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,
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