Datasets:
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 |