<s>
In	O
mathematics	O
,	O
and	O
more	O
specifically	O
in	O
linear	B-Language
algebra	I-Language
,	O
a	O
linear	B-Architecture
map	I-Architecture
(	O
also	O
called	O
a	O
linear	B-Architecture
mapping	I-Architecture
,	O
linear	B-Architecture
transformation	I-Architecture
,	O
vector	B-Architecture
space	I-Architecture
homomorphism	I-Architecture
,	O
or	O
in	O
some	O
contexts	O
linear	O
function	O
)	O
is	O
a	O
mapping	B-Algorithm
between	O
two	O
vector	O
spaces	O
that	O
preserves	O
the	O
operations	O
of	O
vector	O
addition	O
and	O
scalar	O
multiplication	O
.	O
</s>
<s>
If	O
a	O
linear	B-Architecture
map	I-Architecture
is	O
a	O
bijection	B-Algorithm
then	O
it	O
is	O
called	O
a	O
.	O
</s>
<s>
In	O
the	O
case	O
where	O
,	O
a	O
linear	B-Architecture
map	I-Architecture
is	O
called	O
a	O
linear	O
endomorphism	O
.	O
</s>
<s>
Sometimes	O
the	O
term	O
refers	O
to	O
this	O
case	O
,	O
but	O
the	O
term	O
"	O
linear	B-Architecture
operator	I-Architecture
"	O
can	O
have	O
different	O
meanings	O
for	O
different	O
conventions	O
:	O
for	O
example	O
,	O
it	O
can	O
be	O
used	O
to	O
emphasize	O
that	O
and	O
are	O
real	O
vector	O
spaces	O
(	O
not	O
necessarily	O
with	O
)	O
,	O
or	O
it	O
can	O
be	O
used	O
to	O
emphasize	O
that	O
is	O
a	O
function	B-Algorithm
space	I-Algorithm
,	O
which	O
is	O
a	O
common	O
convention	O
in	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
Sometimes	O
the	O
term	O
linear	O
function	O
has	O
the	O
same	O
meaning	O
as	O
linear	B-Architecture
map	I-Architecture
,	O
while	O
in	O
analysis	O
it	O
does	O
not	O
.	O
</s>
<s>
A	O
linear	B-Architecture
map	I-Architecture
from	O
V	O
to	O
W	O
always	O
maps	O
the	O
origin	O
of	O
V	O
to	O
the	O
origin	O
of	O
W	O
.	O
Moreover	O
,	O
it	O
maps	O
linear	O
subspaces	O
in	O
V	O
onto	B-Algorithm
linear	O
subspaces	O
in	O
W	O
(	O
possibly	O
of	O
a	O
lower	O
dimension	O
)	O
;	O
for	O
example	O
,	O
it	O
maps	O
a	O
plane	O
through	O
the	O
origin	O
in	O
V	O
to	O
either	O
a	O
plane	O
through	O
the	O
origin	O
in	O
W	O
,	O
a	O
line	O
through	O
the	O
origin	O
in	O
W	O
,	O
or	O
just	O
the	O
origin	O
in	O
W	O
.	O
Linear	B-Architecture
maps	I-Architecture
can	O
often	O
be	O
represented	O
as	O
matrices	B-Architecture
,	O
and	O
simple	O
examples	O
include	O
rotation	B-Algorithm
and	I-Algorithm
reflection	I-Algorithm
linear	I-Algorithm
transformations	I-Algorithm
.	O
</s>
<s>
In	O
the	O
language	O
of	O
category	O
theory	O
,	O
linear	B-Architecture
maps	I-Architecture
are	O
the	O
morphisms	O
of	O
vector	O
spaces	O
.	O
</s>
<s>
A	O
function	O
is	O
said	O
to	O
be	O
a	O
linear	B-Architecture
map	I-Architecture
if	O
for	O
any	O
two	O
vectors	O
and	O
any	O
scalar	O
the	O
following	O
two	O
conditions	O
are	O
satisfied	O
:	O
</s>
<s>
Thus	O
,	O
a	O
linear	B-Architecture
map	I-Architecture
is	O
said	O
to	O
be	O
operation	O
