<s>
Linear	B-Language
algebra	I-Language
is	O
the	O
branch	O
of	O
mathematics	O
concerning	O
linear	O
equations	O
such	O
as	O
:	O
</s>
<s>
linear	B-Architecture
maps	I-Architecture
such	O
as	O
:	O
</s>
<s>
and	O
their	O
representations	O
in	O
vector	O
spaces	O
and	O
through	O
matrices	B-Architecture
.	O
</s>
<s>
Linear	B-Language
algebra	I-Language
is	O
central	O
to	O
almost	O
all	O
areas	O
of	O
mathematics	O
.	O
</s>
<s>
For	O
instance	O
,	O
linear	B-Language
algebra	I-Language
is	O
fundamental	O
in	O
modern	O
presentations	O
of	O
geometry	O
,	O
including	O
for	O
defining	O
basic	O
objects	O
such	O
as	O
lines	O
,	O
planes	O
and	O
rotations	B-General_Concept
.	O
</s>
<s>
Also	O
,	O
functional	B-Application
analysis	I-Application
,	O
a	O
branch	O
of	O
mathematical	O
analysis	O
,	O
may	O
be	O
viewed	O
as	O
the	O
application	O
of	O
linear	B-Language
algebra	I-Language
to	O
spaces	B-Algorithm
of	I-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
Linear	B-Language
algebra	I-Language
is	O
also	O
used	O
in	O
most	O
sciences	O
and	O
fields	O
of	O
engineering	O
,	O
because	O
it	O
allows	O
modeling	O
many	O
natural	O
phenomena	O
,	O
and	O
computing	O
efficiently	O
with	O
such	O
models	O
.	O
</s>
<s>
For	O
nonlinear	O
systems	O
,	O
which	O
cannot	O
be	O
modeled	O
with	O
linear	B-Language
algebra	I-Language
,	O
it	O
is	O
often	O
used	O
for	O
dealing	O
with	O
first-order	B-Algorithm
approximations	I-Algorithm
,	O
using	O
the	O
fact	O
that	O
the	O
differential	O
of	O
a	O
multivariate	O
function	O
at	O
a	O
point	O
is	O
the	O
linear	B-Architecture
map	I-Architecture
that	O
best	O
approximates	O
the	O
function	O
near	O
that	O
point	O
.	O
</s>
<s>
The	O
procedure	O
(	O
using	O
counting	O
rods	O
)	O
for	O
solving	O
simultaneous	O
linear	O
equations	O
now	O
called	O
Gaussian	B-Algorithm
elimination	I-Algorithm
appears	O
in	O
the	O
ancient	O
Chinese	O
mathematical	O
text	O
Chapter	O
Eight	O
:	O
Rectangular	O
Arrays	O
of	O
The	O
Nine	O
Chapters	O
on	O
the	O
Mathematical	O
Art	O
.	O
</s>
<s>
Later	O
,	O
Gauss	B-Algorithm
further	O
described	O
the	O
method	B-Algorithm
of	I-Algorithm
elimination	I-Algorithm
,	O
which	O
was	O
initially	O
listed	O
as	O
an	O
advancement	O
in	O
geodesy	O
.	O
</s>
<s>
In	O
1844	O
Hermann	O
Grassmann	O
published	O
his	O
"	O
Theory	O
of	O
Extension	O
"	O
which	O
included	O
foundational	O
new	O
topics	O
of	O
what	O
is	O
today	O
called	O
linear	B-Language
algebra	I-Language
.	O
</s>
<s>
In	O
1848	O
,	O
James	O
Joseph	O
Sylvester	O
introduced	O
the	O
term	O
matrix	B-Architecture
,	O
which	O
is	O
Latin	O
for	O
womb	O
.	O
</s>
<s>
Linear	B-Language
algebra	I-Language
grew	O
with	O
ideas	O
noted	O
in	O
the	O
complex	O
plane	O
.	O
</s>
<s>
Arthur	O
Cayley	O
introduced	O
matrix	B-Architecture
multiplication	O
and	O
the	O
inverse	O
matrix	B-Architecture
in	O
1856	O
,	O
making	O
possible	O
the	O
general	O
linear	O
group	O
.	O
</s>
<s>
Crucially	O
,	O
Cayley	O
used	O
a	O
single	O
letter	O
to	O
denote	O
a	O
matrix	B-Architecture
,	O
thus	O
treating	O
a	O
matrix	B-Architecture
as	O
an	O
aggregate	O
object	O
.	O
</s>
<s>
He	O
also	O
realized	O
the	O
connection	O
between	O
matrices	B-Architecture
and	O
determinants	O
,	O
and	O
wrote	O
"	O
There	O
would	O
be	O
many	O
things	O
to	O
say	O
about	O
this	O
theory	O
of	O
matrices	B-Architecture
which	O
should	O
,	O
it	O
seems	O
to	O
me	O
,	O
precede	O
the	O
theory	O
of	O
determinants	O
"	O
.	O
</s>
<s>
The	O
telegraph	O
required	O
an	O
explanatory	O
system	O
,	O
and	O
the	O
