bthis O
is O
called O
a O
linear B-Math
model I-Math
or O
firstorder O
approximatio O
or O
firstorder B-Math
approximation I-Math

it O
also O
provides O
the O
foundation O
and O
theoretical O
framework O
that O
underlies O
the O
fourier B-Math
transform I-Math
and O
related O
method O

for O
example O
the O
collection O
of O
all O
possible O
linear B-Math
combinations I-Math
of O
the O
vectors O
on O
the O
lefthand O
side O
is O
called O
their O
span O
and O
the O
equations O
have O
a O
solution O
just O
when O
the O
righthand O
vector O
is O
within O
that O
spa O
of O
the O
vectors B-Math
on O
the O
lefthand O
side O
is O
called O
their O
span O
and O
the O
equations O
have O
a O
solution O
just O
when O
the O
righthand O
vector O
is O
within O
that O
spa O

the O
coefficients O
of O
this O
linear O
combination O
are O
referred O
to O
as O
components B-Math
or O
coordinates O
of O
the O
vector O
with O
respect O
to O
or O
coordinates B-Math
of O
the O
vector O
with O
respect O
to O
linear B-Math
combination I-Math
are O
referred O
to O
as O
components O
or O
coordinates O
of O
the O
vector O
with O
respect O
to O

for O
nonlinear B-Math
systems I-Math
this O
interaction O
is O
often O
approximated O
by O
linear O
function O
this O
interaction O
is O
often O
approximated O
by O
linear B-Math
functions I-Math

in O
the O
theory O
of O
vector B-Math
spaces I-Math
a O
set O
of O
vectors O
is O
said O
to O
be O
linearly O
independent O
if O
there O
exists O
no O
nontrivial O
linear O
combination O
of O
the O
vectors O
that O
equals O
the O
zero O
vecto O
a O
set O
of O
vectors B-Math
is O
said O
to O
be O
linearly O
independent O
if O
there O
exists O
no O
nontrivial O
linear O
combination O
of O
the O
vectors O
that O
equals O
the O
zero O
vecto O
is O
said O
to O
be O
linearly B-Math
independent I-Math
if O
there O
exists O
no O
nontrivial O
linear O
combination O
of O
the O
vectors O
that O
equals O
the O
zero O
vecto O
if O
there O
exists O
no O
nontrivial O
linear B-Math
combination I-Math
of O
the O
vectors O
that O
equals O
the O
zero O
vecto O

in O
mathematics B-Math
he O
linear O
span O
also O
called O
the O
linear O
hull O
or O
just O
span O
of O
a O
set O
s O
of O
vectors O
from O
a O
vector O
space O
denoted O
spans O
is O
defined O
as O
the O
set O
of O
all O
linear O
combinations O
of O
the O
vectors O
in O
s O
for O
example O
two O
linearly O
independent O
vectors O
span O
a O
plan O
he O
linear B-Math
span I-Math
also O
called O
the O
linear O
hull O
or O
just O
span O
of O
a O
set O
s O
of O
vectors O
from O
a O
vector O
space O
denoted O
spans O
is O
defined O
as O
the O
set O
of O
all O
linear O
combinations O
of O
the O
vectors O
in O
s O
for O
example O
two O
linearly O
independent O
vectors O
span O
a O
plan O
also O
called O
the O
linear B-Math
hull I-Math
or O
just O
span O
of O
a O
set O
s O
of O
vectors O
from O
a O
vector O
space O
denoted O
spans O
is O
defined O
as O
the O
set O
of O
all O
linear O
combinations O
of O
the O
vectors O
in O
s O
for O
example O
two O
linearly O
independent O
vectors O
span O
a O
plan O

however O
many O
of O
the O
principles O
are O
also O
valid O
for O
infinitedimensional O
vector B-Math
spaces I-Math

also O
functional B-Math
analysis I-Math
a O
branch O
of O
mathematical O
analysis O
may O
be O
viewed O
as O
the O
application O
of O
linear O
algebra O
to O
function O
space O
a O
branch O
of O
mathematical O
analysis O
may O
be O
viewed O
as O
the O
application O
of O
linear B-Math
algebra I-Math
to O
function O
space O

in O
mathematics O
physics O
and O
engineering O
a O
euclidean O
vector O
or O
simply O
a O
vector O
sometimes O
called O
a O
geometric B-Math
vector I-Math
or O
spatial O
vector O
is O
a O
geometric O
object O
that O
has O
magnitude O
or O
length O
and O
directio O
or O
spatial O
vector O
is O
a O
geometric O
object O
that O
has O
magnitude B-Attributes
or O
length O
and O
directio O
or O
length B-Attributes
and O
directio O
and O
direction B-Attributes

here O
in O
general O
means O
that O
a O
different O
behavior O
may O
occur O
for O
specific B-Math
values I-Math
of O
the O
coefficients O
of O
the O
equation O