preserving	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
it	O
does	O
not	O
matter	O
whether	O
the	O
linear	B-Architecture
map	I-Architecture
is	O
applied	O
before	O
(	O
the	O
right	O
hand	O
sides	O
of	O
the	O
above	O
examples	O
)	O
or	O
after	O
(	O
the	O
left	O
hand	O
sides	O
of	O
the	O
examples	O
)	O
the	O
operations	O
of	O
addition	O
and	O
scalar	O
multiplication	O
.	O
</s>
<s>
Thus	O
a	O
linear	B-Architecture
map	I-Architecture
is	O
one	O
which	O
preserves	O
linear	O
combinations	O
.	O
</s>
<s>
A	O
linear	B-Architecture
map	I-Architecture
with	O
viewed	O
as	O
a	O
one-dimensional	O
vector	O
space	O
over	O
itself	O
is	O
called	O
a	O
linear	B-Algorithm
functional	I-Algorithm
.	O
</s>
<s>
A	O
prototypical	O
example	O
that	O
gives	O
linear	B-Architecture
maps	I-Architecture
their	O
name	O
is	O
a	O
function	O
,	O
of	O
which	O
the	O
graph	B-Application
is	O
a	O
line	O
through	O
the	O
origin	O
.	O
</s>
<s>
More	O
generally	O
,	O
any	O
homothety	B-Algorithm
where	O
centered	O
in	O
the	O
origin	O
of	O
a	O
vector	O
space	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
.	O
</s>
<s>
The	O
identity	O
map	O
on	O
any	O
module	O
is	O
a	O
linear	B-Architecture
operator	I-Architecture
.	O
</s>
<s>
For	O
real	O
numbers	O
,	O
the	O
map	O
is	O
not	O
linear	O
(	O
but	O
is	O
an	O
affine	B-Algorithm
transformation	I-Algorithm
)	O
.	O
</s>
<s>
If	O
is	O
a	O
real	O
matrix	B-Architecture
,	O
then	O
defines	O
a	O
linear	B-Architecture
map	I-Architecture
from	O
to	O
by	O
sending	O
a	O
column	O
vector	O
to	O
the	O
column	O
vector	O
.	O
</s>
<s>
Conversely	O
,	O
any	O
linear	B-Architecture
map	I-Architecture
between	O
finite-dimensional	O
vector	O
spaces	O
can	O
be	O
represented	O
in	O
this	O
manner	O
;	O
see	O
the	O
,	O
below	O
.	O
</s>
<s>
If	O
is	O
an	O
isometry	O
between	O
real	O
normed	O
spaces	O
such	O
that	O
then	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
.	O
</s>
<s>
Differentiation	B-Algorithm
defines	O
a	O
linear	B-Architecture
map	I-Architecture
from	O
the	O
space	O
of	O
all	O
differentiable	O
functions	O
to	O
the	O
space	O
of	O
all	O
functions	O
.	O
</s>
<s>
It	O
also	O
defines	O
a	O
linear	B-Architecture
operator	I-Architecture
on	O
the	O
space	O
of	O
all	O
smooth	O
functions	O
(	O
a	O
linear	B-Architecture
operator	I-Architecture
is	O
a	O
linear	O
endomorphism	O
,	O
that	O
is	O
,	O
a	O
linear	B-Architecture
map	I-Architecture
with	O
the	O
same	O
domain	B-Algorithm
and	O
codomain	B-Algorithm
)	O
.	O
</s>
<s>
A	O
definite	O
integral	O
over	O
some	O
interval	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
from	O
the	O
space	O
of	O
all	O
real-valued	O
integrable	O
functions	O
on	O
to	O
.	O
</s>
<s>
An	O
indefinite	B-Algorithm
integral	I-Algorithm
(	O
or	O
antiderivative	B-Algorithm
)	O
with	O
a	O
fixed	O
integration	O
starting	O
point	O
defines	O
a	O