1873	O
publication	O
of	O
A	O
Treatise	O
on	O
Electricity	O
and	O
Magnetism	O
instituted	O
a	O
field	O
theory	O
of	O
forces	O
and	O
required	O
differential	B-Language
geometry	I-Language
for	O
expression	O
.	O
</s>
<s>
Linear	B-Language
algebra	I-Language
is	O
flat	O
differential	B-Language
geometry	I-Language
and	O
serves	O
in	O
tangent	O
spaces	O
to	O
manifolds	B-Architecture
.	O
</s>
<s>
Electromagnetic	O
symmetries	O
of	O
spacetime	O
are	O
expressed	O
by	O
the	O
Lorentz	O
transformations	O
,	O
and	O
much	O
of	O
the	O
history	O
of	O
linear	B-Language
algebra	I-Language
is	O
the	O
history	O
of	O
Lorentz	O
transformations	O
.	O
</s>
<s>
The	O
first	O
modern	O
and	O
more	O
precise	O
definition	O
of	O
a	O
vector	O
space	O
was	O
introduced	O
by	O
Peano	O
in	O
1888	O
;	O
by	O
1900	O
,	O
a	O
theory	O
of	O
linear	B-Architecture
transformations	I-Architecture
of	O
finite-dimensional	O
vector	O
spaces	O
had	O
emerged	O
.	O
</s>
<s>
Linear	B-Language
algebra	I-Language
took	O
its	O
modern	O
form	O
in	O
the	O
first	O
half	O
of	O
the	O
twentieth	O
century	O
,	O
when	O
many	O
ideas	O
and	O
methods	O
of	O
previous	O
centuries	O
were	O
generalized	O
as	O
abstract	O
algebra	O
.	O
</s>
<s>
The	O
development	O
of	O
computers	O
led	O
to	O
increased	O
research	O
in	O
efficient	O
algorithms	O
for	O
Gaussian	B-Algorithm
elimination	I-Algorithm
and	O
matrix	B-Architecture
decompositions	O
,	O
and	O
linear	B-Language
algebra	I-Language
became	O
an	O
essential	O
tool	O
for	O
modelling	O
and	O
simulations	O
.	O
</s>
<s>
Until	O
the	O
19th	O
century	O
,	O
linear	B-Language
algebra	I-Language
was	O
introduced	O
through	O
systems	O
of	O
linear	O
equations	O
and	O
matrices	B-Architecture
.	O
</s>
<s>
A	O
vector	O
space	O
over	O
a	O
field	O
(	O
often	O
the	O
field	O
of	O
the	O
real	O
numbers	O
)	O
is	O
a	O
set	O
equipped	O
with	O
two	O
binary	O
operations	O
satisfying	O
the	O
following	O
axioms	B-Algorithm
.	O
</s>
<s>
The	O
axioms	B-Algorithm
that	O
addition	O
and	O
scalar	O
multiplication	O
must	O
satisfy	O
are	O
the	O
following	O
.	O
</s>
<s>
The	O
first	O
four	O
axioms	B-Algorithm
mean	O
that	O
is	O
an	O
abelian	O
group	O
under	O
addition	O
.	O
</s>
<s>
An	O
element	O
of	O
a	O
specific	O
vector	O
space	O
may	O
have	O
various	O
nature	O
;	O
for	O
example	O
,	O
it	O
could	O
be	O
a	O
sequence	O
,	O
a	O
function	O
,	O
a	O
polynomial	O
or	O
a	O
matrix	B-Architecture
.	O
</s>
<s>
Linear	B-Language
algebra	I-Language
is	O
concerned	O
with	O
those	O
properties	O
of	O
such	O
objects	O
that	O
are	O
common	O
to	O
all	O
vector	O
spaces	O
.	O
</s>
<s>
Linear	B-Architecture
maps	I-Architecture
are	O
mappings	B-Algorithm
between	O
vector	O
spaces	O
that	O
preserve	O
the	O
vector-space	O
structure	O
.	O
</s>
<s>
When	O
are	O
the	O
same	O
vector	O
space	O
,	O
a	O
linear	B-Architecture
map	I-Architecture
is	O
also	O
known	O
as	O
a	O
linear	B-Architecture
operator	I-Architecture
on	O
.	O
</s>
<s>
A	O
bijective	B-Architecture
linear	I-Architecture
map	I-Architecture
between	O
two	O
vector	O
spaces	O
(	O
that	O
is	O
,	O
every	O
vector	O
from	O
the	O
second	O
space	O
is	O
associated	O
with	O
exactly	O
one	O
in	O
the	O
first	O
)	O
is	O
an	O
isomorphism	O
.	O
</s>
<s>
Because	O
an	O
isomorphism	O
preserves	O
linear	O
structure	O
,	O
two	O
isomorphic	O
vector	O
spaces	O
are	O
"	O
essentially	O
the	O
same	O
"	O
from	O
the	O
linear	B-Language
algebra	I-Language
point	O
of	O
view	O
,	O
in	O
the	O
sense	O
that	O
they	O
cannot	O
be	O
distinguished	O
by	O
using	O
vector	O
