle = 0.

For an orthogonal basis

\left\:

\langle e_j, e_k\rangle =
\begin
q(e_k) & j = k \\
0      & j \neq k,
\end

where

q

is a

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...

associated with

\langle \cdot, \cdot \rangle:

q(v) = \langle v, v \rangle

(in an inner product space,

q(v) = \, v\, ^2.

). Hence for an orthogonal basis

\left\,

\langle v, w \rangle = \sum_k q(e_k) v^k w^k,

where

v_k

and

w_k

are components of

v

and

w

in the basis.

Quadratic form

The concept of orthogonality may be extended to a vector space (over any field) equipped with a quadratic form

q(v)

. Starting from the observation that, when the characteristic of the underlying field is not 2, the associated symmetric bilinear form

\langle v, w \rangle = \tfrac(q(v+w) - q(v) - q(w))

allows vectors

v

and

w

to be defined as being orthogonal with respect to

q

when

q(v+w) - q(v) - q(w) = 0.

References

* *

External links

* {{Functional analysis Functional analysis Linear algebra de:Orthogonalbasis