linear	B-Architecture
map	I-Architecture
from	O
the	O
space	O
of	O
all	O
real-valued	O
integrable	O
functions	O
on	O
to	O
the	O
space	O
of	O
all	O
real-valued	O
,	O
differentiable	O
functions	O
on	O
.	O
</s>
<s>
Without	O
a	O
fixed	O
starting	O
point	O
,	O
the	O
antiderivative	B-Algorithm
maps	O
to	O
the	O
quotient	O
space	O
of	O
the	O
differentiable	O
functions	O
by	O
the	O
linear	O
space	O
of	O
constant	O
functions	O
.	O
</s>
<s>
If	O
and	O
are	O
finite-dimensional	O
vector	O
spaces	O
over	O
a	O
field	O
,	O
of	O
respective	O
dimensions	O
and	O
,	O
then	O
the	O
function	O
that	O
maps	O
linear	B-Architecture
maps	I-Architecture
to	O
matrices	B-Architecture
in	O
the	O
way	O
described	O
in	O
(	O
below	O
)	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
,	O
and	O
even	O
a	O
linear	B-Architecture
isomorphism	I-Architecture
.	O
</s>
<s>
Often	O
,	O
a	O
linear	B-Architecture
map	I-Architecture
is	O
constructed	O
by	O
defining	O
it	O
on	O
a	O
subset	O
of	O
a	O
vector	O
space	O
and	O
then	O
to	O
the	O
linear	O
span	O
of	O
the	O
domain	B-Algorithm
.	O
</s>
<s>
When	O
the	O
subset	O
is	O
a	O
vector	O
subspace	O
of	O
then	O
a	O
(	O
-valued	O
)	O
linear	O
extension	O
of	O
to	O
all	O
of	O
is	O
guaranteed	O
to	O
exist	O
if	O
(	O
and	O
only	O
if	O
)	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
.	O
</s>
<s>
If	O
and	O
are	O
finite-dimensional	O
vector	O
spaces	O
and	O
a	O
basis	O
is	O
defined	O
for	O
each	O
vector	O
space	O
,	O
then	O
every	O
linear	B-Architecture
map	I-Architecture
from	O
to	O
can	O
be	O
represented	O
by	O
a	O
matrix	B-Architecture
.	O
</s>
<s>
Matrices	B-Architecture
yield	O
examples	O
of	O
linear	B-Architecture
maps	I-Architecture
:	O
if	O
is	O
a	O
real	O
matrix	B-Architecture
,	O
then	O
describes	O
a	O
linear	B-Architecture
map	I-Architecture
(	O
see	O
Euclidean	O
space	O
)	O
.	O
</s>
<s>
If	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
,	O
</s>
<s>
If	O
we	O
put	O
these	O
values	O
into	O
an	O
matrix	B-Architecture
,	O
then	O
we	O
can	O
conveniently	O
use	O
it	O
to	O
compute	O
the	O
vector	O
output	O
of	O
for	O
any	O
vector	O
in	O
.	O
</s>
<s>
To	O
define	O
it	O
more	O
clearly	O
,	O
for	O
some	O
column	O
that	O
corresponds	O
to	O
the	O
mapping	B-Algorithm
,	O
</s>
<s>
where	O
is	O
the	O
matrix	B-Architecture
of	O
.	O
</s>
<s>
A	O
single	O
linear	B-Architecture
map	I-Architecture
may	O
be	O
represented	O
by	O
many	O
matrices	B-Architecture
.	O
</s>
<s>
This	O
is	O
because	O
the	O
values	O
of	O
the	O
elements	O
of	O
a	O
matrix	B-Architecture
depend	O
on	O
the	O
bases	O
chosen	O
.	O
</s>
<s>
The	O
matrices	B-Architecture
of	O
a	O
linear	B-Architecture
transformation	I-Architecture
can	O
be	O
represented	O
visually	O
:	O
</s>
<s>
Matrix	B-Architecture
for	O
relative	O
to	O
:	O
</s>
<s>
Matrix	B-Architecture
for	O
relative	O
to	O
:	O