space	O
properties	O
.	O
</s>
<s>
An	O
essential	O
question	O
in	O
linear	B-Language
algebra	I-Language
is	O
testing	O
whether	O
a	O
linear	B-Architecture
map	I-Architecture
is	O
an	O
isomorphism	O
or	O
not	O
,	O
and	O
,	O
if	O
it	O
is	O
not	O
an	O
isomorphism	O
,	O
finding	O
its	O
range	B-Algorithm
(	O
or	O
image	O
)	O
and	O
the	O
set	O
of	O
elements	O
that	O
are	O
mapped	O
to	O
the	O
zero	O
vector	O
,	O
called	O
the	O
kernel	B-Algorithm
of	O
the	O
map	B-Algorithm
.	O
</s>
<s>
All	O
these	O
questions	O
can	O
be	O
solved	O
by	O
using	O
Gaussian	B-Algorithm
elimination	I-Algorithm
or	O
some	O
variant	O
of	O
this	O
algorithm	O
.	O
</s>
<s>
For	O
example	O
,	O
given	O
a	O
linear	B-Architecture
map	I-Architecture
,	O
the	O
image	O
of	O
,	O
and	O
the	O
inverse	O
image	O
of	O
(	O
called	O
kernel	B-Algorithm
or	O
null	B-Algorithm
space	I-Algorithm
)	O
,	O
are	O
linear	O
subspaces	O
of	O
and	O
,	O
respectively	O
.	O
</s>
<s>
Any	O
two	O
bases	O
of	O
a	O
vector	O
space	O
have	O
the	O
same	O
cardinality	B-Application
,	O
which	O
is	O
called	O
the	O
dimension	O
of	O
;	O
this	O
is	O
the	O
dimension	O
theorem	O
for	O
vector	O
spaces	O
.	O
</s>
<s>
Matrices	B-Architecture
allow	O
explicit	O
manipulation	O
of	O
finite-dimensional	O
vector	O
spaces	O
and	O
linear	B-Architecture
maps	I-Architecture
.	O
</s>
<s>
Their	O
theory	O
is	O
thus	O
an	O
essential	O
part	O
of	O
linear	B-Language
algebra	I-Language
.	O
</s>
<s>
is	O
a	O
bijection	B-Algorithm
from	O
,	O
the	O
set	O
of	O
the	O
sequences	O
of	O
elements	O
of	O
,	O
onto	O
.	O
</s>
<s>
If	O
is	O
another	O
finite	O
dimensional	O
vector	O
space	O
(	O
possibly	O
the	O
same	O
)	O
,	O
with	O
a	O
basis	O
,	O
a	O
linear	B-Architecture
map	I-Architecture
from	O
to	O
is	O
well	O
defined	O
by	O
its	O
values	O
on	O
the	O
basis	O
elements	O
,	O
that	O
is	O
.	O
</s>
<s>
Thus	O
,	O
is	O
well	O
represented	O
by	O
the	O
list	O
of	O
the	O
corresponding	O
column	O
matrices	B-Architecture
.	O
</s>
<s>
Matrix	B-Architecture
multiplication	O
is	O
defined	O
in	O
such	O
a	O
way	O
that	O
the	O
product	O
of	O
two	O
matrices	B-Architecture
is	O
the	O
matrix	B-Architecture
of	O
the	O
composition	B-Application
of	O
the	O
corresponding	O
linear	B-Architecture
maps	I-Architecture
,	O
and	O
the	O
product	O
of	O
a	O
matrix	B-Architecture
and	O
a	O
column	O
matrix	B-Architecture
is	O
the	O
column	O
matrix	B-Architecture
representing	O
the	O
result	O
of	O
applying	O
the	O
represented	O
linear	B-Architecture
map	I-Architecture
to	O
the	O
represented	O
vector	O
.	O
</s>
<s>
It	O
follows	O
that	O
the	O
theory	O
of	O
finite-dimensional	O
vector	O
spaces	O
and	O
the	O
theory	O
of	O
matrices	B-Architecture
are	O
two	O
different	O
languages	O
for	O
expressing	O
exactly	O
the	O
same	O
concepts	O
.	O
</s>
<s>
Two	O
matrices	B-Architecture
that	O
encode	O
the	O
same	O
linear	B-Architecture
transformation	I-Architecture
in	O
different	O
bases	O
are	O
called	O
similar	B-Algorithm
.	O
</s>
<s>
It	O
can	O
be	O
proved	O
that	O
two	O
matrices	B-Architecture
are	O
similar	B-Algorithm
if	O
and	O
only	O
if	O
one	O
can	O
transform	O
one	O
into	O
the	O
other	O
by	O
elementary	O
row	O
and	O
column	O
operations	O
.	O
</s>
<s>
For	O
a	O
matrix	B-Architecture
representing	O
a	O
linear	B-Architecture
map	I-Architecture
from	O
to	O
,	O
the	O
row	O
operations	O
correspond	O
to	O
change	O
of	O
bases	O
in	O
and	O
the	O
column	O
operations	O
correspond	O
to	O
change	O
of	O
bases	O
in	O
.	O
</s>
<s>
Every	O
matrix	B-Architecture
is	O
