equivalently O
a O
set O
b O
is O
a O
basis O
if O
its O
elements O
are O
linearly B-Math
independent I-Math
and O
every O
element O
of O
v O
is O
a O
linear O
combination O
of O
elements O
of O
and O
every O
element O
of O
v O
is O
a O
linear B-Math
combination I-Math
of O
elements O
of O

thus O
the O
solution B-Math
set I-Math
may O
be O
a O
plane O
a O
line O
a O
single O
point O
or O
the O
empty O
se O
may O
be O
a O
plane B-Math
a O
line O
a O
single O
point O
or O
the O
empty O
se O
a O
line B-Math
a O
single O
point O
or O
the O
empty O
se O
a O
single B-Math
point I-Math
or O
the O
empty O
se O
or O
the O
empty B-Math
set I-Math

the O
mathematical O
representation O
of O
a O
physical O
vector O
depends O
on O
the O
coordinate B-Math
system I-Math
used O
to O
describe O
i O
vector B-Math
depends O
on O
the O
coordinate O
system O
used O
to O
describe O
i O

in O
general O
a O
system O
with O
more O
equations B-Math
than O
unknowns O
has O
no O
solutio O
than O
unknowns O
has O
no O
solution B-Math

it O
was O
initially O
a O
subbranch O
of O
linear B-Math
algebra I-Math
but O
soon O
grew O
to O
include O
subjects O
related O
to O
graph O
theory O
algebra O
combinatorics O
and O
statistic O
but O
soon O
grew O
to O
include O
subjects O
related O
to O
graph B-Math
theory I-Math
algebra O
combinatorics O
and O
statistic O
algebra O
combinatorics O
and O
statistics B-Math
algebra B-Math
combinatorics O
and O
statistic O
combinatorics B-Math
and O
statistic O

for O
three O
variables O
each O
linear B-Math
equation I-Math
determines O
a O
plane O
in O
threedimensional O
space O
and O
the O
solution O
set O
is O
the O
intersection O
of O
these O
plane O
determines O
a O
plane B-Math
in O
threedimensional O
space O
and O
the O
solution O
set O
is O
the O
intersection O
of O
these O
plane O
in O
threedimensional O
space O
and O
the O
solution B-Math
set O
is O
the O
intersection O
of O
these O
plane O

matrices B-Math
are O
used O
in O
most O
areas O
of O
mathematics O
and O
most O
scientific O
fields O
either O
directly O
or O
through O
their O
use O
in O
geometry O
and O
numerical O
analysi O
are O
used O
in O
most O
areas O
of O
mathematics B-Math
and O
most O
scientific O
fields O
either O
directly O
or O
through O
their O
use O
in O
geometry O
and O
numerical O
analysi O
and O
most O
scientific O
fields O
either O
directly O
or O
through O
their O
use O
in O
geometry O
and O
numerical B-Math
analysis I-Math

these O
are O
vector B-Math
spaces I-Math
with O
additional O
structure O
such O
as O
hilbert O
space O
with O
additional O
structure O
such O
as O
hilbert B-Math
spaces I-Math

these O
are O
called O
the O
trivial B-Math
subspaces I-Math
f O
the O
vector O
spac O
f O
the O
vector B-Math
space I-Math

many O
vector B-Math
spaces I-Math
that O
are O
considered O
in O
mathematics O
are O
also O
endowed O
with O
other O
structure O
that O
are O
considered O
in O
mathematics B-Math
are O
also O
endowed O
with O
other O
structure O

other O
vectorlike O
objects O
that O
describe O
physical O
quantities O
and O
transform O
in O
a O
similar O
way O
under O
changes O
of O
the O
coordinate O
system O
include O
pseudovectors O
and O
tensors B-Math

analysis O
evolved O
from O
calculus B-Math
which O
involves O
the O
elementary O
concepts O
and O
techniques O
of O
analysi O
analysis B-Math
evolved O
from O
calculus O
which O
involves O
the O
elementary O
concepts O
and O
techniques O
of O
analysi O

this O
line O
of O
inquiry O
naturally O
leads O
to O
the O
idea O
of O
the O
dual B-Math
space I-Math
the O
vector O
space O
v O
consisting O
of O
linear O
maps O
f O
v O
f O
where O
f O
is O
the O
field O
of O
scalar O
the O
vector O
space O
v O
consisting O
of O
linear B-Math
maps I-Math
f O
v O
f O
where O
f O
is O
the O
field O
of O
scalar O
vector B-Math
space I-Math
v O
consisting O
of O
linear O
maps O
f O
v O
f O
where O
f O
is O
the O
field O
of O
scalar O

most O
of O
this O
article O
deals O
with O
linear B-Math
combinations I-Math
in O
the O
context O
of O
a O
vector O
space O
over O
a O
field O
with O
some O
generalizations O
given O
at O
the O
end O
of O
the O
articl O
in O
the O
context O
of O
a O
vector B-Math
space I-Math
over O
a O
field O
with O
some O
generalizations O
given O
at O
the O
end O
of O
the O
articl O

in O
classical B-Math
geometry I-Math
the O
involved O
vector O
spaces O
are O
vector O
spaces O
over O
the O
reals O
but O
the O
constructions O
may O
be O
extended O
to O
vector O
spaces O
over O
any O
field O
allowing O
considering O
geometry O
over O
arbitrary O
fields O
including O
finite O
field O
the O
involved O
vector B-Math
spaces I-Math
are O
vector O
spaces O
over O
the O
reals O
but O
the O
constructions O
may O
be O
extended O
to O
vector O
spaces O
over O
any O
field O
allowing O
considering O
geometry O
over O
arbitrary O
fields O
including O
finite O
field O
are O
vector O
spaces O
over O
the O
reals O
but O
the O
constructions O
may O
be O
extended O
to O
vector O
spaces O
over O
any O
field B-Math
allowing O
considering O
geometry O
over O
arbitrary O
fields O
including O
finite O
field O

this O
is O
also O
the O
case O
of O
homographies O
and O
möbius O
transformations O
when O
considered O
as O
transformations O
of O
a O
projective B-Math
space I-Math

the O
third O
system O
has O
no O
solutions O
since O
the O
three O
lines O
share O
no O
common O
point B-Math

the O
basic O
objects O
of O
geometry B-Math
which O
are O
lines O
and O
planes O
are O
represented O
by O
linear O
equation O
which O
are O
lines B-Math
and O
planes O
are O
represented O
by O
linear O
equation O
and O
planes B-Math
are O
represented O
by O
linear O
equation O
are O
represented O
by O
linear B-Math
equations I-Math

this O
is O
also O
the O
case O
of O
topological B-Math
vector I-Math
spaces I-Math
which O
include O
function O
spaces O
inner O
product O
spaces O
normed O
spaces O
hilbert O
spaces O
and O
banach O
space O
which O
include O
function O
spaces O
inner O
product O
spaces O
normed O
spaces O
hilbert B-Math
spaces I-Math
and O
banach O
space O
and O
banach B-Math
spaces I-Math

the O
linear B-Math
span I-Math
can O
be O
characterized O
either O
as O
the O
intersection O
of O
all O
linear O
subspaces O
that O
contain O
s O
or O
as O
the O
smallest O
subspace O
containing O
s O
the O
linear O
span O
of O
a O
set O
of O
vectors O
is O
therefore O
a O
vector O
space O
itsel O
can O
be O
characterized O
either O
as O
the O
intersection O
of O
all O
linear B-Math
subspaces I-Math
that O
contain O
s O
or O
as O
the O
smallest O
subspace O
containing O
s O
the O
linear O
span O
of O
a O
set O
of O
vectors O
is O
therefore O
a O
vector O
space O
itsel O
that O
contain O
s O
or O
as O
the O
smallest O
subspace O
containing O
s O
the O
linear O
span O
of O
a O
set O
of O
vectors O
is O
therefore O
a O
vector B-Math
space I-Math
itsel O
vectors B-Math
s O
therefore O
a O
vector O
space O
itsel O

