in O
fluid B-Attributes
mechanics I-Attributes
linear O
algebra O
is O
integral O
to O
understanding O
and O
solving O
problems O
related O
to O
the O
behavior O
of O
fluid O
linear B-Math
algebra I-Math
is O
integral O
to O
understanding O
and O
solving O
problems O
related O
to O
the O
behavior O
of O
fluid O

a O
vector B-Math
space I-Math
can O
be O
of O
finite O
dimension O
or O
infinite O
dimension O
depending O
on O
the O
maximum O
number O
of O
linearly O
independent O
vector O
can O
be O
of O
finite B-Math
dimension I-Math
or O
infinite O
dimension O
depending O
on O
the O
maximum O
number O
of O
linearly O
independent O
vector O
or O
infinite B-Math
dimension I-Math
depending O
on O
the O
maximum O
number O
of O
linearly O
independent O
vector O

the O
theory O
of O
matrices B-Math
over O
a O
ring O
is O
similar O
to O
that O
of O
matrices O
over O
a O
field O
except O
that O
determinants O
exist O
only O
if O
the O
ring O
is O
commutative O
and O
that O
a O
square O
matrix O
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
rin O
over O
a O
ring B-Math
is O
similar O
to O
that O
of O
matrices O
over O
a O
field O
except O
that O
determinants O
exist O
only O
if O
the O
ring O
is O
commutative O
and O
that O
a O
square O
matrix O
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
rin O
is O
similar O
to O
that O
of O
matrices O
over O
a O
field B-Math
except O
that O
determinants O
exist O
only O
if O
the O
ring O
is O
commutative O
and O
that O
a O
square O
matrix O
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
rin O
matrices B-Math
over O
a O
field O
except O
that O
determinants O
exist O
only O
if O
the O
ring O
is O
commutative O
and O
that O
a O
square O
matrix O
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
rin O

sometimes O
the O
term O
linear B-Math
operator I-Math
refers O
to O
this O
case O
but O
the O
term O
linear O
operator O
can O
have O
different O
meanings O
for O
different O
conventions O
for O
example 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
not O
necessarily O
with O
citation O
needed O
or O
it O
can O
be O
used O
to O
emphasize O
that O
is O
a O
function O
space O
which O
is O
a O
common O
convention O
in O
functional O
analysi O
refers O
to O
this O
case O
but O
the O
term O
linear O
operator O
can O
have O
different O
meanings O
for O
different O
conventions O
for O
example 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
not O
necessarily O
with O
citation O
needed O
or O
it O
can O
be O
used O
to O
emphasize O
that O
is O
a O
function O
space O
which O
is O
a O
common O
convention O
in O
functional B-Math
analysis I-Math
function B-Math
space I-Math
which O
is O
a O
common O
convention O
in O
functional O
analysi O

a O
vector B-Math
space I-Math
over O
a O
field O
f O
is O
a O
nonempty O
set O
v O
together O
with O
a O
binary O
operation O
and O
a O
binary O
function O
that O
satisfy O
the O
eight O
axioms O
listed O
belo O
over O
a O
field B-Math
is O
a O
nonempty O
set O
v O
together O
with O
a O
binary O
operation O
and O
a O
binary O
function O
that O
satisfy O
the O
eight O
axioms O
listed O
belo O
is O
a O
nonempty B-Math
set I-Math
v O
together O
with O
a O
binary O
operation O
and O
a O
binary O
function O
that O
satisfy O
the O
eight O
axioms O
listed O
belo O

linear B-Math
models I-Math
are O
frequently O
used O
for O
complex O
nonlinear O
realworld O
systems O
because O
it O
makes O
parametrization O
more O
manageabl O
are O
frequently O
used O
for O
complex B-Attributes
nonlinear I-Attributes
realworld I-Attributes
systems I-Attributes
because O
it O
makes O
parametrization O
more O
manageabl O