similar	B-Algorithm
to	O
an	O
identity	B-Algorithm
matrix	I-Algorithm
possibly	O
bordered	O
by	O
zero	O
rows	O
and	O
zero	O
columns	O
.	O
</s>
<s>
In	O
terms	O
of	O
vector	O
spaces	O
,	O
this	O
means	O
that	O
,	O
for	O
any	O
linear	B-Architecture
map	I-Architecture
from	O
to	O
,	O
there	O
are	O
bases	O
such	O
that	O
a	O
part	O
of	O
the	O
basis	O
of	O
is	O
mapped	O
bijectively	O
on	O
a	O
part	O
of	O
the	O
basis	O
of	O
,	O
and	O
that	O
the	O
remaining	O
basis	O
elements	O
of	O
,	O
if	O
any	O
,	O
are	O
mapped	O
to	O
zero	O
.	O
</s>
<s>
Gaussian	B-Algorithm
elimination	I-Algorithm
is	O
the	O
basic	O
algorithm	O
for	O
finding	O
these	O
elementary	O
operations	O
,	O
and	O
proving	O
these	O
results	O
.	O
</s>
<s>
Systems	O
of	O
linear	O
equations	O
form	O
a	O
fundamental	O
part	O
of	O
linear	B-Language
algebra	I-Language
.	O
</s>
<s>
Historically	O
,	O
linear	B-Language
algebra	I-Language
and	O
matrix	B-Architecture
theory	I-Architecture
has	O
been	O
developed	O
for	O
solving	O
such	O
systems	O
.	O
</s>
<s>
In	O
the	O
modern	O
presentation	O
of	O
linear	B-Language
algebra	I-Language
through	O
vector	O
spaces	O
and	O
matrices	B-Architecture
,	O
many	O
problems	O
may	O
be	O
interpreted	O
in	O
terms	O
of	O
linear	O
systems	O
.	O
</s>
<s>
Let	O
be	O
the	O
linear	B-Architecture
transformation	I-Architecture
associated	O
to	O
the	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
solutions	O
of	O
(	O
)	O
are	O
exactly	O
the	O
elements	O
of	O
the	O
kernel	B-Algorithm
of	O
or	O
,	O
equivalently	O
,	O
.	O
</s>
<s>
It	O
follows	O
from	O
this	O
matrix	B-Architecture
interpretation	O
of	O
linear	O
systems	O
that	O
the	O
same	O
methods	O
can	O
be	O
applied	O
for	O
solving	O
linear	O
systems	O
and	O
for	O
many	O
operations	O
on	O
matrices	B-Architecture
and	O
linear	B-Architecture
transformations	I-Architecture
,	O
which	O
include	O
the	O
computation	O
of	O
the	O
ranks	O
,	O
kernels	B-Algorithm
,	O
matrix	B-Architecture
inverses	O
.	O
</s>
<s>
A	O
linear	O
endomorphism	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
that	O
maps	O
a	O
vector	O
space	O
to	O
itself	O
.	O
</s>
<s>
If	O
has	O
a	O
basis	O
of	O
elements	O
,	O
such	O
an	O
endomorphism	O
is	O
represented	O
by	O
a	O
square	O
matrix	B-Architecture
of	O
size	O
.	O
</s>
<s>
With	O
respect	O
to	O
general	O
linear	B-Architecture
maps	I-Architecture
,	O
linear	O
endomorphisms	O
and	O
square	O
matrices	B-Architecture
have	O
some	O
specific	O
properties	O
that	O
make	O
their	O
study	O
an	O
important	O
part	O
of	O
linear	B-Language
algebra	I-Language
,	O
which	O
is	O
used	O
in	O
many	O
parts	O
of	O
mathematics	O
,	O
including	O
geometric	B-Algorithm
transformations	I-Algorithm
,	O
coordinate	O
changes	O
,	O
quadratic	O
forms	O
,	O
and	O
many	O
other	O
part	O
of	O
mathematics	O
.	O
</s>
<s>
where	O
is	O
the	O
group	B-Algorithm
of	I-Algorithm
all	I-Algorithm
permutations	I-Algorithm
of	O
elements	O
,	O
is	O
a	O
permutation	O
,	O
and	O
the	O
parity	O
of	O
the	O
permutation	O
.	O
</s>
<s>
A	O
matrix	B-Architecture
is	O
invertible	O
if	O
and	O
only	O
if	O
the	O
determinant	O
is	O
invertible	O
(	O
i.e.	O
,	O
nonzero	O
if	O
the	O
scalars	O
belong	O
to	O
a	O
field	O
)	O
.	O
</s>
<s>
Cramer	O
's	O
rule	O
is	O
useful	O
for	O
reasoning	O
about	O
the	O
solution	O
,	O
but	O
,	O
except	O
for	O
or	O
,	O
it	O
is	O
rarely	O
used	O
for	O
computing	O
a	O
solution	O
,	O
since	O
Gaussian	B-Algorithm
elimination	I-Algorithm
is	O
a	O
faster	O
algorithm	O
.	O
</s>
<s>
The	O
determinant	O
of	O
an	O