it O
is O
possible O
for O
a O
system O
of O
two O
equations O
and O
two O
unknowns O
to O
have O
no O
solution O
if O
the O
two O
lines O
are O
parallel B-Math
or O
for O
a O
system O
of O
three O
equations O
and O
two O
unknowns O
to O
be O
solvable O
if O
the O
three O
lines O
intersect O
at O
a O
single O
poin O
solution B-Math
if O
the O
two O
lines O
are O
parallel O
or O
for O
a O
system O
of O
three O
equations O
and O
two O
unknowns O
to O
be O
solvable O
if O
the O
three O
lines O
intersect O
at O
a O
single O
poin O
if O
the O
two O
lines O
are O
parallel O
or O
for O
a O
system O
of O
three O
equations O
and O
two O
unknowns O
to O
be O
solvable O
if O
the O
three O
lines O
intersect O
at O
a O
single O
point B-Math

in O
mathematics O
a O
linear B-Math
combination I-Math
is O
an O
expression O
constructed O
from O
a O
set O
of O
terms O
by O
multiplying O
each O
term O
by O
a O
constant O
and O
adding O
the O
results O
e O
is O
an O
expression B-Math
constructed O
from O
a O
set O
of O
terms O
by O
multiplying O
each O
term O
by O
a O
constant O
and O
adding O
the O
results O
e O

real O
vector O
space O
and O
complex O
vector O
space O
are O
kinds O
of O
vector B-Math
spaces I-Math
ased O
on O
different O
kinds O
of O
scalars O
real O
coordinate O
space O
or O
complex O
coordinate O
spac O
ased O
on O
different O
kinds O
of O
scalars O
real B-Math
coordinate I-Math
space I-Math
r O
complex O
coordinate O
spac O
r O
complex B-Math
coordinate I-Math
space I-Math

most O
geometric B-Math
transformation I-Math
such O
as O
translations O
rotations O
reflections O
rigid O
motions O
isometries O
and O
projections O
transform O
lines O
into O
line O
such O
as O
translations B-Attributes
rotations O
reflections O
rigid O
motions O
isometries O
and O
projections O
transform O
lines O
into O
line O
rotations B-Attributes
reflections O
rigid O
motions O
isometries O
and O
projections O
transform O
lines O
into O
line O
reflections B-Attributes
rigid O
motions O
isometries O
and O
projections O
transform O
lines O
into O
line O
rigid B-Attributes
motions I-Attributes
isometries O
and O
projections O
transform O
lines O
into O
line O
isometries B-Attributes
and O
projections O
transform O
lines O
into O
line O
and O
projections B-Attributes
transform I-Attributes
lines O
into O
line O

moreover O
it O
maps O
linear B-Math
subspaces I-Math
in O
onto O
linear O
subspaces O
in O
possibly O
of O
a O
lower O
dimension O
for O
example O
it O
maps O
a O
plane O
through O
the O
origin O
in O
to O
either O
a O
plane O
through O
the O
origin O
in O
a O
line O
through O
the O
origin O
in O
or O
just O
the O
origin O
i O
in O
onto O
linear O
subspaces O
in O
possibly O
of O
a O
lower O
dimension B-Math
for O
example O
it O
maps O
a O
plane O
through O
the O
origin O
in O
to O
either O
a O
plane O
through O
the O
origin O
in O
a O
line O
through O
the O
origin O
in O
or O
just O
the O
origin O
i O

a O
system O
of O
linear B-Math
equations I-Math
behave O
differently O
from O
the O
general O
case O
if O
the O
equations O
are O
linearly O
dependent O
or O
if O
it O
is O
inconsistent O
and O
has O
no O
more O
equations O
than O
unknown O
behave O
differently O
from O
the O
general O
case O
if O
the O
equations O
are O
linearly B-Math
dependent I-Math
or O
if O
it O
is O
inconsistent O
and O
has O
no O
more O
equations O
than O
unknown O

the O
binary O
operation O
called O
vector O
addition O
or O
simply O
addition O
assigns O
to O
any O
two O
vectors B-Math
v O
and O
w O
in O
v O
a O
third O
vector O
in O
v O
which O
is O
commonly O
written O
as O
v O
w O
and O
called O
the O
sum O
of O
these O
two O
vector O

this O
point O
of O
view O
turned O
out O
to O
be O
particularly O
useful O
for O
the O
study O
of O
differential B-Math
and I-Math
integral I-Math
equations I-Math

if O
such O
a O
linear B-Math
combination I-Math
exists O
then O
the O
vectors O
are O
said O
to O
be O
linearly O
dependen O
exists O
then O
the O
vectors O
are O
said O
to O
be O
linearly B-Math
dependent I-Math

this O
is O
the O
case O
with O
mechanics O
and O
robotics O
for O
describing O
rigid B-Attributes
body I-Attributes
dynamics O
geodesy O
for O
describing O
earth O
shape O
perspectivity O
computer O
vision O
and O
computer O
graphics O
for O
describing O
the O
relationship O
between O
a O
scene O
and O
its O
plane O
representation O
and O
many O
other O
scientific O
domain O

many O
other O
physical O
quantities O
can O
be O
usefully O
thought O
of O
as O
vectors B-Math

all O
these O
questions O
can O
be O
solved O
by O
using O
gaussian B-Math
elimination I-Math
or O
some O
variant O
of O
this O
algorith O

most O
physical O
phenomena O
are O
modeled O
by O
partial B-Math
differential I-Math
equations I-Math

the O
binary O
function O
called O
scalar O
multiplicationassigns B-Math
to O
any O
scalar O
a O
in O
f O
and O
any O
vector O
v O
in O
v O
another O
vector O
in O
v O
which O
is O
denoted O
a O
to O
any O
scalar B-Math
a O
in O
f O
and O
any O
vector O
v O
in O
v O
another O
vector O
in O
v O
which O
is O
denoted O
a O
a O
in O
f O
and O
any O
vector B-Math
v O
in O
v O
another O
vector O
in O
v O
which O
is O
denoted O
a O

finitedimensional O
vector B-Math
spaces I-Math
occur O
naturally O
in O
geometry O
and O
related O
area O
occur O
naturally O
in O
geometry B-Math
and O
related O
area O
finitedimensional B-Attributes
vector O
spaces O
occur O
naturally O
in O
geometry O
and O
related O
area O

these O
equations O
often O
complex O
and O
nonlinear O
can O
be O
linearized O
using O
linear B-Math
algebra I-Math
methods O
allowing O
for O
simpler O
solutions O
and O
analyse O

however O
the O
general O
concept O
of O
a O
functional B-Math
had O
previously O
been O
introduced O
in O
by O
the O
italian O
mathematician O
and O
physicist O
vito O
volterr O

this O
means O
that O
for O
two O
vector B-Math
spaces I-Math
over O
a O
given O
field O
and O
with O
the O
same O
dimension O
the O
properties O
that O
depend O
only O
on O
the O
vectorspace O
structure O
are O
exactly O
the O
same O
technically O
the O
vector O
spaces O
are O
isomorphi O
over O
a O
given O
field O
and O
with O
the O
same O
dimension O
the O
properties O
that O
depend O
only O
on O
the O
vectorspace O
structure O
are O
exactly O
the O
same O
technically O
the O
vector O
spaces O
are O
isomorphic B-Attributes
dimension B-Math
he O
properties O
that O
depend O
only O
on O
the O
vectorspace O
structure O
are O
exactly O
the O
same O
technically O
the O
vector O
spaces O
are O
isomorphi O

vector B-Math
spaces I-Math
generalize O
euclidean O
vectors O
which O
allow O
modeling O
of O
physical O
quantities O
such O
as O
forces O
and O
velocity O
that O
have O
not O
only O
a O
magnitude O
but O
also O
a O
directio O
generalize O
euclidean O
vectors O
which O
allow O
modeling O
of O
physical O
quantities O
such O
as O
forces O
and O
velocity O
that O
have O
not O
only O
a O
magnitude B-Attributes
but O
also O
a O
directio O
but O
also O
a O
direction B-Attributes
vectors B-Math
which O
allow O
modeling O
of O
physical O
quantities O
such O
as O
forces O
and O
velocity O
that O
have O
not O
only O
a O
magnitude O
but O
also O
a O
directio O