multilinear B-Math
maps I-Math
t O
vn O
f O
can O
be O
described O
via O
tensor O
products O
of O
elements O
of O
t O
vn O
f O
can O
be O
described O
via O
tensor B-Math
products O
of O
elements O
of O

a O
vector B-Math
is O
what O
is O
needed O
to O
carry O
the O
point O
a O
to O
the O
point O
b O
the O
latin O
word O
vector O
means O
carrie O

the O
application O
of O
linear B-Math
algebra I-Math
in O
this O
context O
is O
vital O
for O
the O
design O
and O
operation O
of O
modern O
power O
systems O
including O
renewable O
energy O
sources O
and O
smart O
grid O

one O
may O
thus O
replace O
the O
field O
of O
scalars O
by O
a O
ring B-Math
r O
and O
this O
gives O
the O
structure O
called O
a O
module O
over O
r O
or O
rmodul O

when O
the O
scalar O
field O
is O
the O
real O
numbers O
the O
vector B-Math
space I-Math
is O
called O
a O
real O
vector O
space O
and O
when O
the O
scalar O
field O
is O
the O
complex O
numbers O
the O
vector O
space O
is O
called O
a O
complex O
vector O
spac O
is O
called O
a O
real O
vector O
space O
and O
when O
the O
scalar O
field O
is O
the O
complex O
numbers O
the O
vector O
space O
is O
called O
a O
complex B-Math
vector I-Math
space I-Math
real B-Math
vector I-Math
space I-Math
and O
when O
the O
scalar O
field O
is O
the O
complex O
numbers O
the O
vector O
space O
is O
called O
a O
complex O
vector O
spac O

computational O
algorithms O
for O
finding O
the O
solutions O
are O
an O
important O
part O
of O
numerical O
linear B-Math
algebra I-Math
and O
play O
a O
prominent O
role O
in O
engineering O
physics O
chemistry O
computer O
science O
and O
economic O
solutions B-Math
are O
an O
important O
part O
of O
numerical O
linear O
algebra O
and O
play O
a O
prominent O
role O
in O
engineering O
physics O
chemistry O
computer O
science O
and O
economic O

in O
the O
first O
case O
the O
dimension B-Math
of O
the O
solution O
set O
is O
in O
general O
equal O
to O
n O
m O
where O
n O
is O
the O
number O
of O
variables O
and O
m O
is O
the O
number O
of O
equation O
of O
the O
solution B-Math
set I-Math
is O
in O
general O
equal O
to O
n O
m O
where O
n O
is O
the O
number O
of O
variables O
and O
m O
is O
the O
number O
of O
equation O
is O
in O
general O
equal O
to O
n O
m O
where O
n O
is O
the O
number O
of O
variables O
and O
m O
is O
the O
number O
of O
equations B-Math
variables B-Math
and O
m O
is O
the O
number O
of O
equation O

in O
the O
case O
of O
linear B-Math
differential I-Math
equations I-Math
this O
means O
that O
there O
are O
no O
constant O
term O

furthermore O
linear B-Math
algebra I-Math
plays O
a O
crucial O
role O
in O
thermal O
energy O
systems O
particularly O
in O
power O
systems O
analysi O
plays O
a O
crucial O
role O
in O
thermal B-Attributes
energy I-Attributes
systems I-Attributes
particularly O
in O
power O
systems O
analysi O

vector B-Math
spaces I-Math
that O
are O
not O
finite O
dimensional O
often O
require O
additional O
structure O
to O
be O
tractabl O

the O
historical O
roots O
of O
functional B-Math
analysis I-Math
lie O
in O
the O
study O
of O
spaces O
of O
functions O
and O
the O
formulation O
of O
properties O
of O
transformations O
of O
functions O
such O
as O
the O
fourier O
transform O
as O
transformations O
defining O
for O
example O
continuous O
or O
unitary O
operators O
between O
function O
space O
lie O
in O
the O
study O
of O
spaces O
of O
functions O
and O
the O
formulation O
of O
properties O
of O
transformations O
of O
functions O
such O
as O
the O
fourier B-Math
transform I-Math
as O
transformations O
defining O
for O
example O
continuous O
or O
unitary O
operators O
between O
function O
space O
as O
transformations O
defining O
for O
example O
continuous O
or O
unitary B-Math
operators I-Math
between O
function O
space O

in O
general O
there O
is O
not O
such O
a O
complete O
classification O
for O
modules B-Math
even O
if O
one O
restricts O
oneself O
to O
finitely O
generated O
module O

there O
is O
a O
strong O
relationship O
between O
linear B-Math
algebra I-Math
and O
geometry O
which O
started O
with O
the O
introduction O
by O
rené O
descartes O
in O
of O
cartesian O
coordinate O
and O
geometry B-Math
which O
started O
with O
the O
introduction O
by O
rené O
descartes O
in O
of O
cartesian O
coordinate O