</s>
<s>
Transition	O
matrix	B-Architecture
from	O
to	O
:	O
</s>
<s>
Transition	O
matrix	B-Architecture
from	O
to	O
:	O
</s>
<s>
In	O
two-dimensional	O
space	O
R2	O
linear	B-Architecture
maps	I-Architecture
are	O
described	O
by	O
2	O
×	O
2	O
matrices	B-Architecture
.	O
</s>
<s>
scaling	B-Algorithm
by	O
2	O
in	O
all	O
directions	O
:	O
</s>
<s>
horizontal	B-Algorithm
shear	I-Algorithm
mapping	I-Algorithm
:	O
</s>
<s>
squeeze	O
mapping	B-Algorithm
:	O
</s>
<s>
projection	B-Algorithm
onto	B-Algorithm
the	O
y	O
axis	O
:	O
</s>
<s>
The	O
composition	O
of	O
linear	B-Architecture
maps	I-Architecture
is	O
linear	O
:	O
if	O
and	O
are	O
linear	O
,	O
then	O
so	O
is	O
their	O
composition	O
.	O
</s>
<s>
The	O
inverse	O
of	O
a	O
linear	B-Architecture
map	I-Architecture
,	O
when	O
defined	O
,	O
is	O
again	O
a	O
linear	B-Architecture
map	I-Architecture
.	O
</s>
<s>
Thus	O
the	O
set	O
of	O
linear	B-Architecture
maps	I-Architecture
from	O
to	O
itself	O
forms	O
a	O
vector	O
space	O
over	O
,	O
sometimes	O
denoted	O
.	O
</s>
<s>
Furthermore	O
,	O
in	O
the	O
case	O
that	O
,	O
this	O
vector	O
space	O
,	O
denoted	O
,	O
is	O
an	O
associative	O
algebra	O
under	O
composition	O
of	O
maps	O
,	O
since	O
the	O
composition	O
of	O
two	O
linear	B-Architecture
maps	I-Architecture
is	O
again	O
a	O
linear	B-Architecture
map	I-Architecture
,	O
and	O
the	O
composition	O
of	O
maps	O
is	O
always	O
associative	O
.	O
</s>
<s>
Given	O
again	O
the	O
finite-dimensional	O
case	O
,	O
if	O
bases	O
have	O
been	O
chosen	O
,	O
then	O
the	O
composition	O
of	O
linear	B-Architecture
maps	I-Architecture
corresponds	O
to	O
the	O
matrix	B-Architecture
multiplication	O
,	O
the	O
addition	O
of	O
linear	B-Architecture
maps	I-Architecture
corresponds	O
to	O
the	O
matrix	B-Architecture
addition	O
,	O
and	O
the	O
multiplication	O
of	O
linear	B-Architecture
maps	I-Architecture
with	O
scalars	O
corresponds	O
to	O
the	O
multiplication	O
of	O
matrices	B-Architecture
with	O
scalars	O
.	O
</s>
<s>
A	O
linear	B-Architecture
transformation	I-Architecture
is	O
an	O
endomorphism	O
of	O
;	O
the	O
set	O
of	O
all	O
such	O
endomorphisms	O
together	O
with	O
addition	O
,	O
composition	O
and	O
scalar	O
multiplication	O
as	O
defined	O
above	O
forms	O
an	O
associative	O
algebra	O
with	O
identity	O
element	O
over	O
the	O
field	O
(	O
and	O
in	O
particular	O
a	O
ring	O
)	O
.	O
</s>
<s>
The	O
composition	O
of	O
two	O
automorphisms	O
is	O
again	O
an	O
automorphism	O
,	O
and	O
the	O
set	O
of	O
all	O
automorphisms	O
of	O
forms	O
a	O
group	O
,	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
which	O
is	O
denoted	O
by	O
or	O
.	O
</s>
<s>
If	O
has	O
finite	O
dimension	O
,	O
then	O
is	O
isomorphic	O
to	O
the	O
associative	O
algebra	O
of	O
all	O
matrices	B-Architecture
with	O
entries	O
in	O
.	O
</s>
<s>
The	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