functional B-Math
analysis I-Math
applies O
the O
methods O
of O
linear O
algebra O
alongside O
those O
of O
mathematical O
analysis O
to O
study O
various O
function O
spaces O
the O
central O
objects O
of O
study O
in O
functional O
analysis O
are O
lp O
spaces O
which O
are O
banach O
spaces O
and O
especially O
the O
l O
space O
of O
square O
integrable O
functions O
which O
is O
the O
only O
hilbert O
space O
among O
the O
applies O
the O
methods O
of O
linear B-Math
algebra I-Math
alongside O
those O
of O
mathematical O
analysis O
to O
study O
various O
function O
spaces O
the O
central O
objects O
of O
study O
in O
functional O
analysis O
are O
lp O
spaces O
which O
are O
banach O
spaces O
and O
especially O
the O
l O
space O
of O
square O
integrable O
functions O
which O
is O
the O
only O
hilbert O
space O
among O
the O
alongside O
those O
of O
mathematical O
analysis O
to O
study O
various O
function O
spaces O
the O
central O
objects O
of O
study O
in O
functional O
analysis O
are O
lp O
spaces O
which O
are O
banach B-Math
spaces I-Math
and O
especially O
the O
l O
space O
of O
square O
integrable O
functions O
which O
is O
the O
only O
hilbert O
space O
among O
the O
and O
especially O
the O
l O
space O
of O
square O
integrable O
functions O
which O
is O
the O
only O
hilbert B-Math
space I-Math
among O
the O

in O
this O
case O
the O
change O
of O
variable O
y O
ux O
leads O
to O
an O
equation B-Math
of O
the O
for O

for O
example O
polynomial B-Math
rings I-Math
are O
countably O
infinitedimensional O
vector O
spaces O
and O
many O
function O
spaces O
have O
the O
cardinality O
of O
the O
continuum O
as O
a O
dimensio O
are O
countably O
infinitedimensional B-Attributes
vector O
spaces O
and O
many O
function O
spaces O
have O
the O
cardinality O
of O
the O
continuum O
as O
a O
dimensio O
vector B-Math
spaces I-Math
and O
many O
function O
spaces O
have O
the O
cardinality O
of O
the O
continuum O
as O
a O
dimensio O

fluid B-Attributes
mechanics I-Attributes
fluid O
dynamics O
and O
thermal O
energy O
system O
fluid B-Attributes
dynamics I-Attributes
and O
thermal O
energy O
system O
and O
thermal B-Attributes
energy I-Attributes
systems I-Attributes

linear B-Math
algebra I-Math
a O
branch O
of O
mathematics O
dealing O
with O
vector O
spaces O
and O
linear O
mappings O
between O
these O
spaces O
plays O
a O
critical O
role O
in O
various O
engineering O
disciplines O
including O
fluid O
mechanics O
fluid O
dynamics O
and O
thermal O
energy O
system O
a O
branch O
of O
mathematics B-Math
dealing O
with O
vector O
spaces O
and O
linear O
mappings O
between O
these O
spaces O
plays O
a O
critical O
role O
in O
various O
engineering O
disciplines O
including O
fluid O
mechanics O
fluid O
dynamics O
and O
thermal O
energy O
system O
dealing O
with O
vector B-Math
spaces I-Math
and O
linear O
mappings O
between O
these O
spaces O
plays O
a O
critical O
role O
in O
various O
engineering O
disciplines O
including O
fluid O
mechanics O
fluid O
dynamics O
and O
thermal O
energy O
system O
and O
linear B-Math
mappings I-Math
between O
these O
spaces O
plays O
a O
critical O
role O
in O
various O
engineering O
disciplines O
including O
fluid O
mechanics O
fluid O
dynamics O
and O
thermal O
energy O
system O
between O
these O
spaces O
plays O
a O
critical O
role O
in O
various O
engineering O
disciplines O
including O
fluid B-Attributes
mechanics I-Attributes
fluid O
dynamics O
and O
thermal O
energy O
system O
fluid B-Attributes
dynamics I-Attributes
and O
thermal O
energy O
system O
and O
thermal B-Attributes
energy I-Attributes
systems I-Attributes

in O
the O
example O
above O
a O
solution B-Math
is O
given O
by O
the O
ordered O
triple O
since O
it O
makes O
all O
three O
equations O
vali O
is O
given O
by O
the O
ordered O
triple O
since O
it O
makes O
all O
three O
equations B-Math
vali O

integer B-Math
linear I-Math
programming I-Math
is O
a O
collection O
of O
methods O
for O
finding O
the O
best O
integer O
solution O
when O
there O
are O
man O

very O
often O
and O
in O
this O
article O
the O
coefficients O
of O
the O
equations O
are O
real O
or O
complex O
numbers O
and O
the O
solutions O
are O
searched O
in O
the O
same O
set O
of O
numbers O
but O
the O
theory O
and O
the O
algorithms O
apply O
for O
coefficients B-Math
and O
solutions O
in O
any O
fiel O
and O
solutions B-Math
in O
any O
fiel O

