<s>
In	O
mathematics	O
,	O
specifically	O
linear	B-Language
algebra	I-Language
,	O
a	O
degenerate	B-Algorithm
bilinear	I-Algorithm
form	I-Algorithm
on	O
a	O
vector	O
space	O
V	O
is	O
a	O
bilinear	O
form	O
such	O
that	O
the	O
map	O
from	O
V	O
to	O
V∗	O
(	O
the	O
dual	O
space	O
of	O
V	O
)	O
given	O
by	O
is	O
not	O
an	O
isomorphism	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
manifold	B-Architecture
with	O
an	O
inner	O
product	O
structure	O
on	O
its	O
tangent	O
spaces	O
is	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
,	O
while	O
relaxing	O
this	O
to	O
a	O
symmetric	O
nondegenerate	O
form	O
yields	O
a	O
pseudo-Riemannian	O
manifold	B-Architecture
.	O
</s>
<s>
If	O
V	O
is	O
finite-dimensional	O
then	O
,	O
relative	O
to	O
some	O
basis	O
for	O
V	O
,	O
a	O
bilinear	O
form	O
is	O
degenerate	O
if	O
and	O
only	O
if	O
the	O
determinant	O
of	O
the	O
associated	O
matrix	B-Architecture
is	O
zero	O
–	O
if	O
and	O
only	O
if	O
the	O
matrix	B-Architecture
is	O
singular	O
,	O
and	O
accordingly	O
degenerate	B-Algorithm
forms	I-Algorithm
are	O
also	O
called	O
singular	O
forms	O
.	O
</s>
<s>
Likewise	O
,	O
a	O
nondegenerate	O
form	O
is	O
one	O
for	O
which	O
the	O
associated	O
matrix	B-Architecture
is	O
non-singular	O
,	O
and	O
accordingly	O
nondegenerate	O
forms	O
are	O
also	O
referred	O
to	O
as	O
non-singular	O
forms	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
manifold	B-Architecture
with	O
an	O
inner	O
product	O
structure	O
on	O
its	O
tangent	O
spaces	O
is	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
,	O
while	O
relaxing	O
this	O
to	O
a	O
symmetric	O
nondegenerate	O
form	O
yields	O
a	O
pseudo-Riemannian	O
manifold	B-Architecture
.	O
</s>
<s>
Note	O
that	O
in	O
an	O
infinite-dimensional	O
space	O
,	O
we	O
can	O
have	O
a	O
bilinear	O
form	O
ƒ	O
for	O
which	O
is	O
injective	O
but	O
not	O
surjective	B-Algorithm
.	O
</s>
<s>
is	O
not	O
surjective	B-Algorithm
:	O
for	O
instance	O
,	O
the	O
Dirac	O
delta	O
functional	O
is	O
in	O
the	O
dual	O
space	O
but	O
not	O
of	O
the	O
required	O
form	O
.	O
</s>
<s>
In	O
such	O
a	O
case	O
where	O
ƒ	O
satisfies	O
injectivity	O
(	O
but	O
not	O
necessarily	O
surjectivity	B-Algorithm
)	O
,	O
ƒ	O
is	O
said	O
to	O
be	O
weakly	O
nondegenerate	O
.	O
</s>