is	O
isomorphic	O
to	O
the	O
general	O
linear	O
group	O
of	O
all	O
invertible	O
matrices	B-Architecture
with	O
entries	O
in	O
.	O
</s>
<s>
If	O
and	O
are	O
finite-dimensional	O
,	O
bases	O
have	O
been	O
chosen	O
and	O
is	O
represented	O
by	O
the	O
matrix	B-Architecture
,	O
then	O
the	O
rank	O
and	O
nullity	O
of	O
are	O
equal	O
to	O
the	O
rank	O
and	O
nullity	O
of	O
the	O
matrix	B-Architecture
,	O
respectively	O
.	O
</s>
<s>
This	O
is	O
the	O
dual	O
notion	O
to	O
the	O
kernel	B-Algorithm
:	O
just	O
as	O
the	O
kernel	B-Algorithm
is	O
a	O
subspace	O
of	O
the	O
domain	B-Algorithm
,	O
the	O
co-kernel	O
is	O
a	O
quotient	O
space	O
of	O
the	O
target	O
.	O
</s>
<s>
the	O
kernel	B-Algorithm
is	O
the	O
space	O
of	O
solutions	O
to	O
the	O
homogeneous	O
equation	O
f(v )	O
=	O
0	O
,	O
and	O
its	O
dimension	O
is	O
the	O
number	O
of	O
degrees	O
of	O
freedom	O
in	O
the	O
space	O
of	O
solutions	O
,	O
if	O
it	O
is	O
not	O
empty	O
;	O
</s>
<s>
the	O
co-kernel	O
is	O
the	O
space	O
of	O
constraints	O
that	O
the	O
solutions	O
must	O
satisfy	O
,	O
and	O
its	O
dimension	O
is	O
the	O
maximal	O
number	O
of	O
independent	O
constraints	O
.	O
</s>
<s>
The	O
dimension	O
of	O
the	O
co-kernel	O
and	O
the	O
dimension	O
of	O
the	O
image	O
(	O
the	O
rank	O
)	O
add	O
up	O
to	O
the	O
dimension	O
of	O
the	O
target	O
space	O
.	O
</s>
<s>
The	O
kernel	B-Algorithm
may	O
be	O
expressed	O
as	O
the	O
subspace	O
(	O
x	O
,	O
0	O
)	O
<	O
V	O
:	O
the	O
value	O
of	O
x	O
is	O
the	O
freedom	O
in	O
a	O
solution	O
–	O
while	O
the	O
cokernel	O
may	O
be	O
expressed	O
via	O
the	O
map	O
W	O
→	O
R	O
,	O
:	O
given	O
a	O
vector	O
(	O
a	O
,	O
b	O
)	O
,	O
the	O
value	O
of	O
a	O
is	O
the	O
obstruction	O
to	O
there	O
being	O
a	O
solution	O
.	O
</s>
<s>
Thus	O
,	O
whereas	O
its	O
kernel	B-Algorithm
has	O
dimension	O
0	O
(	O
it	O
maps	O
only	O
the	O
zero	O
sequence	O
to	O
the	O
zero	O
sequence	O
)	O
,	O
its	O
co-kernel	O
has	O
dimension	O
1	O
.	O
</s>
<s>
Since	O
the	O
domain	B-Algorithm
and	O
the	O
target	O
space	O
are	O
the	O
same	O
,	O
the	O
rank	O
and	O
the	O
dimension	O
of	O
the	O
kernel	B-Algorithm
add	O
up	O
to	O
the	O
same	O
sum	O
as	O
the	O
rank	O
and	O
the	O
dimension	O
of	O
the	O
co-kernel	O
(	O
)	O
,	O
but	O
in	O
the	O
infinite-dimensional	O
case	O
it	O
cannot	O
be	O
inferred	O
that	O
the	O
kernel	B-Algorithm
and	O
the	O
co-kernel	O
of	O
an	O
endomorphism	O
have	O
the	O
same	O
dimension	O
(	O
0	O
≠	O
1	O
)	O
.	O
</s>
<s>
Its	O
image	O
is	O
the	O
entire	O
target	O
space	O
,	O
and	O
hence	O
its	O
co-kernel	O
has	O
dimension	O
0	O
,	O
but	O
since	O
it	O
maps	O
all	O
sequences	O
in	O
which	O
only	O
the	O
first	O
element	O
is	O
non-zero	O
to	O
the	O
zero	O
sequence	O
,	O
its	O
kernel	B-Algorithm
has	O
dimension	O
1	O
.	O
</s>
<s>
For	O
a	O
linear	B-Architecture
operator	I-Architecture
with	O
