<s>
In	O
mathematics	O
,	O
the	O
kernel	B-Algorithm
of	O
a	O
linear	B-Architecture
map	I-Architecture
,	O
also	O
known	O
as	O
the	O
null	B-Algorithm
space	I-Algorithm
or	O
nullspace	B-Algorithm
,	O
is	O
the	O
linear	O
subspace	O
of	O
the	O
domain	B-Algorithm
of	O
the	O
map	O
which	O
is	O
mapped	O
to	O
the	O
zero	O
vector	O
.	O
</s>
<s>
That	O
is	O
,	O
given	O
a	O
linear	B-Architecture
map	I-Architecture
between	O
two	O
vector	O
spaces	O
and	O
,	O
the	O
kernel	B-Algorithm
of	O
is	O
the	O
vector	O
space	O
of	O
all	O
elements	O
of	O
such	O
that	O
,	O
where	O
denotes	O
the	O
zero	O
vector	O
in	O
,	O
or	O
more	O
symbolically	O
:	O
</s>
<s>
The	O
kernel	B-Algorithm
of	O
is	O
a	O
linear	O
subspace	O
of	O
the	O
domain	B-Algorithm
.	O
</s>
<s>
In	O
the	O
linear	B-Architecture
map	I-Architecture
two	O
elements	O
of	O
have	O
the	O
same	O
image	O
in	O
if	O
and	O
only	O
if	O
their	O
difference	O
lies	O
in	O
the	O
kernel	B-Algorithm
of	O
,	O
that	O
is	O
,	O
</s>
<s>
From	O
this	O
,	O
it	O
follows	O
that	O
the	O
image	O
of	O
is	O
isomorphic	O
to	O
the	O
quotient	O
of	O
by	O
the	O
kernel	B-Algorithm
:	O
</s>
<s>
where	O
the	O
term	O
refers	O
the	O
dimension	O
of	O
the	O
image	O
of	O
,	O
while	O
refers	O
to	O
the	O
dimension	O
of	O
the	O
kernel	B-Algorithm
of	O
,	O
</s>
<s>
When	O
is	O
an	O
inner	O
product	O
space	O
,	O
the	O
quotient	O
can	O
be	O
identified	O
with	O
the	O
orthogonal	B-Algorithm
complement	I-Algorithm
in	O
of	O
This	O
is	O
the	O
generalization	O
to	O
linear	B-Architecture
operators	I-Architecture
of	O
the	O
row	O
space	O
,	O
or	O
coimage	O
,	O
of	O
a	O
matrix	O
.	O
</s>
<s>
The	O
notion	O
of	O
kernel	B-Algorithm
also	O
makes	O
sense	O
for	O
homomorphisms	O
of	O
modules	O
,	O
which	O
are	O
generalizations	O
of	O
vector	O
spaces	O
where	O
the	O
scalars	O
are	O
elements	O
of	O
a	O
ring	O
,	O
rather	O
than	O
a	O
field	O
.	O
</s>
<s>
The	O
domain	B-Algorithm
of	O
the	O
mapping	O
is	O
a	O
module	O
,	O
with	O
the	O
kernel	B-Algorithm
constituting	O
a	O
submodule	O
.	O
</s>
<s>
If	O
V	O
and	O
W	O
are	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
such	O
that	O
W	O
is	O
finite-dimensional	O
,	O
then	O
a	O
linear	B-Architecture
operator	I-Architecture
L:V	O
→	O
W	O
is	O
continuous	B-Algorithm
if	O
and	O
only	O
if	O
the	O
kernel	B-Algorithm
of	O
L	O
is	O
a	O
closed	O
subspace	O
of	O
V	O
.	O
</s>
<s>
Consider	O
a	O
linear	B-Architecture
map	I-Architecture
represented	O
as	O
a	O
m	O
×	O
n	O
matrix	O
A	O
with	O
coefficients	O
in	O
a	O
field	O
K	O
(	O
typically	O
or	O
)	O
,	O
that	O
is	O
operating	O