this O
is O
important O
because O
if O
we O
have O
m O
independent B-Math
vectors I-Math
a O
solution O
is O
guaranteed O
regardless O
of O
the O
righthand O
side O
and O
otherwise O
not O
guarantee O

many O
algebraic B-Math
operations I-Math
on O
real O
numbers O
such O
as O
addition O
subtraction O
multiplication O
and O
negation O
have O
close O
analogues O
for O
vectors O
operations O
which O
obey O
the O
familiar O
algebraic O
laws O
of O
commutativity O
associativity O
and O
distributivit O
on O
real O
numbers O
such O
as O
addition O
subtraction B-Attributes
multiplication O
and O
negation O
have O
close O
analogues O
for O
vectors O
operations O
which O
obey O
the O
familiar O
algebraic O
laws O
of O
commutativity O
associativity O
and O
distributivit O
addition B-Attributes
subtraction O
multiplication O
and O
negation O
have O
close O
analogues O
for O
vectors O
operations O
which O
obey O
the O
familiar O
algebraic O
laws O
of O
commutativity O
associativity O
and O
distributivit O
subtraction O
multiplication B-Attributes
and O
negation O
have O
close O
analogues O
for O
vectors O
operations O
which O
obey O
the O
familiar O
algebraic O
laws O
of O
commutativity O
associativity O
and O
distributivit O
and O
negation B-Attributes
have O
close O
analogues O
for O
vectors O
operations O
which O
obey O
the O
familiar O
algebraic O
laws O
of O
commutativity O
associativity O
and O
distributivit O
have O
close O
analogues O
for O
vectors O
operations O
which O
obey O
the O
familiar O
algebraic O
laws O
of O
commutativity B-Attributes
associativity O
and O
distributivit O
associativity B-Attributes
and O
distributivit O
and O
distributivity B-Attributes

the O
magnitude O
of O
the O
vector B-Math
is O
the O
distance O
between O
the O
two O
points O
and O
the O
direction O
refers O
to O
the O
direction O
of O
displacement O
from O
a O
to O
is O
the O
distance B-Attributes
between O
the O
two O
points O
and O
the O
direction O
refers O
to O
the O
direction O
of O
displacement O
from O
a O
to O
between O
the O
two O
points O
and O
the O
direction B-Attributes
refers O
to O
the O
direction O
of O
displacement O
from O
a O
to O

in O
mathematics B-Math
a O
matrix O
p O
a O
matrix B-Math
p O

the O
concept O
of O
vector B-Math
spaces I-Math
is O
fundamental O
for O
linear O
algebra O
together O
with O
the O
concept O
of O
matrices O
which O
allows O
computing O
in O
vector O
space O
is O
fundamental O
for O
linear B-Math
algebra I-Math
together O
with O
the O
concept O
of O
matrices O
which O
allows O
computing O
in O
vector O
space O
together O
with O
the O
concept O
of O
matrices B-Math
which O
allows O
computing O
in O
vector O
space O

this O
allows O
all O
the O
language O
and O
theory O
of O
vector B-Math
spaces I-Math
or O
more O
generally O
modules O
to O
be O
brought O
to O
bea O

the O
solutions O
of O
any O
linear B-Math
ordinary I-Math
differential I-Math
equation I-Math
of O
any O
order O
may O
be O
deduced O
by O
integration O
from O
the O
solution O
of O
the O
homogeneous O
equation O
obtained O
by O
removing O
the O
constant O
ter O
of O
any O
order O
may O
be O
deduced O
by O
integration O
from O
the O
solution O
of O
the O
homogeneous O
equation O
obtained O
by O
removing O
the O
constant B-Math
term I-Math
homogeneous B-Math
equation I-Math
obtained O
by O
removing O
the O
constant O
ter O

gröbner O
basis O
theory O
provides O
algorithms B-Math
when O
coefficients O
and O
unknowns O
are O
polynomial O

a O
complete B-Math
metric I-Math
space I-Math
along O
with O
the O
additional O
structure O
of O
an O
inner O
product O
a O
conjugate O
symmetric O
sesquilinear O
form O
is O
known O
as O
a O
hilbert O
space O
which O
is O
in O
some O
sense O
a O
particularly O
wellbehaved O
banach O
spac O
along O
with O
the O
additional O
structure O
of O
an O
inner O
product O
a O
conjugate O
symmetric O
sesquilinear O
form O
is O
known O
as O
a O
hilbert B-Math
space I-Math
which O
is O
in O
some O
sense O
a O
particularly O
wellbehaved O
banach O
spac O

linear B-Math
algebra I-Math
is O
central O
to O
almost O
all O
areas O
of O
mathematic O
is O
central O
to O
almost O
all O
areas O
of O
mathematics B-Math

the O
definition O
of O
linear B-Math
dependence I-Math
and O
the O
ability O
to O
determine O
whether O
a O
subset O
of O
vectors O
in O
a O
vector O
space O
is O
linearly O
dependent O
are O
central O
to O
determining O
the O
dimension O
of O
a O
vector O
spac O
and O
the O
ability O
to O
determine O
whether O
a O
subset O
of O
vectors O
in O
a O
vector B-Math
space I-Math
is O
linearly O
dependent O
are O
central O
to O
determining O
the O
dimension O
of O
a O
vector O
spac O

scalars B-Math
are O
often O
real O
numbers O
but O
can O
be O
complex O
numbers O
or O
more O
generally O
elements O
of O
any O
fiel O
are O
often O
real O
numbers O
but O
can O
be O
complex O
numbers O
or O
more O
generally O
elements O
of O
any O
field B-Math

a O
differential B-Math
equation I-Math
can O
be O
homogeneous O
in O
either O
of O
two O
respect O
can O
be O
homogeneous B-Math
in O
either O
of O
two O
respect O

linear B-Math
maps I-Math
can O
often O
be O
represented O
as O
matrices O
and O
simple O
examples O
include O
rotation O
and O
reflection O
linear O
transformation O
can O
often O
be O
represented O
as O
matrices B-Math
and O
simple O
examples O
include O
rotation O
and O
reflection O
linear O
transformation O
and O
simple O
examples O
include O
rotation B-Attributes
and O
reflection O
linear O
transformation O
and O
reflection O
linear B-Math
transformations I-Math

presently O
most O
textbooks O
introduce O
geometric O
spaces O
from O
linear B-Math
algebra I-Math
and O
geometry O
is O
often O
presented O
at O
elementary O
level O
as O
a O
subfield O
of O
linear O
algebr O
geometric B-Math
spaces I-Math
from O
linear O
algebra O
and O
geometry O
is O
often O
presented O
at O
elementary O
level O
as O
a O
subfield O
of O
linear O
algebr O
from O
linear O
algebra O
and O
geometry B-Math
is O
often O
presented O
at O
elementary O
level O
as O
a O
subfield O
of O
linear O
algebr O
is O
often O
presented O
at O
elementary O
level O
as O
a O
subfield O
of O
linear B-Math
algebra I-Math

in O
general O
a O
system O
with O
the O
same O
number O
of O
equations B-Math
and O
unknowns O
has O
a O
single O
unique O
solutio O
and O
unknowns O
has O
a O
single O
unique O
solution B-Math

around O
this O
date O
it O
appeared O
that O
one O
may O
also O
define O
geometric B-Math
spaces I-Math
by O
constructions O
involving O
vector O
spaces O
see O
for O
example O
projective O
space O
and O
affine O
spac O
by O
constructions O
involving O
vector B-Math
spaces I-Math
see O
for O
example O
projective O
space O
and O
affine O
spac O
see O
for O
example O
projective B-Math
space I-Math
and O
affine O
spac O
and O
affine B-Math
space I-Math

the B-Math
theory I-Math
of I-Math
nonlinear I-Math
functionals I-Math
as O
continued O
by O
students O
of O
hadamard O
in O
particular O
fréchet O
and O
lév O

this O
article O
focuses O
on O
matrices B-Math
related O
to O
linear O
algebra O
and O
unless O
otherwise O
specified O
all O
matrices O
represent O
linear O
maps O
or O
may O
be O
viewed O
as O
suc O
related O
to O
linear B-Math
algebra I-Math
and O
unless O
otherwise O
specified O
all O
matrices O
represent O
linear O
maps O
or O
may O
be O
viewed O
as O
suc O
and O
unless O
otherwise O
specified O
all O
matrices B-Math
represent O
linear O
maps O
or O
may O
be O
viewed O
as O
suc O
represent O
linear B-Math
maps I-Math
or O
may O
be O
viewed O
as O
suc O