if O
v O
has O
a O
basis B-Math
of O
n O
elements O
such O
an O
endomorphism O
is O
represented O
by O
a O
square O
matrix O
of O
size O
of O
n O
elements O
such O
an O
endomorphism O
is O
represented O
by O
a O
square B-Math
matrix I-Math
of O
size O

not O
all O
matrices B-Math
are O
related O
to O
linear O
algebr O
are O
related O
to O
linear B-Math
algebra I-Math

bases B-Attributes
if O
every O
element O
of O
v O
may O
be O
written O
in O
a O
unique O
way O
as O
a O
finite O
linear O
combination O
of O
elements O
of O
if O
every O
element O
of O
v O
may O
be O
written O
in O
a O
unique O
way O
as O
a O
finite O
linear B-Math
combination I-Math
of O
elements O
of O

otherwise O
a O
differential B-Math
equation I-Math
is O
homogeneous O
if O
it O
is O
a O
homogeneous O
function O
of O
the O
unknown O
function O
and O
its O
derivative O
is O
homogeneous O
if O
it O
is O
a O
homogeneous B-Math
function I-Math
of O
the O
unknown O
function O
and O
its O
derivative O

in O
this O
article O
vectors B-Math
are O
represented O
in O
boldface O
to O
distinguish O
them O
from O
scalar O
are O
represented O
in O
boldface O
to O
distinguish O
them O
from O
scalars B-Math

in O
general O
a O
system O
with O
fewer O
equations B-Math
than O
unknowns O
has O
infinitely O
many O
solutions O
but O
it O
may O
have O
no O
solutio O

it O
has O
been O
shown O
that O
the O
two O
approaches O
are O
essentially O
equivalent B-Math

most O
of O
the O
theory O
of O
abelian B-Math
groups I-Math
may O
be O
extended O
to O
modules O
over O
a O
principal O
ideal O
domai O
may O
be O
extended O
to O
modules B-Math
over O
a O
principal O
ideal O
domai O

nearly O
all O
scientific O
computations O
involve O
linear B-Math
algebra I-Math

the O
modeling O
of O
ambient O
space O
is O
based O
on O
geometry B-Math

in O
this O
new O
at O
that O
time O
geometry B-Math
now O
called O
cartesian O
geometry O
points O
are O
represented O
by O
cartesian O
coordinates O
which O
are O
sequences O
of O
three O
real O
numbers O
in O
the O
case O
of O
the O
usual O
threedimensional O
spac O
now O
called O
cartesian B-Math
geometry I-Math
points O
are O
represented O
by O
cartesian O
coordinates O
which O
are O
sequences O
of O
three O
real O
numbers O
in O
the O
case O
of O
the O
usual O
threedimensional O
spac O

for O
more O
details O
see O
linear B-Math
equation I-Math
over O
a O
rin O
over O
a O
ring B-Math

hadamard O
also O
founded O
the O
modern O
school O
of O
linear B-Math
functional I-Math
analysis I-Math
further O
developed O
by O
riesz O
and O
the O
group O
of O
polish O
mathematicians O
around O
stefan O
banac O

to O
have O
a O
vector B-Math
space I-Math
the O
eight O
following O
axioms O
must O
be O
satisfied O
for O
every O
u O
v O
and O
w O
in O
v O
and O
a O
and O
b O
in O

these O
two O
cases O
are O
the O
most O
common O
ones O
but O
vector B-Math
spaces I-Math
with O
scalars O
in O
an O
arbitrary O
field O
f O
are O
also O
commonly O
considere O
with O
scalars B-Math
in O
an O
arbitrary O
field O
f O
are O
also O
commonly O
considere O