on	O
column	O
vectors	O
x	O
with	O
n	O
components	O
over	O
K	O
.	O
</s>
<s>
The	O
kernel	B-Algorithm
of	O
this	O
linear	B-Architecture
map	I-Architecture
is	O
the	O
set	O
of	O
solutions	O
to	O
the	O
equation	O
,	O
where	O
0	O
is	O
understood	O
as	O
the	O
zero	O
vector	O
.	O
</s>
<s>
The	O
dimension	O
of	O
the	O
kernel	B-Algorithm
of	O
A	O
is	O
called	O
the	O
nullity	O
of	O
A	O
.	O
</s>
<s>
In	O
set-builder	B-Language
notation	I-Language
,	O
</s>
<s>
Thus	O
the	O
kernel	B-Algorithm
of	O
A	O
is	O
the	O
same	O
as	O
the	O
solution	O
set	O
to	O
the	O
above	O
homogeneous	O
equations	O
.	O
</s>
<s>
The	O
kernel	B-Algorithm
of	O
a	O
matrix	O
A	O
over	O
a	O
field	O
K	O
is	O
a	O
linear	O
subspace	O
of	O
Kn	O
.	O
</s>
<s>
That	O
is	O
,	O
the	O
kernel	B-Algorithm
of	O
A	O
,	O
the	O
set	O
Null(A )	O
,	O
has	O
the	O
following	O
three	O
properties	O
:	O
</s>
<s>
It	O
follows	O
that	O
x	O
is	O
in	O
the	O
kernel	B-Algorithm
of	O
A	O
,	O
if	O
and	O
only	O
if	O
x	O
is	O
orthogonal	B-Application
(	O
or	O
perpendicular	O
)	O
to	O
each	O
of	O
the	O
row	O
vectors	O
of	O
A	O
(	O
since	O
orthogonality	B-Application
is	O
defined	O
as	O
having	O
a	O
dot	O
product	O
of	O
0	O
)	O
.	O
</s>
<s>
By	O
the	O
above	O
reasoning	O
,	O
the	O
kernel	B-Algorithm
of	O
A	O
is	O
the	O
orthogonal	B-Algorithm
complement	I-Algorithm
to	O
the	O
row	O
space	O
.	O
</s>
<s>
That	O
is	O
,	O
a	O
vector	O
x	O
lies	O
in	O
the	O
kernel	B-Algorithm
of	O
A	O
,	O
if	O
and	O
only	O
if	O
it	O
is	O
perpendicular	O
to	O
every	O
vector	O
in	O
the	O
row	O
space	O
of	O
A	O
.	O
</s>
<s>
The	O
dimension	O
of	O
the	O
row	O
space	O
of	O
A	O
is	O
called	O
the	O
rank	O
of	O
A	O
,	O
and	O
the	O
dimension	O
of	O
the	O
kernel	B-Algorithm
of	O
A	O
is	O
called	O
the	O
nullity	O
of	O
A	O
.	O
</s>
<s>
The	O
left	O
null	B-Algorithm
space	I-Algorithm
,	O
or	O
cokernel	O
,	O
of	O
a	O
matrix	O
A	O
consists	O
of	O
all	O
column	O
vectors	O
x	O
such	O
that	O
xTA	O
=	O
0T	O
,	O
where	O
T	O
denotes	O
the	O
transpose	O
of	O
a	O
matrix	O
.	O
</s>
<s>
The	O
left	O
null	B-Algorithm
space	I-Algorithm
of	O
A	O
is	O
the	O
same	O
as	O
the	O
kernel	B-Algorithm
of	O
AT	O
.	O
</s>
<s>
The	O
left	O
null	B-Algorithm
space	I-Algorithm
of	O
A	O
is	O
the	O
orthogonal	B-Algorithm
complement	I-Algorithm
to	O
the	O
column	O
space	O
of	O
A	O
,	O
and	O
is	O
dual	O
to	O
the	O
cokernel	O
of	O
the	O
associated	O
linear	B-Architecture
transformation	I-Architecture
.	O
</s>
<s>
The	O
kernel	B-Algorithm
,	O
the	O
row	O
space	O
,	O
the	O
column	O
space	O
,	O
and	O
the	O
left	O
null	B-Algorithm
space	I-Algorithm