endomorphism	O
is	O
the	O
determinant	O
of	O
the	O
matrix	B-Architecture
representing	O
the	O
endomorphism	O
in	O
terms	O
of	O
some	O
ordered	O
basis	O
.	O
</s>
<s>
As	O
is	O
supposed	O
to	O
be	O
nonzero	O
,	O
this	O
means	O
that	O
is	O
a	O
singular	O
matrix	B-Architecture
,	O
and	O
thus	O
that	O
its	O
determinant	O
equals	O
zero	O
.	O
</s>
<s>
If	O
is	O
of	O
dimension	O
,	O
this	O
is	O
a	O
monic	O
polynomial	O
of	O
degree	O
,	O
called	O
the	O
characteristic	O
polynomial	O
of	O
the	O
matrix	B-Architecture
(	O
or	O
of	O
the	O
endomorphism	O
)	O
,	O
and	O
there	O
are	O
,	O
at	O
most	O
,	O
eigenvalues	O
.	O
</s>
<s>
If	O
a	O
basis	O
exists	O
that	O
consists	O
only	O
of	O
eigenvectors	O
,	O
the	O
matrix	B-Architecture
of	O
on	O
this	O
basis	O
has	O
a	O
very	O
simple	O
structure	O
:	O
it	O
is	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
such	O
that	O
the	O
entries	O
on	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
are	O
eigenvalues	O
,	O
and	O
the	O
other	O
entries	O
are	O
zero	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
endomorphism	O
and	O
the	O
matrix	B-Architecture
are	O
said	O
to	O
be	O
diagonalizable	B-Algorithm
.	O
</s>
<s>
More	O
generally	O
,	O
an	O
endomorphism	O
and	O
a	O
matrix	B-Architecture
are	O
also	O
said	O
diagonalizable	B-Algorithm
,	O
if	O
they	O
become	O
diagonalizable	B-Algorithm
after	O
extending	O
the	O
field	O
of	O
scalars	O
.	O
</s>
<s>
In	O
this	O
extended	O
sense	O
,	O
if	O
the	O
characteristic	O
polynomial	O
is	O
square-free	O
,	O
then	O
the	O
matrix	B-Architecture
is	O
diagonalizable	B-Algorithm
.	O
</s>
<s>
A	O
symmetric	B-Algorithm
matrix	I-Algorithm
is	O
always	O
diagonalizable	B-Algorithm
.	O
</s>
<s>
(	O
it	O
cannot	O
be	O
diagonalizable	B-Algorithm
since	O
its	O
square	O
is	O
the	O
zero	B-Algorithm
matrix	I-Algorithm
,	O
and	O
the	O
square	O
of	O
a	O
nonzero	O
diagonal	B-Algorithm
matrix	I-Algorithm
is	O
never	O
zero	O
)	O
.	O
</s>
<s>
When	O
an	O
endomorphism	O
is	O
not	O
diagonalizable	B-Algorithm
,	O
there	O
are	O
bases	O
on	O
which	O
it	O
has	O
a	O
simple	O
form	O
,	O
although	O
not	O
as	O
simple	O
as	O
the	O
diagonal	O
form	O
.	O
</s>
<s>
The	O
Frobenius	O
normal	B-Algorithm
form	O
does	O
not	O
need	O
of	O
extending	O
the	O
field	O
of	O
scalars	O
and	O
makes	O
the	O
characteristic	O
polynomial	O
immediately	O
readable	O
on	O
the	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
Jordan	O
normal	B-Algorithm
form	O
requires	O
to	O
extend	O
the	O
field	O
of	O
scalar	O
for	O
containing	O
all	O
eigenvalues	O
,	O
and	O
differs	O
from	O
the	O
diagonal	O
form	O
only	O
by	O
some	O
entries	O
that	O
are	O
just	O
above	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
and	O
are	O
equal	O
to	O
1	O
.	O
</s>
<s>
A	O
linear	B-Algorithm
form	I-Algorithm
is	O
a	O
linear	B-Architecture
map	I-Architecture
from	O
a	O
vector	O
space	O
over	O
a	O
field	O
to	O
the	O
field	O
of	O
scalars	O
,	O
viewed	O
as	O
a	O
vector	O
space	O
over	O
itself	O
.	O
</s>
<s>
Equipped	O
by	O
pointwise	O
addition	O
and	O
multiplication	O
by	O
a	O
scalar	O
,	O
the	O
linear	B-Algorithm
forms	I-Algorithm
form	O
a	O
vector	O
space	O
,	O
called	O
the	O
dual	O
space	O
of	O
,	O
and	O
usually	O
denoted	O
or	O
.	O
</s>
<s>
If	O
is	O
a	O
basis	O
of	O
(	O
this	O
implies	O
that	O
is	O
finite-dimensional	O
)	O
,	O
then	O
one	O
can	O
define	O
,	O
for	O
,	O
a	O
linear	B-Architecture
map	I-Architecture
such	O
that	O
and	O
if	O
.	O
</s>
<s>
These	O
linear	B-Architecture
maps	I-Architecture
form	O
a	O
basis	O
