8 1. Linea r Algebr a Revie w

Definition 1.6. A subspace o f a vector space V is a nonempty subse t

E o f V which i s close d unde r th e additio n an d scala r multiplicatio n

operations fro m V . Tha t is , i t i s require d tha t x - f y an d ax ar e

contained i n E wheneve r x,y G E an d a G F .

In othe r words , a subspac e o f V is just a nonempt y subse t o f V

which i s a vector spac e i n it s ow n right unde r th e additio n an d scala r

multiplication operation s i t inherit s a s a subse t o f V .

Prom thi s definitio n i t i s clea r tha t F

n

[a, b] sits naturall y a s a

subspace o f C[a, b] by simpl e set-inclusion , an d similarl y C[a, b] is a

subspace o f F[a , b]. In fact , ever y functio n spac e o n th e domai n [a , b]

is a subspac e o f F[a , b].

From the abov e definition, th e entire space V is also a subspace of

V. Th e subse t o f V consisting o f the zer o vector alon e is a subspace of

V called the zero or trivial subspace, an d i t is written {0} . A subspac e

is calle d proper, o r nontrivial, i f i t i s neither th e entir e spac e no r th e

zero subspace . A super space o f a vecto r spac e V i s a large r vecto r

space W whic h contain s V as a prope r subspace .

1.2. Base s fo r Vecto r Space s

A fundamenta l ide a i n th e stud y o f vecto r space s i s th e practic e o f

combining certai n element s togethe r t o for m others . Ther e ar e a

variety o f interestin g question s tha t arise :

• Whic h vector s "ca n b e obtaine d from " a specifi c collectio n

of vectors ?

• Ho w man y vector s ar e neede d t o "create " al l th e vector s i n

the space ?

• I s ther e mor e tha n on e wa y t o "get " a vecto r fro m a give n

set o f vectors ?

In thi s sectio n w e wil l mak e thi s notio n o f "combinin g element s

to for m othe r vectors " precise .

Definition 1.7. Le t x\, #2 , • • • %k b e vector s i n a vecto r spac e V . A

linear combination o f vector s #i , x2, • • •, %k i s a su m o f th e form :

x = c\X\ + c

2

x2 H h c

k

xk,