<s>
In	O
mathematics	O
,	O
particularly	O
linear	B-Language
algebra	I-Language
,	O
an	O
orthogonal	B-Algorithm
basis	I-Algorithm
for	O
an	O
inner	O
product	O
space	O
is	O
a	O
basis	O
for	O
whose	O
vectors	O
are	O
mutually	O
orthogonal	O
.	O
</s>
<s>
If	O
the	O
vectors	O
of	O
an	O
orthogonal	B-Algorithm
basis	I-Algorithm
are	O
normalized	O
,	O
the	O
resulting	O
basis	O
is	O
an	O
orthonormal	O
basis	O
.	O
</s>
<s>
Any	O
orthogonal	B-Algorithm
basis	I-Algorithm
can	O
be	O
used	O
to	O
define	O
a	O
system	O
of	O
orthogonal	O
coordinates	O
Orthogonal	O
(	O
not	O
necessarily	O
orthonormal	O
)	O
bases	O
are	O
important	O
due	O
to	O
their	O
appearance	O
from	O
curvilinear	O
orthogonal	O
coordinates	O
in	O
Euclidean	O
spaces	O
,	O
as	O
well	O
as	O
in	O
Riemannian	B-Architecture
and	O
pseudo-Riemannian	O
manifolds	O
.	O
</s>
<s>
In	O
functional	B-Application
analysis	I-Application
,	O
an	O
orthogonal	B-Algorithm
basis	I-Algorithm
is	O
any	O
basis	O
obtained	O
from	O
an	O
orthonormal	O
basis	O
(	O
or	O
Hilbert	O
basis	O
)	O
using	O
multiplication	O
by	O
nonzero	O
scalars	O
.	O
</s>
<s>
The	O
concept	O
of	O
orthogonality	B-Application
may	O
be	O
extended	O
to	O
a	O
vector	O
space	O
(	O
over	O
any	O
field	O
)	O
equipped	O
with	O
a	O
quadratic	O
form	O
.	O
</s>