of	O
,	O
called	O
the	O
dual	O
basis	O
of	O
.	O
</s>
<s>
is	O
a	O
linear	B-Algorithm
form	I-Algorithm
on	O
.	O
</s>
<s>
This	O
defines	O
the	O
canonical	O
linear	B-Architecture
map	I-Architecture
from	O
into	O
,	O
the	O
dual	O
of	O
,	O
called	O
the	O
bidual	O
of	O
.	O
</s>
<s>
This	O
canonical	O
map	B-Algorithm
is	O
an	O
isomorphism	O
if	O
is	O
finite-dimensional	O
,	O
and	O
this	O
allows	O
identifying	O
with	O
its	O
bidual	O
.	O
</s>
<s>
(	O
In	O
the	O
infinite	O
dimensional	O
case	O
,	O
the	O
canonical	O
map	B-Algorithm
is	O
injective	O
,	O
but	O
not	O
surjective	O
.	O
)	O
</s>
<s>
be	O
a	O
linear	B-Architecture
map	I-Architecture
.	O
</s>
<s>
For	O
every	O
linear	B-Algorithm
form	I-Algorithm
on	O
,	O
the	O
composite	O
function	O
is	O
a	O
linear	B-Algorithm
form	I-Algorithm
on	O
.	O
</s>
<s>
If	O
and	O
are	O
finite	O
dimensional	O
,	O
and	O
is	O
the	O
matrix	B-Architecture
of	O
in	O
terms	O
of	O
some	O
ordered	O
bases	O
,	O
then	O
the	O
matrix	B-Architecture
of	O
over	O
the	O
dual	O
bases	O
is	O
the	O
transpose	O
of	O
,	O
obtained	O
by	O
exchanging	O
rows	O
and	O
columns	O
.	O
</s>
<s>
Besides	O
these	O
basic	O
concepts	O
,	O
linear	B-Language
algebra	I-Language
also	O
studies	O
vector	O
spaces	O
with	O
additional	O
structure	O
,	O
such	O
as	O
an	O
inner	O
product	O
.	O
</s>
<s>
that	O
satisfies	O
the	O
following	O
three	O
axioms	B-Algorithm
for	O
all	O
vectors	O
in	O
and	O
all	O
scalars	O
in	O
:	O
</s>
<s>
In	O
,	O
it	O
is	O
symmetric	B-Algorithm
.	O
</s>
<s>
Given	O
any	O
finite-dimensional	O
vector	O
space	O
,	O
an	O
orthonormal	O
basis	O
could	O
be	O
found	O
by	O
the	O
Gram	B-Algorithm
–	I-Algorithm
Schmidt	I-Algorithm
procedure	O
.	O
</s>
<s>
If	O
satisfies	O
,	O
we	O
call	O
normal	B-Algorithm
.	O
</s>
<s>
It	O
turns	O
out	O
that	O
normal	B-Algorithm
matrices	I-Algorithm
are	O
precisely	O
the	O
matrices	B-Architecture
that	O
have	O
an	O
orthonormal	O
system	O
of	O
eigenvectors	O
that	O
span	O
.	O
</s>
<s>
There	O
is	O
a	O
strong	O
relationship	O
between	O
linear	B-Language
algebra	I-Language
and	O
geometry	O
,	O
which	O
started	O
with	O
the	O
introduction	O
by	O
René	O
Descartes	O
,	O
in	O
1637	O
,	O
of	O
Cartesian	O
coordinates	O
.	O
</s>
<s>
This	O
was	O
one	O
of	O
the	O
main	O
motivations	O
for	O
developing	O
linear	B-Language
algebra	I-Language
.	O
</s>
<s>
Most	O
geometric	B-Algorithm
transformation	I-Algorithm
,	O
such	O
as	O
translations	B-Algorithm
,	O
rotations	B-General_Concept
,	O
reflections	O
,	O
rigid	B-Algorithm
motions	I-Algorithm
,	O
isometries	O
,	O
and	O
projections	O
transform	O
lines	O
into	O
lines	O
.	O
</s>
<s>
It	O
follows	O
that	O
they	O
can	O
be	O
defined	O
,	O
specified	O
and	O
studied	O
in	O
terms	O
of	O
linear	B-Architecture
maps	I-Architecture
.	O
</s>
<s>
This	O
is	O
also	O
the	O
case	O
of	O
homographies	B-Algorithm
and	O
Möbius	O
transformations	O
,	O
when	O
considered	O
as	O
transformations	O
of	O
a	O
projective	O
space	O
.	O
</s>
<s>
Until	O
the	O
end	O
of	O
the	O
19th	O
century	O
,	O
geometric	O
spaces	O
were	O
defined	O
by	O
axioms	B-Algorithm
relating	O
points	O
,	O
lines	O
and	O
planes	O
(	O
synthetic	O
geometry	O
)	O
.	O
</s>
<s>
Presently	O
,	O
most	O
textbooks	O
,	O
introduce	O
geometric	O
spaces	O
from	O
linear	B-Language
algebra	I-Language
,	O
and	O
geometry	O
is	O
often	O
presented	O
,	O
at	O
elementary	O
level	O
,	O
as	O
a	O
subfield	O
of	O
linear	B-Language
algebra	I-Language
.	O
</s>
<s>
Linear	B-Language
algebra	I-Language
is	O
used	O
in	O
almost	O
all	O
areas	O