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 O
defining O
a O
vector B-Math
space I-Math

matrices B-Math
is O
a O
rectangular O
array O
or O
table O
of O
numbers O
symbols O
or O
expressions O
arranged O
in O
rows O
and O
columns O
which O
is O
used O
to O
represent O
a O
mathematical O
object O
or O
a O
property O
of O
such O
an O
objec O

in O
all O
these O
applications O
synthetic B-Math
geometry I-Math
is O
often O
used O
for O
general O
descriptions O
and O
a O
qualitative O
approach O
but O
for O
the O
study O
of O
explicit O
situations O
one O
must O
compute O
with O
coordinate O
is O
often O
used O
for O
general O
descriptions O
and O
a O
qualitative O
approach O
but O
for O
the O
study O
of O
explicit O
situations O
one O
must O
compute O
with O
coordinates B-Math

a O
bijective O
linear O
map O
between O
two O
vector O
spaces O
that O
is 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
is O
an O
isomorphism B-Math

in O
mathematics B-Math
and O
more O
specifically O
in O
linear O
algebra O
a O
linear O
subspace O
or O
vector O
subspacenote O
is O
a O
vector O
space O
that O
is O
a O
subset O
of O
some O
larger O
vector O
spac O
and O
more O
specifically O
in O
linear B-Math
algebra I-Math
a O
linear O
subspace O
or O
vector O
subspacenote O
is O
a O
vector O
space O
that O
is O
a O
subset O
of O
some O
larger O
vector O
spac O
a O
linear B-Math
subspace I-Math
or O
vector O
subspacenote O
is O
a O
vector O
space O
that O
is O
a O
subset O
of O
some O
larger O
vector O
spac O
or O
vector B-Math
subspacenote I-Math
is O
a O
vector O
space O
that O
is O
a O
subset O
of O
some O
larger O
vector O
spac O
is O
a O
vector O
space O
that O
is O
a O
subset O
of O
some O
larger O
vector B-Math
space I-Math

this O
requires O
the O
heavy O
use O
of O
linear B-Math
algebra I-Math

for O
nonlinear O
systems O
which O
can O
not O
be O
modeled O
with O
linear O
algebra O
it O
is O
often O
used O
for O
dealing O
with O
firstorder O
approximations 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-Math
map I-Math
that O
best O
approximates O
the O
function O
near O
that O
poin O
linear B-Math
algebra I-Math
it O
is O
often O
used O
for O
dealing O
with O
firstorder O
approximations 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 O
map O
that O
best O
approximates O
the O
function O
near O
that O
poin O

the O
metric O
also O
allows O
for O
a O
definition O
of O
limits O
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 B-Math
space I-Math
metric B-Math
space I-Math
that O
is O
complete O
is O
known O
as O
a O
banach O
spac O

it O
follows O
that O
they O
can O
be O
defined O
specified O
and O
studied O
in O
terms O
of O
linear B-Math
maps I-Math

in O
the O
field O
of O
fluid O
dynamics O
linear B-Math
algebra I-Math
finds O
its O
application O
in O
computational O
fluid O
dynamics O
cfd O
a O
branch O
that O
uses O
numerical O
analysis O
and O
data O
structures O
to O
solve O
and O
analyze O
problems O
involving O
fluid O
flow O

the O
first O
system O
has O
infinitely O
many O
solutions O
namely O
all O
of O
the O
points B-Math
on O
the O
blue O
lin O
solutions B-Math
namely O
all O
of O
the O
points O
on O
the O
blue O
lin O

weather O
forecasting O
or O
more O
specifically O
parametrization O
for O
atmospheric O
modeling B-Attributes
is O
a O
typical O
example O
of O
a O
realworld O
application O
where O
the O
whole O
earth O
atmosphere O
is O
divided O
into O
cells O
of O
say O
km O
of O
width O
and O
km O
of O
heigh O

a O
vector B-Math
space I-Math
is O
finitedimensional O
if O
its O
dimension O
is O
a O
natural O
numbe O
is O
finitedimensional B-Attributes
if O
its O
dimension O
is O
a O
natural O
numbe O

to O
solve O
them O
one O
usually O
decomposes O
the O
space O
in O
which O
the O
solutions B-Math
are O
searched O
into O
small O
mutually O
interacting O
cell O

for O
example O
as O
three O
parallel O
planes O
do O
not O
have O
a O
common O
point O
the O
solution O
set O
of O
their O
equations O
is O
empty O
the O
solution O
set O
of O
the O
equations O
of O
three O
planes O
intersecting O
at O
a O
point O
is O
single O
point O
if O
three O
planes O
pass O
through O
two O
points O
their O
equations O
have O
at O
least O
two O
common O
solutions O
in O
fact O
the O
solution B-Math
set I-Math
is O
infinite O
and O
consists O
in O
all O
the O
line O
passing O
through O
these O
point O

in O
mathematics O
differential B-Math
refers O
to O
several O
related O
notions O
derived O
from O
the O
early O
days O
of O
calculus O
put O
on O
a O
rigorous O
footing O
such O
as O
infinitesimal O
differences O
and O
the O
derivatives O
of O
function O
refers O
to O
several O
related O
notions O
derived O
from O
the O
early O
days O
of O
calculus O
put O
on O
a O
rigorous O
footing O
such O
as O
infinitesimal B-Attributes
differences I-Attributes
and O
the O
derivatives O
of O
function O
and O
the O
derivatives B-Math
of O
function O

in O
contrast O
linear B-Math
algebra I-Math
deals O
mostly O
with O
finitedimensional O
spaces O
and O
does O
not O
use O
topolog O
deals O
mostly O
with O
finitedimensional B-Math
spaces I-Math
and O
does O
not O
use O
topolog O
and O
does O
not O
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topology B-Math

this O
is O
in O
particular O
the O
case O
in O
graph B-Math
theory I-Math
f O
incidence O
matrices O
and O
adjacency O
matrice O
f O
incidence O
matrices B-Math
and O
adjacency O
matrice O

if O
every O
vector B-Math
within O
that O
span O
has O
exactly O
one O
expression O
as O
a O
linear O
combination O
of O
the O
given O
lefthand O
vectors O
then O
any O
solution O
is O
uniqu O
within O
that O
span O
has O
exactly O
one O
expression O
as O
a O
linear B-Math
combination I-Math
of O
the O
given O
lefthand O
vectors O
then O
any O
solution O
is O
uniqu O

these O
concepts O
are O
central O
to O
the O
definition O
of O
dimension B-Math

in O
mathematics O
a O
set O
b O
of O
vectors O
in O
a O
vector O
space O
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called O
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basis B-Attributes
p O
vector B-Math
space I-Math
v O
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called O
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in O
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called O
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p O