of	O
A	O
are	O
the	O
four	O
fundamental	O
subspaces	O
associated	O
to	O
the	O
matrix	O
A	O
.	O
</s>
<s>
The	O
kernel	B-Algorithm
also	O
plays	O
a	O
role	O
in	O
the	O
solution	O
to	O
a	O
nonhomogeneous	O
system	O
of	O
linear	O
equations	O
:	O
</s>
<s>
Thus	O
,	O
the	O
difference	O
of	O
any	O
two	O
solutions	O
to	O
the	O
equation	O
Ax	O
=	O
b	O
lies	O
in	O
the	O
kernel	B-Algorithm
of	O
A	O
.	O
</s>
<s>
It	O
follows	O
that	O
any	O
solution	O
to	O
the	O
equation	O
Ax	O
=	O
b	O
can	O
be	O
expressed	O
as	O
the	O
sum	O
of	O
a	O
fixed	O
solution	O
v	O
and	O
an	O
arbitrary	O
element	O
of	O
the	O
kernel	B-Algorithm
.	O
</s>
<s>
Geometrically	O
,	O
this	O
says	O
that	O
the	O
solution	O
set	O
to	O
Ax	O
=	O
b	O
is	O
the	O
translation	B-Algorithm
of	O
the	O
kernel	B-Algorithm
of	O
A	O
by	O
the	O
vector	O
v	O
.	O
See	O
also	O
Fredholm	O
alternative	O
and	O
flat	O
(	O
geometry	O
)	O
.	O
</s>
<s>
The	O
following	O
is	O
a	O
simple	O
illustration	O
of	O
the	O
computation	O
of	O
the	O
kernel	B-Algorithm
of	I-Algorithm
a	I-Algorithm
matrix	I-Algorithm
(	O
see	O
,	O
below	O
for	O
methods	O
better	O
suited	O
to	O
more	O
complex	O
calculations	O
)	O
.	O
</s>
<s>
The	O
illustration	O
also	O
touches	O
on	O
the	O
row	O
space	O
and	O
its	O
relation	O
to	O
the	O
kernel	B-Algorithm
.	O
</s>
<s>
Through	O
Gauss	B-Algorithm
–	I-Algorithm
Jordan	I-Algorithm
elimination	I-Algorithm
,	O
the	O
matrix	O
can	O
be	O
reduced	O
to	O
:	O
</s>
<s>
The	O
elements	O
of	O
the	O
kernel	B-Algorithm
can	O
be	O
further	O
expressed	O
in	O
parametric	O
vector	O
form	O
,	O
as	O
follows	O
:	O
</s>
<s>
The	O
kernel	B-Algorithm
of	O
A	O
is	O
precisely	O
the	O
solution	O
set	O
to	O
these	O
equations	O
(	O
in	O
this	O
case	O
,	O
a	O
line	O
through	O
the	O
origin	O
in	O
R3	O
)	O
.	O
</s>
<s>
Here	O
,	O
since	O
the	O
vector	O
( −1	O
,	O
−26	O
,	O
16	O
)	O
T	O
constitutes	O
a	O
basis	O
of	O
the	O
kernel	B-Algorithm
of	O
A	O
.	O
</s>
<s>
which	O
illustrates	O
that	O
vectors	O
in	O
the	O
kernel	B-Algorithm
of	O
A	O
are	O
orthogonal	B-Application
to	O
each	O
of	O
the	O
row	O
vectors	O
of	O
A	O
.	O
</s>
<s>
These	O
two	O
(	O
linearly	O
independent	O
)	O
row	O
vectors	O
span	O
the	O
row	O
space	O
of	O
A	O
—	O
a	O
plane	O
orthogonal	B-Application
to	O
the	O
vector	O
( −1	O
,	O
−26	O
,	O
16	O
)	O
T	O
.	O
</s>
<s>
If	O
,	O
then	O
the	O
kernel	B-Algorithm
of	O
L	O
is	O
the	O
solution	O
set	O
to	O
a	O
homogeneous	O
system	O
of	O
linear	O
equations	O
.	O
</s>
<s>
Let	O
C[0,1]	O
denote	O
the	O
vector	O
space	O
of	O
all	O
continuous	B-Algorithm
real-valued	O