of	O
mathematics	O
,	O
thus	O
making	O
it	O
relevant	O
in	O
almost	O
all	O
scientific	O
domains	O
that	O
use	O
mathematics	O
.	O
</s>
<s>
This	O
is	O
the	O
case	O
with	O
mechanics	O
and	O
robotics	O
,	O
for	O
describing	O
rigid	O
body	O
dynamics	O
;	O
geodesy	O
for	O
describing	O
Earth	O
shape	O
;	O
perspectivity	B-Application
,	O
computer	B-Application
vision	I-Application
,	O
and	O
computer	O
graphics	O
,	O
for	O
describing	O
the	O
relationship	O
between	O
a	O
scene	O
and	O
its	O
plane	O
representation	O
;	O
and	O
many	O
other	O
scientific	O
domains	O
.	O
</s>
<s>
This	O
requires	O
the	O
heavy	O
use	O
of	O
linear	B-Language
algebra	I-Language
.	O
</s>
<s>
Functional	B-Application
analysis	I-Application
studies	O
function	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
Linear	B-Language
algebra	I-Language
is	O
thus	O
a	O
fundamental	O
part	O
of	O
functional	B-Application
analysis	I-Application
and	O
its	O
applications	O
,	O
which	O
include	O
,	O
in	O
particular	O
,	O
quantum	O
mechanics	O
(	O
wave	O
functions	O
)	O
.	O
</s>
<s>
To	O
solve	O
them	O
,	O
one	O
usually	O
decomposes	O
the	O
space	O
in	O
which	O
the	O
solutions	O
are	O
searched	O
into	O
small	O
,	O
mutually	O
interacting	O
cells	B-Algorithm
.	O
</s>
<s>
For	O
nonlinear	O
systems	O
,	O
this	O
interaction	O
is	O
often	O
approximated	O
by	O
linear	O
functions.This	O
is	O
called	O
a	O
linear	O
model	O
or	O
first-order	B-Algorithm
approximation	I-Algorithm
.	O
</s>
<s>
Linear	O
models	O
are	O
frequently	O
used	O
for	O
complex	O
nonlinear	O
real-world	O
systems	O
because	O
it	O
makes	O
parametrization	B-Application
more	O
manageable	O
.	O
</s>
<s>
In	O
both	O
cases	O
,	O
very	O
large	O
matrices	B-Architecture
are	O
generally	O
involved	O
.	O
</s>
<s>
Weather	O
forecasting	O
(	O
or	O
more	O
specifically	O
,	O
parametrization	B-Application
for	I-Application
atmospheric	I-Application
modeling	I-Application
)	O
is	O
a	O
typical	O
example	O
of	O
a	O
real-world	O
application	O
,	O
where	O
the	O
whole	O
Earth	O
atmosphere	O
is	O
divided	O
into	O
cells	B-Algorithm
of	O
,	O
say	O
,	O
100km	O
of	O
width	O
and	O
100km	O
of	O
height	O
.	O
</s>
<s>
Nearly	O
all	O
scientific	O
computations	O
involve	O
linear	B-Language
algebra	I-Language
.	O
</s>
<s>
Consequently	O
,	O
linear	B-Language
algebra	I-Language
algorithms	O
have	O
been	O
highly	O
optimized	O
.	O
</s>
<s>
BLAS	B-Application
and	O
LAPACK	B-Application
are	O
the	O
best	O
known	O
implementations	O
.	O
</s>
<s>
For	O
improving	O
efficiency	O
,	O
some	O
of	O
them	O
configure	O
the	O
algorithms	O
automatically	O
,	O
at	O
run	O
time	O
,	O
for	O
adapting	O
them	O
to	O
the	O
specificities	O
of	O
the	O
computer	O
(	O
cache	B-General_Concept
size	O
,	O
number	O
of	O
available	O
cores	B-Architecture
,...	O
)	O
.	O
</s>
<s>
Some	O
processors	O
,	O
typically	O
graphics	B-Architecture
processing	I-Architecture
units	I-Architecture
(	O
GPU	B-Architecture
)	O
,	O
are	O
designed	O
with	O
a	O
matrix	B-Architecture
structure	O
,	O
for	O
optimizing	O
the	O
operations	O
of	O
linear	B-Language
algebra	I-Language
.	O
</s>
<s>
This	O
section	O
presents	O
several	O
related	O
topics	O
that	O
do	O
not	O
appear	O
generally	O
in	O
elementary	O
textbooks	O
on	O
linear	B-Language
algebra	I-Language
,	O
but	O
are	O
commonly	O
considered	O
,	O
in	O
advanced	O
mathematics	O
,	O
as	O
parts	O
of	O
linear	B-Language
algebra	I-Language
.	O
</s>
<s>
The	O
existence	O
of	O
multiplicative	O
inverses	O
in	O
fields	O
is	O
not	O
involved	O
in	O