with O
respect O
to O
general O
linear B-Math
maps I-Math
linear O
endomorphisms O
and O
square O
matrices O
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 O
algebra O
which O
is O
used O
in O
many O
parts O
of O
mathematics O
including O
geometric O
transformations O
coordinate O
changes O
quadratic O
forms O
and O
many O
other O
part O
of O
mathematic O
linear B-Math
endomorphisms I-Math
and O
square O
matrices O
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some O
specific O
properties O
that O
make O
their O
study O
an O
important O
part O
of O
linear O
algebra O
which O
is O
used O
in O
many O
parts O
of O
mathematics O
including O
geometric O
transformations O
coordinate O
changes O
quadratic O
forms O
and O
many O
other O
part O
of O
mathematic O
and O
square B-Math
matrices I-Math
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 O
algebra O
which O
is O
used O
in O
many O
parts O
of O
mathematics O
including O
geometric O
transformations O
coordinate O
changes O
quadratic O
forms O
and O
many O
other O
part O
of O
mathematic O
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 O
algebra O
which O
is O
used O
in O
many O
parts O
of O
mathematics O
including O
geometric O
transformations O
coordinate B-Math
changes I-Math
quadratic O
forms O
and O
many O
other O
part O
of O
mathematic O
quadratic B-Math
forms I-Math
and O
many O
other O
part O
of O
mathematic O
geometric B-Math
transformations I-Math
coordinate O
changes O
quadratic O
forms O
and O
many O
other O
part O
of O
mathematic O

sciences O
concerned O
with O
this O
space O
use O
geometry B-Math
widel O
sciences B-Attributes
concerned O
with O
this O
space O
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geometry O
widel O

in O
both O
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very O
large O
matrices B-Math
are O
generally O
involve O

the O
following O
pictures O
illustrate O
this O
trichotomy O
in O
the O
case O
of O
two O
variables B-Math

vectors B-Math
play O
an O
important O
role O
in O
physics O
the O
velocity O
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forces O
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can O
all O
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play O
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in O
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of O
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it O
assists O
in O
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modeling O
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providing O
essential O
tools O
for O
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analysis O
of O
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the O
term O
is O
used O
in O
various O
branches O
of O
mathematics O
such O
as O
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differential I-Math
geometry I-Math
algebraic O
geometry O
and O
algebraic O
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geometry I-Math
and O
algebraic O
topolog O
and O
algebraic B-Math
topology I-Math

a O
linear B-Math
combination I-Math
of O
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and O
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any O
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to O
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vector O
space O
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subset O
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spans O
v O
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v O
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is O
a O
generator O
or O
generator O
set O
of O

spans O
can O
be O
generalized O
to O
matroids B-Attributes
and O
module O
and O
modules B-Attributes
spans B-Math
can O
be O
generalized O
to O
matroids O
and O
module O

the O
elements O
of O
a O
basis B-Attributes
are O
called O
basis O
vector O
are O
called O
basis B-Math
vectors I-Math

f O
a O
linear B-Math
map I-Math
is O
a O
bijection O
then O
it O
is O
called O
a O
linear O
isomorphis O
is O
a O
bijection O
then O
it O
is O
called O
a O
linear B-Math
isomorphism I-Math
bijection B-Attributes
then O
it O
is O
called O
a O
linear O
isomorphis O

in O
mathematics B-Math
and O
physics O
a O
vector O
space O
also O
called O
a O
linear O
space O
is O
a O
set O
whose O
elements O
often O
called O
vectors O
may O
be O
added O
together O
and O
multiplied O
scaled O
by O
numbers O
called O
scalar O
and O
physics O
a O
vector B-Math
space I-Math
also O
called O
a O
linear O
space O
is O
a O
set O
whose O
elements O
often O
called O
vectors O
may O
be O
added O
together O
and O
multiplied O
scaled O
by O
numbers O
called O
scalar O
also O
called O
a O
linear B-Math
space I-Math
is O
a O
set O
whose O
elements O
often O
called O
vectors O
may O
be O
added O
together O
and O
multiplied O
scaled O
by O
numbers O
called O
scalar O
is O
a O
set O
whose O
elements O
often O
called O
vectors B-Math
may O
be O
added O
together O
and O
multiplied O
scaled O
by O
numbers O
called O
scalar O

the O
same O
names O
and O
the O
same O
definition O
are O
also O
used O
for O
the O
more O
general O
case O
of O
modules O
over O
a O
ring O
see O
module B-Math
homomorphism I-Math

because O
a O
solution O
to O
a O
linear B-Math
system I-Math
must O
satisfy O
all O
of O
the O
equations O
the O
solution O
set O
is O
the O
intersection O
of O
these O
lines O
and O
is O
hence O
either O
a O
line O
a O
single O
point O
or O
the O
empty O
se O

for O
improving O
efficiency O
some O
of O
them O
configure O
the O
algorithms B-Attributes
automatically O
at O
run O
time O
for O
adapting O
them O
to O
the O
specificities O
of O
the O
computer O
cache O
size O
number O
of O
available O
core O

an O
important O
part O
of O
functional O
analysis O
is O
the O
extension O
of O
the O
theories O
of O
measure B-Math
integration I-Math
and O
probability O
to O
infinite O
dimensional O
spaces O
also O
known O
as O
infinite O
dimensional O
analysi O
and O
probability B-Math
to O
infinite O
dimensional O
spaces O
also O
known O
as O
infinite O
dimensional O
analysi O
functional B-Math
analysis I-Math
is O
the O
extension O
of O
the O
theories O
of O
measure O
integration O
and O
probability O
to O
infinite O
dimensional O
spaces O
also O
known O
as O
infinite O
dimensional O
analysi O

functional B-Math
analysis I-Math
is O
of O
particular O
importance O
to O
quantum O
mechanics O
the O
theory O
of O
partial O
differential O
equations O
digital O
signal O
processing O
and O
electrical O
engineerin O
is O
of O
particular O
importance O
to O
quantum O
mechanics O
the O
theory O
of O
partial B-Math
differential I-Math
equations I-Math
digital O
signal O
processing O
and O
electrical O
engineerin O

for O
example O
the O
navierstokes O
equations O
fundamental O
in O
fluid O
dynamics O
are O
often O
solved O
using O
techniques O
derived O
from O
linear B-Math
algebra I-Math

analysis B-Math
may O
be O
distinguished O
from O
geometry O
however O
it O
can O
be O
applied O
to O
any O
space O
of O
mathematical O
objects O
that O
has O
a O
definition O
of O
nearness O
a O
topological O
space O
or O
specific O
distances O
between O
objects O
a O
metric O
spac O
may O
be O
distinguished O
from O
geometry B-Math
however O
it O
can O
be O
applied O
to O
any O
space O
of O
mathematical O
objects O
that O
has O
a O
definition O
of O
nearness O
a O
topological O
space O
or O
specific O
distances O
between O
objects O
a O
metric O
spac O
however O
it O
can O
be O
applied O
to O
any O
space O
of O
mathematical O
objects O
that O
has O
a O
definition O
of O
nearness O
a O
topological B-Math
space I-Math
or O
specific O
distances O
between O
objects O
a O
metric O
spac O

the O
concept O
of O
linear B-Math
combinations I-Math
is O
central O
to O
linear O
algebra O
and O
related O
fields O
of O
mathematic O
is O
central O
to O
linear B-Math
algebra I-Math
and O
related O
fields O
of O
mathematic O
and O
related O
fields O
of O
mathematics B-Math

an O
essential O
question O
in O
linear B-Math
algebra I-Math
is O
testing O
whether O
a O
linear O
map O
is O
an O
isomorphism O
or O
not O
and O
if O
it O
is O
not O
an O
isomorphism O
finding O
its O
range O
or O
image 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
called O
the O
kernel O
of O
the O
ma O
is O
testing O
whether O
a O
linear B-Math
map I-Math
is O
an O
isomorphism O
or O
not O
and O
if O
it O
is O
not O
an O
isomorphism O
finding O
its O
range O
or O
image 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
called O
the O
kernel O
of O
the O
ma O
is O
an O
isomorphism B-Attributes
or O
not O
and O
if O
it O
is O
not O
an O
isomorphism O
finding O
its O
range O
or O
image 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
called O
the O
kernel O
of O
the O
ma O
or O
not O
and O
if O
it O
is O
not O
an O
isomorphism O
finding O
its O
range O
or O
image 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
called O
the O
kernel B-Math
of O
the O
ma O