a O
linear B-Math
endomorphism I-Math
is O
a O
linear O
map O
that O
maps O
a O
vector O
space O
v O
to O
itsel O
is O
a O
linear B-Math
map I-Math
that O
maps O
a O
vector O
space O
v O
to O
itsel O
that O
maps O
a O
vector B-Math
space I-Math
v O
to O
itsel O

the O
determinant B-Math
of O
a O
square O
matrix O
is O
a O
number O
associated O
to O
the O
matrix O
which O
is O
fundamental O
for O
the O
study O
of O
a O
square O
matrix O
for O
example O
a O
square O
matrix O
is O
invertible O
if O
and O
only O
if O
it O
has O
a O
nonzero O
determinant O
and O
the O
eigenvalues O
of O
a O
square O
matrix O
are O
the O
roots O
of O
a O
polynomial O
determinan O
of O
a O
square B-Math
matrix I-Math
is O
a O
number O
associated O
to O
the O
matrix O
which O
is O
fundamental O
for O
the O
study O
of O
a O
square O
matrix O
for O
example O
a O
square O
matrix O
is O
invertible O
if O
and O
only O
if O
it O
has O
a O
nonzero O
determinant O
and O
the O
eigenvalues O
of O
a O
square O
matrix O
are O
the O
roots O
of O
a O
polynomial O
determinan O
is O
a O
number O
associated O
to O
the O
matrix O
which O
is O
fundamental O
for O
the O
study O
of O
a O
square O
matrix O
for O
example O
a O
square O
matrix O
is O
invertible O
if O
and O
only O
if O
it O
has O
a O
nonzero O
determinant O
and O
the O
eigenvalues B-Math
of O
a O
square O
matrix O
are O
the O
roots O
of O
a O
polynomial O
determinan O
of O
a O
square B-Math
matrix I-Math
are O
the O
roots O
of O
a O
polynomial O
determinan O
are O
the O
roots O
of O
a O
polynomial B-Math
determinant I-Math

in O
this O
context O
the O
elements O
of O
v O
are O
commonly O
called O
vectors B-Math
and O
the O
elements O
of O
f O
are O
called O
scalar O
and O
the O
elements O
of O
f O
are O
called O
scalars B-Math

matrices B-Math
are O
used O
to O
represent O
linear O
maps O
and O
allow O
explicit O
computations O
in O
linear O
algebr O
are O
used O
to O
represent O
linear B-Math
maps I-Math
and O
allow O
explicit O
computations O
in O
linear O
algebr O
and O
allow O
explicit O
computations O
in O
linear B-Math
algebra I-Math

such O
a O
system O
is O
also O
known O
as O
an O
overdetermined B-Math
system I-Math

because O
an O
isomorphism B-Math
preserves O
linear O
structure O
two O
isomorphic O
vector O
spaces O
are O
essentially O
the O
same O
from O
the O
linear O
algebra O
point O
of O
view O
in O
the O
sense O
that O
they O
can O
not O
be O
distinguished O
by O
using O
vector O
space O
propertie O
preserves O
linear O
structure O
two O
isomorphic O
vector O
spaces O
are O
essentially O
the O
same O
from O
the O
linear B-Math
algebra I-Math
point O
of O
view O
in O
the O
sense O
that O
they O
can O
not O
be O
distinguished O
by O
using O
vector O
space O
propertie O
isomorphic B-Math
vector I-Math
spaces I-Math
are O
essentially O
the O
same O
from O
the O
linear O
algebra O
point O
of O
view O
in O
the O
sense O
that O
they O
can O
not O
be O
distinguished O
by O
using O
vector O
space O
propertie O

sometimes O
the O
term O
linear O
function O
has O
the O
same O
meaning O
as O
linear B-Math
map I-Math
while O
in O
analysis O
it O
does O
no O
linear B-Math
function I-Math
has O
the O
same O
meaning O
as O
linear O
map O
while O
in O
analysis O
it O
does O
no O

infinitedimensional O
vector B-Math
spaces I-Math
occur O
in O
many O
areas O
of O
mathematic O
occur O
in O
many O
areas O
of O
mathematics B-Math
infinitedimensional B-Attributes
vector O
spaces O
occur O
in O
many O
areas O
of O
mathematic O

in O
particular O
over O
a O
principal O
ideal O
domain O
every O
submodule B-Math
of O
a O
free O
module O
is O
free O
and O
the O
fundamental O
theorem O
of O
finitely O
generated O
abelian O
groups O
may O
be O
extended O
straightforwardly O
to O
finitely O
generated O
modules O
over O
a O
principal O
rin O
of O
a O
free O
module B-Math
is O
free O
and O
the O
fundamental O
theorem O
of O
finitely O
generated O
abelian O
groups O
may O
be O
extended O
straightforwardly O
to O
finitely O
generated O
modules O
over O
a O
principal O
rin O

module O
homomorphisms O
between O
finitely O
generated O
free O
modules O
may O
be O
represented O
by O
matrices B-Math