finite-dimensional	O
kernel	B-Algorithm
and	O
co-kernel	O
,	O
one	O
may	O
define	O
index	O
as	O
:	O
</s>
<s>
This	O
gives	O
an	O
indication	O
of	O
how	O
many	O
solutions	O
or	O
how	O
many	O
constraints	O
one	O
has	O
:	O
if	O
mapping	B-Algorithm
from	O
a	O
larger	O
space	O
to	O
a	O
smaller	O
one	O
,	O
the	O
map	O
may	O
be	O
onto	B-Algorithm
,	O
and	O
thus	O
will	O
have	O
degrees	O
of	O
freedom	O
even	O
without	O
constraints	O
.	O
</s>
<s>
Conversely	O
,	O
if	O
mapping	B-Algorithm
from	O
a	O
smaller	O
space	O
to	O
a	O
larger	O
one	O
,	O
the	O
map	O
cannot	O
be	O
onto	B-Algorithm
,	O
and	O
thus	O
one	O
will	O
have	O
constraints	O
even	O
without	O
degrees	O
of	O
freedom	O
.	O
</s>
<s>
No	O
classification	O
of	O
linear	B-Architecture
maps	I-Architecture
could	O
be	O
exhaustive	O
.	O
</s>
<s>
Let	O
and	O
denote	O
vector	O
spaces	O
over	O
a	O
field	O
and	O
let	O
be	O
a	O
linear	B-Architecture
map	I-Architecture
.	O
</s>
<s>
is	O
said	O
to	O
be	O
injective	O
or	O
a	O
monomorphism	B-Application
if	O
any	O
of	O
the	O
following	O
equivalent	O
conditions	O
are	O
true	O
:	O
</s>
<s>
is	O
monic	O
or	O
left-cancellable	O
,	O
which	O
is	O
to	O
say	O
,	O
for	O
any	O
vector	O
space	O
and	O
any	O
pair	O
of	O
linear	B-Architecture
maps	I-Architecture
and	O
,	O
the	O
equation	O
implies	O
.	O
</s>
<s>
is	O
left-invertible	O
,	O
which	O
is	O
to	O
say	O
there	O
exists	O
a	O
linear	B-Architecture
map	I-Architecture
such	O
that	O
is	O
the	O
identity	O
map	O
on	O
.	O
</s>
<s>
is	O
said	O
to	O
be	O
surjective	B-Algorithm
or	O
an	O
epimorphism	O
if	O
any	O
of	O
the	O
following	O
equivalent	O
conditions	O
are	O
true	O
:	O
</s>
<s>
is	O
onto	B-Algorithm
as	O
a	O
map	O
of	O
sets	O
.	O
</s>
<s>
is	O
epic	O
or	O
right-cancellable	O
,	O
which	O
is	O
to	O
say	O
,	O
for	O
any	O
vector	O
space	O
and	O
any	O
pair	O
of	O
linear	B-Architecture
maps	I-Architecture
and	O
,	O
the	O
equation	O
implies	O
.	O
</s>
<s>
is	O
right-invertible	O
,	O
which	O
is	O
to	O
say	O
there	O
exists	O
a	O
linear	B-Architecture
map	I-Architecture
such	O
that	O
is	O
the	O
identity	O
map	O
on	O
.	O
</s>
<s>
This	O
is	O
equivalent	O
to	O
being	O
both	O
one-to-one	B-Algorithm
and	I-Algorithm
onto	I-Algorithm
(	O
a	O
bijection	B-Algorithm
of	O
sets	O
)	O
or	O
also	O
to	O
being	O
both	O
epic	O
and	O
monic	O
,	O
and	O
so	O
being	O
a	O
bimorphism	O
.	O
</s>
<s>
If	O
,	O
where	O
is	O
some	O
scalar	O
,	O
then	O
is	O
said	O
to	O
be	O
a	O
scaling	B-Algorithm
transformation	O
or	O
scalar	O
multiplication	O
map	O
;	O
see	O
scalar	O
matrix	B-Architecture
.	O
</s>
<s>
Given	O
a	O
linear	B-Architecture
map	I-Architecture
which	O
is	O
an	O
endomorphism	O
whose	O
matrix	B-Architecture
is	O
A	O
,	O
in	O
the	O
basis	O
B	O
of	O
the	O
space	O
it	O
transforms	O
vector	O
coordinates	O
 [ u ] 	O
as	O
 [ v ] 	O
=	O
