¥tu "
goal : V a V- s .
, B a basis
"
"
theorem ( www.gueuessrepreoeui =
-atvou tleeoseeu) let Beefy , -Tn } be a basis of a vector space V . there for each TeV
these exist unique scalars a , . . ,
on such that runner
T= aTi t - - t CnTn Cx)
preet since B is a basis for V, it generates B . Hence we can
find ocelots 9 - Cn Oo that Ct) is satisfied .
We have to prove they ate eeugue!
Assume I can be also written as Te dit , t - e oben .
Now we need to prove that a -- di , - - r , Cn edn
D= T -Te Citi t - - tcnvn - (divi t - - tdnvn) = (Ci- di) Jit - - -
t kn- dn)Tn
but Ji - VI ate linearly eudep . ( since B is a bans) .
So if
(cc - di) ti t -t (Cn -dnt -Vn= I ⇒ Ci-di so - - . Cn- dnio
⇒ Ccs di - - Cn--dn
DI let Be { k . , -,
In } be a bass for a v. s . V and TEV .
the coordiuatesofTrelatiuetot6eba# 03 , or the B - coordinates
of T , ate thee weighs co
- on such that
these are real numbers .
.
let a - Cn be the
coordinates of T relative to Be . We define the vector EET E IR " as
[ TIB -
- n' ) EIR?
-
Remark V a v. s . .
let B. = {To , - In } a basis . then we have a map lRn→V
run
so to each n-tuple of vectors in IR "
we can associate a vector in Y provided thee bases Ps is fixed .
"
,
.
Provided that D= {To , - - In} is a bass for V .
Moral Duce a basis Be Fbi , - ,bis } for Vis fixed ⇒ V E IR? run '
we say V is to IR"]
Excepts
Be { Tsi - Toa } a basis for IR ' Ts , = (f ) Toa = ( L ) .
Suppose I in 1122 has the coordinate vector EET • = (3) . Find a .
s n j
we want these !
I
E : = {e , Ey et -- ( t) ez -- K) is the standard bans for IR?
I -- (d) EIR? then CEI = ( f) . Indeed .
(f) = I - (f) + 6- (?) = I -E t 6 -Ez
let 53 be the standard bass for B. , te, B={ lit ; ti t'} .
p E Pg has the form pct)= Ao t act tazttag E3 mmmm
this means that
emos .
so Hee coordinate map is IB a p i-> Ep] pg E ? is
well CPTB -_ ( off) is a vector wi a component, ⇒ 1124 what's this ?
So the space of polynomials of degree E3 is the same as a 4 -deer .
Space !
Heeded let B. = {bi - In } be a basis for V . Hear the coordinate
nmapping V z t [Edge IR is a one-to - one linear transformation from V to IR?
proofs ( later , only if theres time
• linearity we need to dear that if it , TeV , then
mm
a- = C.To, t - - - ten In T -
- di bi t .. tdnbn ; Feige (I') EIB - foe!)
aet t ft = accts , t - - t acnb-ntfod.T.si t . - t fdntsn =
= (act t foe ,)-b, t - - t (acntfdn)bT
E.¥:-C:÷÷:'t. it. :L . . t.ec.⇒ . = a Fei B
t f EDB .
• one- to - dee check it for yourself . mm
.