functions	O
on	O
the	O
interval	O
[0,1],	O
and	O
define	O
by	O
the	O
rule	O
Then	O
the	O
kernel	B-Algorithm
of	O
L	O
consists	O
of	O
all	O
functions	O
for	O
which	O
.	O
</s>
<s>
Let	O
C∞( R	O
)	O
be	O
the	O
vector	O
space	O
of	O
all	O
infinitely	O
differentiable	O
functions	O
,	O
and	O
let	O
be	O
the	O
differentiation	O
operator	O
:	O
Then	O
the	O
kernel	B-Algorithm
of	O
D	O
consists	O
of	O
all	O
functions	O
in	O
whose	O
derivatives	O
are	O
zero	O
,	O
i.e.	O
</s>
<s>
Let	O
be	O
the	O
direct	O
product	O
of	O
infinitely	O
many	O
copies	O
of	O
,	O
and	O
let	O
be	O
the	O
shift	B-Algorithm
operator	I-Algorithm
Then	O
the	O
kernel	B-Algorithm
of	O
s	O
is	O
the	O
one-dimensional	O
subspace	O
consisting	O
of	O
all	O
vectors	O
.	O
</s>
<s>
If	O
is	O
an	O
inner	O
product	O
space	O
and	O
is	O
a	O
subspace	O
,	O
the	O
kernel	B-Algorithm
of	O
the	O
orthogonal	B-Algorithm
projection	I-Algorithm
is	O
the	O
orthogonal	B-Algorithm
complement	I-Algorithm
to	O
in	O
.	O
</s>
<s>
A	O
basis	O
of	O
the	O
kernel	B-Algorithm
of	I-Algorithm
a	I-Algorithm
matrix	I-Algorithm
may	O
be	O
computed	O
by	O
Gaussian	B-Algorithm
elimination	I-Algorithm
.	O
</s>
<s>
For	O
this	O
purpose	O
,	O
given	O
an	O
m	O
×	O
n	O
matrix	O
A	O
,	O
we	O
construct	O
first	O
the	O
row	O
augmented	B-Algorithm
matrix	I-Algorithm
where	O
is	O
the	O
n	O
×	O
n	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Computing	O
its	O
column	O
echelon	O
form	O
by	O
Gaussian	B-Algorithm
elimination	I-Algorithm
(	O
or	O
any	O
other	O
suitable	O
method	O
)	O
,	O
we	O
get	O
a	O
matrix	O
A	O
basis	O
of	O
the	O
kernel	B-Algorithm
of	O
A	O
consists	O
in	O
the	O
non-zero	O
columns	O
of	O
C	O
such	O
that	O
the	O
corresponding	O
column	O
of	O
B	O
is	O
a	O
zero	B-Algorithm
column	I-Algorithm
.	O
</s>
<s>
are	O
a	O
basis	O
of	O
the	O
kernel	B-Algorithm
of	O
A	O
.	O
</s>
<s>
Proof	O
that	O
the	O
method	O
computes	O
the	O
kernel	B-Algorithm
:	O
Since	O
column	O
operations	O
correspond	O
to	O
post-multiplication	O
by	O
invertible	O
matrices	O
,	O
the	O
fact	O
that	O
reduces	O
to	O
means	O
that	O
there	O
exists	O
an	O
invertible	O
matrix	O
such	O
that	O
with	O
in	O
column	O
echelon	O
form	O
.	O
</s>
<s>
The	O
problem	O
of	O
computing	O
the	O
kernel	B-Algorithm
on	O
a	O
computer	O
depends	O
on	O
the	O
nature	O
of	O
the	O
coefficients	O
.	O
</s>
<s>
If	O
the	O
coefficients	O
of	O
the	O
matrix	O
are	O
exactly	O
given	O
numbers	O
,	O
the	O
column	O
echelon	O
form	O
of	O
the	O
matrix	O
may	O
be	O
computed	O
by	O
Bareiss	B-Algorithm
algorithm	I-Algorithm
more	O
efficiently	O
than	O
