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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 |