A[u]	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
matrix	B-Architecture
in	O
the	O
new	O
basis	O
is	O
A′	O
=	O
B−1AB	O
,	O
being	O
B	O
the	O
matrix	B-Architecture
of	O
the	O
given	O
basis	O
.	O
</s>
<s>
Therefore	O
,	O
linear	B-Architecture
maps	I-Architecture
are	O
said	O
to	O
be	O
1-co	O
-	O
1-contra-variant	O
objects	O
,	O
or	O
type	O
(	O
1	O
,	O
1	O
)	O
tensors	B-Device
.	O
</s>
<s>
A	O
linear	B-Architecture
transformation	I-Architecture
between	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
,	O
for	O
example	O
normed	O
spaces	O
,	O
may	O
be	O
continuous	O
.	O
</s>
<s>
If	O
its	O
domain	B-Algorithm
and	O
codomain	B-Algorithm
are	O
the	O
same	O
,	O
it	O
will	O
then	O
be	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
A	O
linear	B-Architecture
operator	I-Architecture
on	O
a	O
normed	O
linear	O
space	O
is	O
continuous	O
if	O
and	O
only	O
if	O
it	O
is	O
bounded	O
,	O
for	O
example	O
,	O
when	O
the	O
domain	B-Algorithm
is	O
finite-dimensional	O
.	O
</s>
<s>
An	O
infinite-dimensional	O
domain	B-Algorithm
may	O
have	O
discontinuous	B-Algorithm
linear	I-Algorithm
operators	I-Algorithm
.	O
</s>
<s>
An	O
example	O
of	O
an	O
unbounded	O
,	O
hence	O
discontinuous	O
,	O
linear	B-Architecture
transformation	I-Architecture
is	O
differentiation	B-Algorithm
on	O
the	O
space	O
of	O
smooth	O
functions	O
equipped	O
with	O
the	O
supremum	O
norm	O
(	O
a	O
function	O
with	O
small	O
values	O
can	O
have	O
a	O
derivative	B-Algorithm
with	O
large	O
values	O
,	O
while	O
the	O
derivative	B-Algorithm
of	O
0	O
is	O
0	O
)	O
.	O
</s>
<s>
For	O
a	O
specific	O
example	O
,	O
converges	O
to	O
0	O
,	O
but	O
its	O
derivative	B-Algorithm
does	O
not	O
,	O
so	O
differentiation	B-Algorithm
is	O
not	O
continuous	O
at	O
0	O
(	O
and	O
by	O
a	O
variation	O
of	O
this	O
argument	O
,	O
it	O
is	O
not	O
continuous	O
anywhere	O
)	O
.	O
</s>
<s>
A	O
specific	O
application	O
of	O
linear	B-Architecture
maps	I-Architecture
is	O
for	O
geometric	B-Algorithm
transformations	I-Algorithm
,	O
such	O
as	O
those	O
performed	O
in	O
computer	O
graphics	O
,	O
where	O
the	O
translation	O
,	O
rotation	B-General_Concept
and	O
scaling	B-Algorithm
of	O
2D	O
or	O
3D	O
objects	O
is	O
performed	O
by	O
the	O
use	O
of	O
a	O
transformation	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Linear	B-Architecture
mappings	I-Architecture
also	O
are	O
used	O
as	O
a	O
mechanism	O
for	O
describing	O
change	O
:	O
for	O
example	O
in	O
calculus	O
correspond	O
to	O
derivatives	B-Algorithm
;	O
or	O
in	O
relativity	O
,	O
used	O
as	O
a	O
device	O
to	O
keep	O
track	O
of	O
the	O
local	O
transformations	O
of	O
reference	O
frames	O
.	O
</s>
<s>
Another	O
application	O
of	O
these	O
transformations	O
is	O
in	O
compiler	B-Application
optimizations	I-Application
of	O
nested-loop	O
code	O
,	O
and	O
in	O
parallelizing	O
compiler	O
techniques	O
.	O
</s>