in O
mathematics O
and O
more O
specifically O
in O
linear B-Math
algebra I-Math
a O
linear O
map O
also O
called O
a O
linear O
mapping O
linear O
transformation O
vector O
space O
homomorphism O
or O
in O
some O
contexts O
linear O
function O
is O
a O
mapping O
between O
two O
vector O
spaces O
that O
preserves O
the O
operations O
of O
vector O
addition O
and O
scalar O
multiplicatio O
a O
linear B-Math
map I-Math
also O
called O
a O
linear O
mapping O
linear O
transformation O
vector O
space O
homomorphism O
or O
in O
some O
contexts O
linear O
function O
is O
a O
mapping O
between O
two O
vector O
spaces O
that O
preserves O
the O
operations O
of O
vector O
addition O
and O
scalar O
multiplicatio O
also O
called O
a O
linear B-Math
mapping I-Math
inear O
transformation O
vector O
space O
homomorphism O
or O
in O
some O
contexts O
linear O
function O
is O
a O
mapping O
between O
two O
vector O
spaces O
that O
preserves O
the O
operations O
of O
vector O
addition O
and O
scalar O
multiplicatio O
linear B-Math
transformation I-Math
vector O
space O
homomorphism O
or O
in O
some O
contexts O
linear O
function O
is O
a O
mapping O
between O
two O
vector O
spaces O
that O
preserves O
the O
operations O
of O
vector O
addition O
and O
scalar O
multiplicatio O
vector B-Math
space I-Math
homomorphism I-Math
or O
in O
some O
contexts O
linear O
function O
is O
a O
mapping O
between O
two O
vector O
spaces O
that O
preserves O
the O
operations O
of O
vector O
addition O
and O
scalar O
multiplicatio O

overall O
the O
application O
of O
linear B-Math
algebra I-Math
in O
fluid O
mechanics O
fluid O
dynamics O
and O
thermal O
energy O
systems O
is O
an O
example O
of O
the O
profound O
interconnection O
between O
mathematics O
and O
engineerin O
in O
fluid B-Attributes
mechanics I-Attributes
fluid O
dynamics O
and O
thermal O
energy O
systems O
is O
an O
example O
of O
the O
profound O
interconnection O
between O
mathematics O
and O
engineerin O
fluid B-Attributes
dynamics I-Attributes
and O
thermal O
energy O
systems O
is O
an O
example O
of O
the O
profound O
interconnection O
between O
mathematics O
and O
engineerin O

therefore O
the O
study O
of O
matrices B-Math
is O
a O
large O
part O
of O
linear O
algebra O
and O
most O
properties O
and O
operations O
of O
abstract O
linear O
algebra O
can O
be O
expressed O
in O
terms O
of O
matrice O
is O
a O
large O
part O
of O
linear B-Math
algebra I-Math
and O
most O
properties O
and O
operations O
of O
abstract O
linear O
algebra O
can O
be O
expressed O
in O
terms O
of O
matrice O
and O
most O
properties O
and O
operations O
of O
abstract O
linear B-Math
algebra I-Math
can O
be O
expressed O
in O
terms O
of O
matrice O
can O
be O
expressed O
in O
terms O
of O
matrices B-Math

until O
the O
end O
of O
the O
th O
century O
geometric B-Math
spaces I-Math
were O
defined O
by O
axioms O
relating O
points O
lines O
and O
planes O
synthetic O
geometr O
were O
defined O
by O
axioms O
relating O
points O
lines O
and O
planes O
synthetic B-Math
geometry I-Math

for O
n O
variables O
each O
linear O
equation O
determines O
a O
hyperplane O
in O
ndimensional B-Math
space I-Math
determines B-Math
a O
hyperplane O
in O
ndimensional O
spac O
variables B-Math
each O
linear O
equation O
determines O
a O
hyperplane O
in O
ndimensional O
spac O

these O
theories O
are O
usually O
studied O
in O
the O
context O
of O
real O
and O
complex O
numbers O
and O
functions B-Math

functional B-Math
analysis I-Math
studies O
function O
space O
studies O
function B-Math
spaces I-Math

as O
a O
corollary O
all O
vector O
spaces O
are O
equipped O
with O
at O
least O
two O
possibly O
different O
linear O
subspaces O
the O
zero O
vector O
space O
consisting O
of O
the O
zero O
vector O
alone O
and O
the O
entire O
vector B-Math
space I-Math
itsel O
linear B-Math
subspaces I-Math
the O
zero O
vector O
space O
consisting O
of O
the O
zero O
vector O
alone O
and O
the O
entire O
vector O
space O
itsel O
the O
zero B-Math
vector I-Math
space I-Math
consisting O
of O
the O
zero O
vector O
alone O
and O
the O
entire O
vector O
space O
itsel O

the O
second O
system O
has O
a O
single O
unique O
solution O
namely O
the O
intersection B-Math
of O
the O
two O
line O
single B-Math
unique I-Math
solution I-Math
namely O
the O
intersection O
of O
the O
two O
line O

linear B-Math
algebra I-Math
is O
thus O
a O
fundamental O
part O
of O
functional O
analysis O
and O
its O
applications O
which O
include O
in O
particular O
quantum O
mechanics O
wave O
functions O
and O
fourier O
analysis O
orthogonal O
basi O
is O
thus O
a O
fundamental O
part O
of O
functional B-Math
analysis I-Math
and O
its O
applications O
which O
include O
in O
particular O
quantum O
mechanics O
wave O
functions O
and O
fourier O
analysis O
orthogonal O
basi O
and O
its O
applications O
which O
include O
in O
particular O
quantum O
mechanics O
wave O
functions O
and O
fourier B-Math
analysis I-Math
orthogonal I-Math
basis I-Math

this O
provides O
a O
concise O
and O
synthetic O
way O
for O
manipulating O
and O
studying O
systems O
of O
linear B-Math
equations I-Math

for O
example O
matrix B-Math
multiplication O
represents O
the O
composition O
of O
linear O
map O
multiplication O
represents O
the O
composition O
of O
linear B-Math
maps I-Math

linear O
systems O
are O
the O
basis O
and O
a O
fundamental O
part O
of O
linear B-Math
algebra I-Math
a O
subject O
used O
in O
most O
modern O
mathematic O
a O
subject O
used O
in O
most O
modern O
mathematics B-Math
linear B-Math
systems I-Math
are O
the O
basis O
and O
a O
fundamental O
part O
of O
linear O
algebra O
a O
subject O
used O
in O
most O
modern O
mathematic O

for O
linear B-Math
systems I-Math
this O
interaction O
involves O
linear O
function O
this O
interaction O
involves O
linear B-Math
functions I-Math

in O
the O
case O
where O
a O
linear B-Math
map I-Math
is O
called O
a O
linear O
endomorphis O
is O
called O
a O
linear B-Math
endomorphism I-Math

some O
processors O
typically O
graphics O
processing O
units O
gpu O
are O
designed O
with O
a O
matrix B-Attributes
structure I-Attributes
for O
optimizing O
the O
operations O
of O
linear O
algebr O
for O
optimizing O
the O
operations O
of O
linear B-Math
algebra I-Math

square B-Math
matrices I-Math
matrices O
with O
the O
same O
number O
of O
rows O
and O
columns O
play O
a O
major O
role O
in O
matrix O
theor O
matrices O
with O
the O
same O
number O
of O
rows O
and O
columns O
play O
a O
major O
role O
in O
matrix B-Math
theory I-Math

cfd O
relies O
heavily O
on O
linear B-Math
algebra I-Math
for O
the O
computation O
of O
fluid O
flow O
and O
heat O
transfer O
in O
various O
application O

linear O
algebraic O
concepts O
such O
as O
matrix B-Math
operations O
and O
eigenvalue O
problems O
are O
employed O
to O
enhance O
the O
efficiency O
reliability O
and O
economic O
performance O
of O
power O
system O
operations O
and O
eigenvalue B-Math
problems O
are O
employed O
to O
enhance O
the O
efficiency O
reliability O
and O
economic O
performance O
of O
power O
system O
linear B-Math
algebraic I-Math
concepts O
such O
as O
matrix O
operations O
and O
eigenvalue O
problems O
are O
employed O
to O
enhance O
the O
efficiency O
reliability O
and O
economic O
performance O
of O
power O
system O