functional B-Math
analysis I-Math
is O
a O
branch O
of O
mathematical O
analysis O
the O
core O
of O
which O
is O
formed O
by O
the O
study O
of O
vector O
spaces O
endowed O
with O
some O
kind O
of O
limitrelated O
structure O
for O
example O
inner O
product O
norm O
or O
topology O
and O
the O
linear O
functions O
defined O
on O
these O
spaces O
and O
suitably O
respecting O
these O
structure O
is O
a O
branch O
of O
mathematical B-Math
analysis I-Math
the O
core O
of O
which O
is O
formed O
by O
the O
study O
of O
vector O
spaces O
endowed O
with O
some O
kind O
of O
limitrelated O
structure O
for O
example O
inner O
product O
norm O
or O
topology O
and O
the O
linear O
functions O
defined O
on O
these O
spaces O
and O
suitably O
respecting O
these O
structure O
the O
core O
of O
which O
is O
formed O
by O
the O
study O
of O
vector B-Math
spaces I-Math
endowed O
with O
some O
kind O
of O
limitrelated O
structure O
for O
example O
inner O
product O
norm O
or O
topology O
and O
the O
linear O
functions O
defined O
on O
these O
spaces O
and O
suitably O
respecting O
these O
structure O
endowed O
with O
some O
kind O
of O
limitrelated O
structure O
for O
example O
inner B-Math
product I-Math
norm O
or O
topology O
and O
the O
linear O
functions O
defined O
on O
these O
spaces O
and O
suitably O
respecting O
these O
structure O
norm B-Math
or O
topology O
and O
the O
linear O
functions O
defined O
on O
these O
spaces O
and O
suitably O
respecting O
these O
structure O
or O
topology B-Math
and O
the O
linear O
functions O
defined O
on O
these O
spaces O
and O
suitably O
respecting O
these O
structure O
and O
the O
linear B-Math
functions I-Math
defined O
on O
these O
spaces O
and O
suitably O
respecting O
these O
structure O

the O
usage O
of O
the O
word O
functional O
as O
a O
noun O
goes O
back O
to O
the O
calculus O
of O
variations B-Math
implying O
a O
function O
whose O
argument O
is O
a O
functio O
implying O
a O
function O
whose O
argument O
is O
a O
function B-Math

if O
v O
is O
a O
vector O
space O
over O
a O
field O
k O
and O
if O
w O
is O
a O
subset O
of O
v O
then O
w O
is O
a O
linear O
subspace O
of O
v O
if O
under O
the O
operations O
of O
v O
w O
is O
a O
vector O
space O
over O
k O
equivalently O
a O
nonempty O
subset O
w O
is O
a O
linear B-Math
subspace I-Math
of O
v O
if O
whenever O
w O
w O
are O
elements O
of O
w O
and O
α O
β O
are O
elements O
of O
k O
it O
follows O
that O
αw O
βw O
is O
in O
vector B-Math
space I-Math
ver O
a O
field O
k O
and O
if O
w O
is O
a O
subset O
of O
v O
then O
w O
is O
a O
linear O
subspace O
of O
v O
if O
under O
the O
operations O
of O
v O
w O
is O
a O
vector O
space O
over O
k O
equivalently O
a O
nonempty O
subset O
w O
is O
a O
linear O
subspace O
of O
v O
if O
whenever O
w O
w O
are O
elements O
of O
w O
and O
α O
β O
are O
elements O
of O
k O
it O
follows O
that O
αw O
βw O
is O
in O

square B-Math
matrices I-Math
of O
a O
given O
dimension O
form O
a O
noncommutative O
ring O
which O
is O
one O
of O
the O
most O
common O
examples O
of O
a O
noncommutative O
rin O
of O
a O
given O
dimension B-Math
form O
a O
noncommutative O
ring O
which O
is O
one O
of O
the O
most O
common O
examples O
of O
a O
noncommutative O
rin O
form O
a O
noncommutative B-Math
ring I-Math
which O
is O
one O
of O
the O
most O
common O
examples O
of O
a O
noncommutative O
rin O

for O
instance O
linear B-Math
algebraic I-Math
techniques O
are O
used O
to O
solve O
systems O
of O
differential O
equations O
that O
describe O
fluid O
motio O
techniques O
are O
used O
to O
solve O
systems O
of O
differential O
equations O
that O
describe O
fluid B-Attributes
motion I-Attributes

linear B-Math
algebra I-Math
is O
used O
in O
almost O
all O
areas O
of O
mathematics O
thus O
making O
it O
relevant O
in O
almost O
all O
scientific O
domains O
that O
use O
mathematic O
is O
used O
in O
almost O
all O
areas O
of O
mathematics B-Math
thus O
making O
it O
relevant O
in O
almost O
all O
scientific O
domains O
that O
use O
mathematic O

however O
every O
module B-Math
is O
a O
cokernel O
of O
a O
homomorphism O
of O
free O
module O