10 1. Linea r Algebr a Revie w
of vector s fro m S. Tha t is :
span(«S) = {c\Xi + + CkXk : xi G 5, ci G F} .
A spanning set fo r a vector spac e V is a set S whos e linea r spa n i s V.
If S i an d S2 ar e subset s o f V , w e us e th e notatio n span(«Si,S2 )
to denot e span(S i U £2). W e us e analogou s notatio n fo r th e spa n o f
three o r mor e sets .
Definition 1.11A . vector spac e i s called finite-dimensional i f it ha s
a finite spannin g set . Otherwise , i t i s calle d a n infinite-dimensional
vector space .
In th e remainde r o f thi s section , V i s assume d t o b e a finite-
dimensional vecto r space . Th e results , however , ar e stil l vali d i n th e
infinite-dimensional setting .
Lemma 1.12 . Suppose that {xi}^
=1
is a spanning set for the vector
space V, and let w be a vector in V . Then the set {w, x\, X2, . •, Xk}
is linearly dependent.
Proof. Exercis e 2 .
The followin g lemm a i s mos t usefu l i n constructin g an d workin g
with subspaces .
Lemma 1.13 . The linear span of a finite set of vectors in a vector
space V is a subspace ofV.
Proof. Le t S = {xi,X2,.. . ,xn} C V be a give n se t o f vectors , an d
let W = span(S) . Le t u an d v b e vector s i n W , an d writ e u =
c\X\ + C2X2 + + c
n
xn an d v = diXi + c^a ^ + * *' + d
n
xn. The n
u
+
v

Y17=i(ci
~^~ di)Xi, whic h i s a linea r combinatio n o f th e vector s
X{, so u + v i s a n elemen t W , thu s provin g tha t W i s close d unde r
addition. I f a i s a scalar , the n au ac\X\ + ac2X2 + + ac nxn
is agai n a linea r combinatio n o f {xi , X2, •, xn } . Sinc e W i s close d
under additio n an d scala r multiplication , i t i s a subspac e o f V .
Definition 1.14 . A basis fo r a nontrivia l finite-dimensional vecto r
space V is a linearly independent spannin g set for V. (W e note that fo r
an infinite-dimensiona l vecto r space , a linearly independen t spannin g
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