vector B-Math
spaces I-Math
are O
characterized O
by O
their O
dimension O
which O
roughly O
speaking O
specifies O
the O
number O
of O
independent O
directions O
in O
the O
spac O
are O
characterized O
by O
their O
dimension B-Attributes
which O
roughly O
speaking O
specifies O
the O
number O
of O
independent O
directions O
in O
the O
spac O

in O
geometry O
matrices B-Math
are O
widely O
used O
for O
specifying O
and O
representing O
geometric O
transformations O
for O
example O
rotations O
and O
coordinate O
change O
are O
widely O
used O
for O
specifying O
and O
representing O
geometric O
transformations O
for O
example O
rotations B-Attributes
and O
coordinate O
change O
and O
coordinate B-Attributes
changes I-Attributes

there O
are O
many O
rings O
for O
which O
there O
are O
algorithms O
for O
solving O
linear B-Math
equations I-Math
and O
systems O
of O
linear O
equation O
rings B-Math
for O
which O
there O
are O
algorithms O
for O
solving O
linear O
equations O
and O
systems O
of O
linear O
equation O

the O
norm B-Math
induces O
a O
metric O
which O
measures O
the O
distance O
between O
elements O
and O
induces O
a O
topology O
which O
allows O
for O
a O
definition O
of O
continuous O
map O
induces O
a O
metric O
which O
measures O
the O
distance B-Attributes
between O
elements O
and O
induces O
a O
topology O
which O
allows O
for O
a O
definition O
of O
continuous O
map O

for O
instance O
linear B-Math
algebra I-Math
is O
fundamental O
in O
modern O
presentations O
of O
geometry O
including O
for O
defining O
basic O
objects O
such O
as O
lines O
planes O
and O
rotation O
is O
fundamental O
in O
modern O
presentations O
of O
geometry B-Math
including O
for O
defining O
basic O
objects O
such O
as O
lines O
planes O
and O
rotation O
including O
for O
defining O
basic O
objects O
such O
as O
lines B-Math
planes I-Math
and O
rotation O
and O
rotations B-Math

linear B-Math
algebra I-Math
is O
also O
used O
in O
most O
sciences O
and O
fields O
of O
engineering O
because O
it O
allows O
modeling O
many O
natural O
phenomena O
and O
computing O
efficiently O
with O
such O
model O

a O
euclidean O
vector B-Math
is O
frequently O
represented O
by O
a O
directed O
line O
segment O
or O
graphically O
as O
an O
arrow O
connecting O
an O
initial O
point O
a O
with O
a O
terminal O
point O
b O
and O
denoted O
b O

where O
f O
and O
g O
are O
homogeneous B-Math
functions I-Math
of O
the O
same O
degree O
of O
x O
and O

in O
other O
words O
a O
basis B-Attributes
is O
a O
linearly O
independent O
spanning O
se O
is O
a O
linearly B-Math
independent I-Math
spanning I-Math
set I-Math

in O
the O
language O
of O
category B-Math
theory I-Math
linear O
maps O
are O
the O
morphisms O
of O
vector O
space O
linear B-Math
maps I-Math
are O
the O
morphisms O
of O
vector O
space O
are O
the O
morphisms O
of O
vector B-Math
spaces I-Math

a O
system O
of O
nonlinear B-Math
equations I-Math
can O
often O
be O
approximated O
by O
a O
linear O
system O
see O
linearization O
a O
helpful O
technique O
when O
making O
a O
mathematical O
model O
or O
computer O
simulation O
of O
a O
relatively O
complex O
syste O
can O
often O
be O
approximated O
by O
a O
linear B-Math
system I-Math
see O
linearization O
a O
helpful O
technique O
when O
making O
a O
mathematical O
model O
or O
computer O
simulation O
of O
a O
relatively O
complex O
syste O

a O
linear B-Math
subspace I-Math
is O
usually O
simply O
called O
a O
subspace O
when O
the O
context O
serves O
to O
distinguish O
it O
from O
other O
types O
of O
subspace O
is O
usually O
simply O
called O
a O
subspace B-Math
when O
the O
context O
serves O
to O
distinguish O
it O
from O
other O
types O
of O
subspace O

the O
solution B-Math
set I-Math
is O
the O
intersection O
of O
these O
hyperplanes O
and O
is O
a O
flat O
which O
may O
have O
any O
dimension O
lower O
than O
is O
the O
intersection O
of O
these O
hyperplanes O
and O
is O
a O
flat O
which O
may O
have O
any O
dimension B-Math
lower O
than O
hyperplanes B-Math
nd O
is O
a O
flat O
which O
may O
have O
any O
dimension O
lower O
than O

this O
was O
one O
of O
the O
main O
motivations O
for O
developing O
linear B-Math
algebra I-Math

this O
includes O
the O
use O
of O
matrices B-Math
and O
vectors O
to O
represent O
and O
manipulate O
fluid O
flow O
field O
and O
vectors B-Math
to O
represent O
and O
manipulate O
fluid O
flow O
field O
to O
represent O
and O
manipulate O
fluid B-Attributes
flow I-Attributes
fields I-Attributes

this O
is O
the O
case O
of O
algebras O
which O
include O
field O
extensions B-Math
polynomial I-Math
rings I-Math
associative O
algebras O
and O
lie O
algebra O

vector B-Math
spaces I-Math
are O
completely O
characterized O
by O
their O
dimension O
up O
to O
an O
isomorphis O
are O
completely O
characterized O
by O
their O
dimension B-Math
up O
to O
an O
isomorphis O
up O
to O
an O
isomorphism B-Math

a O
vector B-Math
space I-Math
can O
have O
several O
bases O
however O
all O
the O
bases O
have O
the O
same O
number O
of O
elements O
called O
the O
dimension O
of O
the O
vector O
spac O
can O
have O
several O
bases B-Attributes
however O
all O
the O
bases O
have O
the O
same O
number O
of O
elements O
called O
the O
dimension O
of O
the O
vector O
spac O
however O
all O
the O
bases O
have O
the O
same O
number O
of O
elements O
called O
the O
dimension B-Math
of O
the O
vector O
spac O

for O
a O
system O
involving O
two O
variables O
x O
and O
y O
each O
linear B-Math
equation I-Math
determines O
a O
line O
on O
the O
xyplan O

tropical B-Math
geometry I-Math
is O
another O
example O
of O
linear O
algebra O
in O
a O
more O
exotic O
structur O
is O
another O
example O
of O
linear B-Math
algebra I-Math
in O
a O
more O
exotic O
structur O

in O
numerical B-Math
analysis I-Math
many O
computational O
problems O
are O
solved O
by O
reducing O
them O
to O
a O
matrix O
computation O
and O
this O
often O
involves O
computing O
with O
matrices O
of O
huge O
dimensio O
many O
computational O
problems O
are O
solved O
by O
reducing O
them O
to O
a O
matrix B-Math
omputation O
and O
this O
often O
involves O
computing O
with O
matrices O
of O
huge O
dimensio O

in O
general O
the O
behavior O
of O
a O
linear B-Math
system I-Math
is O
determined O
by O
the O
relationship O
between O
the O
number O
of O
equations O
and O
the O
number O
of O
unknown O

geometric B-Math
interpretatio O

a B-Math
first I-Math
order I-Math
differential I-Math
equation I-Math
is O
said O
to O
be O
homogeneous O
if O
it O
may O
be O
writte O
is O
said O
to O
be O
homogeneous B-Math
if O
it O
may O
be O
writte O

in O
mathematics B-Math
a O
system O
of O
linear O
equations O
or O
linear O
system O
is O
a O
collection O
of O
one O
or O
more O
linear O
equations O
involving O
the O
same O
variable O
a O
system O
of O
linear B-Math
equations I-Math
or O
linear O
system O
is O
a O
collection O
of O
one O
or O
more O
linear O
equations O
involving O
the O
same O
variable O
or O
linear B-Math
system I-Math
is O
a O
collection O
of O
one O
or O
more O
linear O
equations O
involving O
the O
same O
variable O