with	O
Gaussian	B-Algorithm
elimination	I-Algorithm
.	O
</s>
<s>
It	O
is	O
even	O
more	O
efficient	O
to	O
use	O
modular	O
arithmetic	O
and	O
Chinese	O
remainder	O
theorem	O
,	O
which	O
reduces	O
the	O
problem	O
to	O
several	O
similar	O
ones	O
over	O
finite	O
fields	O
(	O
this	O
avoids	O
the	O
overhead	O
induced	O
by	O
the	O
non-linearity	O
of	O
the	O
computational	B-General_Concept
complexity	I-General_Concept
of	O
integer	O
multiplication	O
)	O
.	O
</s>
<s>
For	O
coefficients	O
in	O
a	O
finite	O
field	O
,	O
Gaussian	B-Algorithm
elimination	I-Algorithm
works	O
well	O
,	O
but	O
for	O
the	O
large	O
matrices	O
that	O
occur	O
in	O
cryptography	O
and	O
Gröbner	O
basis	O
computation	O
,	O
better	O
algorithms	O
are	O
known	O
,	O
which	O
have	O
roughly	O
the	O
same	O
computational	B-General_Concept
complexity	I-General_Concept
,	O
but	O
are	O
faster	O
and	O
behave	O
better	O
with	O
modern	O
computer	B-Architecture
hardware	I-Architecture
.	O
</s>
<s>
For	O
matrices	O
whose	O
entries	O
are	O
floating-point	B-Algorithm
numbers	I-Algorithm
,	O
the	O
problem	O
of	O
computing	O
the	O
kernel	B-Algorithm
makes	O
sense	O
only	O
for	O
matrices	O
such	O
that	O
the	O
number	O
of	O
rows	O
is	O
equal	O
to	O
their	O
rank	O
:	O
because	O
of	O
the	O
rounding	B-Algorithm
errors	I-Algorithm
,	O
a	O
floating-point	B-Algorithm
matrix	O
has	O
almost	O
always	O
a	O
full	O
rank	O
,	O
even	O
when	O
it	O
is	O
an	O
approximation	O
of	O
a	O
matrix	O
of	O
a	O
much	O
smaller	O
rank	O
.	O
</s>
<s>
Even	O
for	O
a	O
full-rank	O
matrix	O
,	O
it	O
is	O
possible	O
to	O
compute	O
its	O
kernel	B-Algorithm
only	O
if	O
it	O
is	O
well	B-Algorithm
conditioned	I-Algorithm
,	O
i.e.	O
</s>
<s>
it	O
has	O
a	O
low	O
condition	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
Even	O
for	O
a	O
well	B-Algorithm
conditioned	I-Algorithm
full	O
rank	O
matrix	O
,	O
Gaussian	B-Algorithm
elimination	I-Algorithm
does	O
not	O
behave	O
correctly	O
:	O
it	O
introduces	O
rounding	B-Algorithm
errors	I-Algorithm
that	O
are	O
too	O
large	O
for	O
getting	O
a	O
significant	O
result	O
.	O
</s>
<s>
As	O
the	O
computation	O
of	O
the	O
kernel	B-Algorithm
of	I-Algorithm
a	I-Algorithm
matrix	I-Algorithm
is	O
a	O
special	O
instance	O
of	O
solving	O
a	O
homogeneous	O
system	O
of	O
linear	O
equations	O
,	O
the	O
kernel	B-Algorithm
may	O
be	O
computed	O
by	O
any	O
of	O
the	O
various	O
algorithms	O
designed	O
to	O
solve	O
homogeneous	O
systems	O
.	O
</s>
<s>
A	O
state	O
of	O
the	O
art	O
software	O
for	O
this	O
purpose	O
is	O
the	O
Lapack	B-Application
library	O
.	O
</s>
