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In
mathematics , particularly
linear algebra , an orthogonal basis for an
inner product space
V
{\displaystyle V}
is a
basis for
V
{\displaystyle V}
whose vectors are mutually
orthogonal . If the vectors of an orthogonal basis are
normalized , the resulting basis is an
orthonormal basis .
As coordinates
Any orthogonal basis can be used to define a system of
orthogonal coordinates
V
.
{\displaystyle V.}
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
In
functional analysis , 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
V
{\displaystyle V}
(over any
field ) equipped with a
symmetric bilinear form
⟨
⋅
,
⋅
⟩
{\displaystyle \langle \cdot ,\cdot \rangle }
, where
orthogonality of two vectors
v
{\displaystyle v}
and
w
{\displaystyle w}
means
⟨
v
,
w
⟩
=
0
{\displaystyle \langle v,w\rangle =0}
. For an orthogonal basis
{
e
k
}
{\displaystyle \left\{e_{k}\right\}}
:
⟨
e
j
,
e
k
⟩
=
{
q
(
e
k
)
j
=
k
0
j
≠
k
,
{\displaystyle \langle e_{j},e_{k}\rangle ={\begin{cases}q(e_{k})&j=k\\0&j\neq k,\end{cases}}}
where
q
{\displaystyle q}
is a
quadratic form associated with
⟨
⋅
,
⋅
⟩
:
{\displaystyle \langle \cdot ,\cdot \rangle :}
q
(
v
)
=
⟨
v
,
v
⟩
{\displaystyle q(v)=\langle v,v\rangle }
(in an inner product space,
q
(
v
)
=
‖
v
‖
2
{\displaystyle q(v)=\Vert v\Vert ^{2}}
).
Hence for an orthogonal basis
{
e
k
}
{\displaystyle \left\{e_{k}\right\}}
,
⟨
v
,
w
⟩
=
∑
k
q
(
e
k
)
v
k
w
k
,
{\displaystyle \langle v,w\rangle =\sum _{k}q(e_{k})v^{k}w^{k},}
where
v
k
{\displaystyle v_{k}}
and
w
k
{\displaystyle w_{k}}
are components of
v
{\displaystyle v}
and
w
{\displaystyle w}
in the basis.
Quadratic form
The concept of orthogonality may be extended to a vector space over any field of characteristic not 2 equipped with a quadratic form
q
(
v
)
{\displaystyle q(v)}
. Starting from the observation that, when the characteristic of the underlying field is not 2, the associated symmetric bilinear form
⟨
v
,
w
⟩
=
1
2
(
q
(
v
+
w
)
−
q
(
v
)
−
q
(
w
)
)
{\displaystyle \langle v,w\rangle ={\tfrac {1}{2}}(q(v+w)-q(v)-q(w))}
allows vectors
v
{\displaystyle v}
and
w
{\displaystyle w}
to be defined as being orthogonal with respect to
q
{\displaystyle q}
when
q
(
v
+
w
)
−
q
(
v
)
−
q
(
w
)
=
0
{\displaystyle q(v+w)-q(v)-q(w)=0}
.
See also
References
Lang, Serge (2004), Algebra ,
Graduate Texts in Mathematics , vol. 211 (Corrected fourth printing, revised third ed.), New York: Springer-Verlag, pp. 572–585,
ISBN
978-0-387-95385-4
Milnor, J. ; Husemoller, D. (1973). Symmetric Bilinear Forms .
Ergebnisse der Mathematik und ihrer Grenzgebiete . Vol. 73.
Springer-Verlag . p. 6.
ISBN
3-540-06009-X .
Zbl
0292.10016 .
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