for O
solutions O
in O
an O
integral O
domain O
like O
the O
ring O
of O
the O
integers O
or O
in O
other O
algebraic O
structures O
other O
theories O
have O
been O
developed O
see O
linear B-Math
equation I-Math
over O
a O
rin O

modules B-Math
over O
the O
integers O
can O
be O
identified O
with O
abelian O
groups O
since O
the O
multiplication O
by O
an O
integer O
may O
be O
identified O
to O
a O
repeated O
additio O
over O
the O
integers O
can O
be O
identified O
with O
abelian B-Math
groups I-Math
since O
the O
multiplication O
by O
an O
integer O
may O
be O
identified O
to O
a O
repeated O
additio O
since O
the O
multiplication B-Attributes
by O
an O
integer O
may O
be O
identified O
to O
a O
repeated O
additio O
by O
an O
integer O
may O
be O
identified O
to O
a O
repeated O
addition B-Attributes
integers B-Math
can O
be O
identified O
with O
abelian O
groups O
since O
the O
multiplication O
by O
an O
integer O
may O
be O
identified O
to O
a O
repeated O
additio O

thus O
computing O
intersections O
of O
lines O
and O
planes O
amounts O
to O
solving O
systems O
of O
linear B-Math
equations I-Math
lines B-Math
and O
planes O
amounts O
to O
solving O
systems O
of O
linear O
equation O
and O
planes B-Math
amounts O
to O
solving O
systems O
of O
linear O
equation O

the O
concepts O
of O
linear B-Math
independence I-Math
span I-Math
basis I-Math
and O
linear O
maps O
also O
called O
module O
homomorphisms O
are O
defined O
for O
modules O
exactly O
as O
for O
vector O
spaces O
with O
the O
essential O
difference O
that O
if O
r O
is O
not O
a O
field O
there O
are O
modules O
that O
do O
not O
have O
any O
basi O
and O
linear B-Math
maps I-Math
also O
called O
module O
homomorphisms O
are O
defined O
for O
modules O
exactly O
as O
for O
vector O
spaces O
with O
the O
essential O
difference O
that O
if O
r O
is O
not O
a O
field O
there O
are O
modules O
that O
do O
not O
have O
any O
basi O
also O
called O
module B-Math
homomorphisms I-Math
are O
defined O
for O
modules O
exactly O
as O
for O
vector O
spaces O
with O
the O
essential O
difference O
that O
if O
r O
is O
not O
a O
field O
there O
are O
modules O
that O
do O
not O
have O
any O
basi O

in O
any O
event O
the O
span O
has O
a O
basis O
of O
linearly O
independent O
vectors O
that O
do O
guarantee O
exactly O
one O
expression O
and O
the O
number O
of O
vectors B-Math
in O
that O
basis O
its O
dimension O
can O
not O
be O
larger O
than O
m O
or O
n O
but O
it O
can O
be O
smalle O

consequently O
linear B-Math
algebra I-Math
algorithms O
have O
been O
highly O
optimize O

these O
operations O
and O
associated O
laws O
qualify O
euclidean O
vectors O
as O
an O
example O
of O
the O
more O
generalized O
concept O
of O
vectors O
defined O
simply O
as O
elements O
of O
a O
vector B-Math
space I-Math

a O
linear B-Math
map I-Math
from O
to O
always O
maps O
the O
origin O
of O
to O
the O
origin O
o O

in O
modern O
introductory O
texts O
on O
functional B-Math
analysis I-Math
the O
subject O
is O
seen O
as O
the O
study O
of O
vector O
spaces O
endowed O
with O
a O
topology O
in O
particular O
infinitedimensional O
space O
the O
subject O
is O
seen O
as O
the O
study O
of O
vector B-Math
spaces I-Math
endowed O
with O
a O
topology O
in O
particular O
infinitedimensional O
space O
endowed O
with O
a O
topology B-Math
in O
particular O
infinitedimensional O
space O

this O
article O
deals O
mainly O
with O
finitedimensional O
vector B-Math
spaces I-Math

matrix B-Math
theory I-Math
is O
the O
branch O
of O
mathematics O
that O
focuses O
on O
the O
study O
of O
matrice O
is O
the O
branch O
of O
mathematics B-Math
that O
focuses O
on O
the O
study O
of O
matrice O
that O
focuses O
on O
the O
study O
of O
matrices B-Math

a O
solution B-Math
to O
a O
linear O
system O
is O
an O
assignment O
of O
values O
to O
the O
variables O
such O
that O
all O
the O
equations O
are O
simultaneously O
satisfie O
to O
a O
linear O
system O
is O
an O
assignment O
of O
values O
to O
the O
variables B-Math
uch O
that O
all O
the O
equations O
are O
simultaneously O
satisfie O
uch O
that O
all O
the O
equations O
are O
simultaneously B-Math
satisfie O

in O
multilinear B-Math
algebra I-Math
one O
considers O
multivariable O
linear O
transformations O
that O
is O
mappings O
that O
are O
linear O
in O
each O
of O
a O
number O
of O
different O
variable O
one O
considers O
multivariable O
linear B-Math
transformations I-Math
that O
is O
mappings O
that O
are O
linear O
in O
each O
of O
a O
number O
of O
different O
variable O

otherwise O
it O
is O
infinitedimensional B-Attributes
and O
its O
dimension O
is O
an O
infinite O
cardina O

a O
normed B-Math
vector I-Math
space I-Math
is O
a O
vector O
space O
along O
with O
a O
function O
called O
a O
norm O
which O
measures O
the O
size O
of O
element O
is O
a O
vector B-Math
space I-Math
along O
with O
a O
function O
called O
a O
norm O
which O
measures O
the O
size O
of O
element O

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

if O
in O
addition O
to O
vector B-Math
addition I-Math
and O
scalar O
multiplication O
there O
is O
a O
bilinear O
vector O
product O
v O
v O
v O
the O
vector O
space O
is O
called O
an O
algebra O
for O
instance O
associative O
algebras O
are O
algebras O
with O
an O
associate O
vector O
product O
like O
the O
algebra O
of O
square O
matrices O
or O
the O
algebra O
of O
polynomial O
and O
scalar B-Math
multiplication I-Math
there O
is O
a O
bilinear O
vector O
product O
v O
v O
v O
the O
vector O
space O
is O
called O
an O
algebra O
for O
instance O
associative O
algebras O
are O
algebras O
with O
an O
associate O
vector O
product O
like O
the O
algebra O
of O
square O
matrices O
or O
the O
algebra O
of O
polynomial O
there O
is O
a O
bilinear B-Math
vector I-Math
product I-Math
v O
v O
v O
the O
vector O
space O
is O
called O
an O
algebra O
for O
instance O
associative O
algebras O
are O
algebras O
with O
an O
associate O
vector O
product O
like O
the O
algebra O
of O
square O
matrices O
or O
the O
algebra O
of O
polynomial O
v O
v O
v O
the O
vector B-Math
space I-Math
is O
called O
an O
algebra O
for O
instance O
associative O
algebras O
are O
algebras O
with O
an O
associate O
vector O
product O
like O
the O
algebra O
of O
square O
matrices O
or O
the O
algebra O
of O
polynomial O

analysis B-Math
is O
the O
branch O
of O
mathematics O
dealing O
with O
continuous O
functions O
limits O
and O
related O
theories O
such O
as O
differentiation O
integration O
measure O
infinite O
sequences O
series O
and O
analytic O
function O
is O
the O
branch O
of O
mathematics O
dealing O
with O
continuous B-Math
functions I-Math
limits I-Math
and O
related O
theories O
such O
as O
differentiation O
integration O
measure O
infinite O
sequences O
series O
and O
analytic O
function O
and O
related O
theories O
such O
as O
differentiation O
integration O
measure O
infinite O
sequences O
series O
and O
analytic B-Math
functions I-Math

vectors O
can O
be O
added O
to O
other O
vectors O
according O
to O
vector B-Math
algebra I-Math
vectors B-Math
can O
be O
added O
to O
other O
vectors O
according O
to O
vector O
algebr O