the	O
axioms	B-Algorithm
defining	O
a	O
vector	O
space	O
.	O
</s>
<s>
The	O
concepts	O
of	O
linear	O
independence	O
,	O
span	O
,	O
basis	O
,	O
and	O
linear	B-Architecture
maps	I-Architecture
(	O
also	O
called	O
module	O
homomorphisms	O
)	O
are	O
defined	O
for	O
modules	O
exactly	O
as	O
for	O
vector	O
spaces	O
,	O
with	O
the	O
essential	O
difference	O
that	O
,	O
if	O
is	O
not	O
a	O
field	O
,	O
there	O
are	O
modules	O
that	O
do	O
not	O
have	O
any	O
basis	O
.	O
</s>
<s>
Module	O
homomorphisms	O
between	O
finitely	O
generated	O
free	O
modules	O
may	O
be	O
represented	O
by	O
matrices	B-Architecture
.	O
</s>
<s>
The	O
theory	O
of	O
matrices	B-Architecture
over	O
a	O
ring	O
is	O
similar	B-Algorithm
to	O
that	O
of	O
matrices	B-Architecture
over	O
a	O
field	O
,	O
except	O
that	O
determinants	O
exist	O
only	O
if	O
the	O
ring	O
is	O
commutative	O
,	O
and	O
that	O
a	O
square	O
matrix	B-Architecture
over	O
a	O
commutative	O
ring	O
is	O
invertible	O
only	O
if	O
its	O
determinant	O
has	O
a	O
multiplicative	O
inverse	O
in	O
the	O
ring	O
.	O
</s>
<s>
However	O
,	O
these	O
algorithms	O
have	O
generally	O
a	O
computational	O
complexity	O
that	O
is	O
much	O
higher	O
than	O
the	O
similar	B-Algorithm
algorithms	O
over	O
a	O
field	O
.	O
</s>
<s>
In	O
multilinear	O
algebra	O
,	O
one	O
considers	O
multivariable	O
linear	B-Architecture
transformations	I-Architecture
,	O
that	O
is	O
,	O
mappings	B-Algorithm
that	O
are	O
linear	O
in	O
each	O
of	O
a	O
number	O
of	O
different	O
variables	O
.	O
</s>
<s>
This	O
line	O
of	O
inquiry	O
naturally	O
leads	O
to	O
the	O
idea	O
of	O
the	O
dual	O
space	O
,	O
the	O
vector	O
space	O
consisting	O
of	O
linear	B-Architecture
maps	I-Architecture
where	O
F	O
is	O
the	O
field	O
of	O
scalars	O
.	O
</s>
<s>
If	O
,	O
in	O
addition	O
to	O
vector	O
addition	O
and	O
scalar	O
multiplication	O
,	O
there	O
is	O
a	O
bilinear	O
vector	O
product	O
,	O
the	O
vector	O
space	O
is	O
called	O
an	O
algebra	O
;	O
for	O
instance	O
,	O
associative	O
algebras	O
are	O
algebras	O
with	O
an	O
associate	O
vector	O
product	O
(	O
like	O
the	O
algebra	O
of	O
square	O
matrices	B-Architecture
,	O
or	O
the	O
algebra	O
of	O
polynomials	O
)	O
.	O
</s>
<s>
The	O
metric	O
also	O
allows	O
for	O
a	O
definition	O
of	O
limits	B-Algorithm
and	O
completeness	O
a	O
metric	O
space	O
that	O
is	O
complete	O
is	O
known	O
as	O
a	O
Banach	O
space	O
.	O
</s>
<s>
A	O
complete	O
metric	O
space	O
along	O
with	O
the	O
additional	O
structure	O
of	O
an	O
inner	O
product	O
(	O
a	O
conjugate	O
symmetric	B-Algorithm
sesquilinear	B-Algorithm
form	I-Algorithm
)	O
is	O
known	O
as	O
a	O
Hilbert	O
space	O
,	O
which	O
is	O
in	O
some	O
sense	O
a	O
particularly	O
well-behaved	O
Banach	O
space	O
.	O
</s>
<s>
Functional	B-Application
analysis	I-Application
applies	O
the	O
methods	O
of	O
linear	B-Language
algebra	I-Language
alongside	O
those	O
of	O
mathematical	O
analysis	O
to	O
study	O
various	O
function	B-Algorithm
spaces	I-Algorithm
;	O
the	O
central	O
objects	O
of	O
study	O
in	O
functional	B-Application
analysis	I-Application
are	O
spaces	O
,	O
which	O
are	O
Banach	O
spaces	O
,	O
and	O
especially	O
the	O
space	O
of	O
square	O
integrable	O
functions	O
,	O
which	O
is	O
the	O
only	O
Hilbert	O
space	O
among	O
them	O
.	O
</s>
<s>
Functional	B-Application
analysis	I-Application
is	O
of	O
particular	O
importance	O
to	O
quantum	O
mechanics	O
,	O
the	O
theory	O
of	O
partial	O
differential	O
equations	O
,	O
digital	O
signal	O
processing	O
,	O
and	O
electrical	O
engineering	O
.	O
</s>
