1.2. Base s fo r Vecto r Space s
9
where ci , C2,..., Ck ar e scalars .
Definition 1.8. A finite, nonempt y se t o f vector s {xi}!?
=1
i n V i s
called linearly dependent i f ther e exis t scalar s ci,C2,.. . ,Cfc, whic h
are no t al l zero , suc h tha t ]T^= i c^ = 0 . A finite nonempt y se t
of vector s i s define d t o b e linearly independent i f i t i s no t linearl y
dependent. I n othe r words , {xi}^
=1
i s linearl y independen t i f th e
only se t o f scalar s {ci , C2,..., c&} which give s 5 ^
= 1
c ixi =
0
1S
^
n e
zero se t c i = C 2 = c ^ = 0 .
We remark tha t th e empt y se t 0 of vectors is defined t o be neithe r
linearly dependen t no r linearl y independent , an d th e trivia l subse t
{0} containin g th e zer o vecto r alon e i s considere d t o b e a linearl y
dependent set .
It follow s fro m th e definitio n tha t a nonempt y se t o f a t leas t tw o
vectors i s linearly dependen t i f an d onl y i f a t leas t on e o f the vector s
can be written a s a linear combination o f others in the set (Exercis e 1).
We ca n no w exten d thes e definition s t o infinite set s o f vectors .
Definition 1.9. A nonempty infinit e se t o f vectors i n a vecto r spac e
is defined t o b e linearly dependent i f at leas t on e o f the vector s i n th e
set ca n b e writte n a s a linear combinatio n o f a finite numbe r o f othe r
vectors i n th e set , an d i t i s calle d linearly independent otherwise .
It i s eas y t o see tha t linea r dependenc e i s a hereditar y propert y
in th e sens e tha t i f a give n se t ha s a linearl y dependen t subset , the n
it itsel f i s linearl y dependent . Fo r instance , th e se t o f polynomial s
{l,x,x 2 ,2 x + l,x 3 } i s easil y see n t o b e linearl y dependen t simpl y
because th e fourt h polynomia l i s clearl y a linea r combinatio n o f th e
first two . Th e othe r polynomial s don' t hav e t o b e checked . O n th e
other hand , linea r independenc e i s a hereditar y propert y i n the exac t
opposite sense : i f a give n se t i s know n t o b e linearl y independent ,
then ever y nonempt y subse t o f i t i s als o linearl y independent .
From th e abov e discussion s i t shoul d b e clea r tha t linearl y in -
dependent set s ar e special , wherea s linearl y dependen t set s ar e ver y
common.
Definition 1.10 . Le t S b e a subse t o f a vecto r spac e V . W e defin e
the linear span of S t o b e th e se t o f al l th e finite linea r combination s
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